#simplify(a_ and b_):=simplify(a) and simplify(b);


simplify(a_=b_):=simplify(a)-simplify(b)=0;

simplify(ds(ds(y_,x_),t_)):=d(d(y,x),t);

simplify(taylors(x_=>f_,x_,x0_,n_)):=taylor(f,x,x0,n);
simplify(taylors(x_=>f_,x_)):=taylor(f,x);
simplify(ds(x_=>f_,x_)):=d(f,x);
simplify(d(x_=>f_,x_)):=d(f,x);
simplify(integrates(x_=>f_,x_)):=integrate(f,x);
simplify(integrals(x_=>f_,x_)):=integral(f,x);
simplify(inte(x_=>f_,x_)):=integrate(f,x);

simplify(log(x_+sqrt(1+x_^2))):=asinh(x);
simplify(log(x_+sqrt(x_^2-1))):=acosh(x);
simplify(log((1+x_)/(x_-1))):= 2acoth(x);
simplify(log((1+x_)/(1-x_))):= 2atanh(x);
#simplify(ln(n_,x)):= If(n>0 and isinteger(n), -(-1)^n*(n-1)!/x^n);
simplify(ln(n_,x_)):= If(n>0,-(-1)^n*(n-1)!/x^n, (log(x)+psi(1)-psi(1-n))/gamma(1-n)*pow(x,-n) );

simplify(log(Gamma(x_))):=logGamma(x);
#simplify(log(gamma(x_))):=loggamma(x);
simplify(Gamma(x_-1)/Gamma(x_)):=1/(x-1);
simplify(Gamma(x_)/Gamma(x_-1)):= x-1;

simplify(sin(x_)^n_):=if(n<0,csc(x)^(-n));
simplify(cos(x_)^n_):=if(n<0,sec(x)^(-n));
simplify(tan(x_)^n_):=if(n<0,cot(x)^(-n));
simplify(cot(x_)^n_):=if(n<0,tan(x)^(-n));
simplify(csc(x_)^n_):=if(n<0,sin(x)^(-n));
simplify(sec(x_)^n_):=if(n<0,cos(x)^(-n));

simplify(1/cos(x_)*sin(x_)):=tan(x);
simplify(cos(x_)/sin(x_)):=cot(x);

simplify(asin(1/x_)):=acsc(x);
simplify(acos(1/x_)):=asec(x);
simplify(atan(1/x_)):=acot(x);
simplify(acot(1/x_)):=atan(x);

simplify(a_*exp(-x_)+b_*exp(x_)):= If(a==-b, 2b*sinh(x));
simplify(a_*exp(-x_)-a_*exp(x_)):= -2a*sinh(x);
simplify(a_*exp(x_)-a_*exp(-x_)):= 2a*sinh(x);
simplify(a_*exp(-x_)+a_*exp(x_)):= 2a*cosh(x);
simplify(exp(-x_)+exp(x_)):= 2cosh(x);
simplify(exp(x_)-exp(-x_)):= 2sinh(x);
simplify(exp(-x_)-exp(x_)):= -2sinh(x);

simplify(a_*exp(-i*x_)-a_*exp(i*x_)):= -2a*i*sin(x);
simplify(a_*exp(-i*x_)+(-a_)*exp(i*x_)):= 2a*i*sin(x);
simplify(a_*exp(-i*x_)+a_*exp(i*x_)):= 2a*cos(x);

simplify(1/(exp(a_)+exp(b_))):=exp(-a)/(exp(b-a)+1);
simplify(1/(exp(-x_)+exp(x_))) := sech(x);
simplify(1/(exp(x_)-exp(-x_))) := csch(x);

simplify(1/sinh(x_)):=csch(x);
simplify(1/cosh(x_)):=sech(x);
simplify(1/tanh(x_)):=coth(x);
simplify(1/coth(x_)):=tanh(x);
simplify(1/csch(x_)):=sinh(x);
simplify(1/sech(x_)):=cosh(x);
simplify(1/cosh(x_)*sinh(x_)):=tanh(x);
simplify(cosh(x_)/sinh(x_)):=coth(x);

simplify(log(x_+(1+x_^2)^(1/2))):=asinh(x);
simplify(log(x_+(x_^2-1)^(1/2))):=acosh(x);

simplify(asinh(1/x_)):=acsch(x);
simplify(acosh(1/x_)):=asech(x);
simplify(atanh(1/x_)):=acoth(x);
simplify(acoth(1/x_)):=atanh(x);

simplify(x_+(1+x_^2)^(1/2)):=exp(asinh(x));
simplify(x_+(x_^2-1)^(1/2)):=exp(acosh(x));

simplify(sin(i*x_)):= -i*sinh(x);
simplify(cos(i*x_)):= cosh(x);
simplify(tan(i*x_)):= i*tanh(x);
simplify(asin(i*x_)):= -i*asinh(-x);
simplify(acos(i*x_)):= -i*acosh(-x);
simplify(atan(i*x_)):= i*atanh(x);

simplify(sinh(i*x_)):= -i*sin(x);
simplify(cosh(i*x_)):= cos(x);
simplify(tanh(i*x_)):= i*tan(x);
simplify(asinh(i*x_)):= -i*asin(-x);
simplify(acosh(i*x_)):= -i*acos(-x);
simplify(atanh(i*x_)):= i*atan(x);

simplify(sgn(x_)*abs(x_)):=x;
simplify(x_/abs(x_)):=sgn(x);
simplify(x_/sgn(x_)):=abs(x);

simplify(exp(a_*log(x_))):=x^a;
simplify(exp(log(x_))):=x;

simplify(cos(x_)*sin(x_)) := sin(2x)/2;
simplify(sin(a_)*cos(b_)+cos(a_)*sin(b_)):=sin(a+b);
simplify(sin(a_)*cos(b_)-cos(a_)*sin(b_)):=sin(a-b);
simplify(cos(a_)*cos(b_)- sin(a_)*sin(b_)):=cos(a+b);
simplify(cos(a_)*cos(b_)+ sin(a_)*sin(b_)):=cos(a-b);

simplify((x_^n_-1)/(x_-1)):=If(n>0, sum(x^k,k,0,n-1,1));
simplify((x_^n_-1)/(1-x_)):=If(n>0, sum(-x^k,k,0,n-1,1));
simplify((1-x_^n_)/(x_-1)):=If(n>0, sum(-x^k,k,0,n-1,1));
simplify((1-x_^n_)/(1-x_)):=If(n>0, sum(x^k,k,0,n-1,1));
simplify(x_/(2-3*x_+x_^2)):=2/(x-2)-1/(x-1);

simplify(cosh(x_)+sinh(x_)):=exp(x);
simplify(cosh(x_)-sinh(x_)):=exp(-x);
simplify(cosh(x_)^2-sinh(x_)^2):=1;

simplify(cos(x_)^2+sin(x_)^2):=1;
simplify(sec(x_)^2-tan(x_)^2):=1;
simplify(csc(x_)^2-cot(x_)^2):=1;
simplify(tan(x_)^2-sec(x_)^2):= -1;

simplify(-1+sin(x_)^2):= -cos(x)^2;
simplify(-1+cos(x_)^2):= -sin(x)^2;

simplify((-a_)^n_):=if(iseven(n),a^n, if(isodd(n),-a^n ));
simplify(sqrt(x_)):=x^(1/2);
simplify(cbrt(x_)):=x^(1/3);

simplify(mittag(a_,b_)*mittag(a_,c_)):=mittag(a,b+c);
simplify(mittag(a_,b_)/mittag(a_,c_)):=mittag(a,b-c);
simplify(mittag(2,x_^2)):=cosh(x);
simplify(mittag(2,-x_^2)):=cos(x);
simplify(mittag(a_,x_)^n_):=mittag(a,n*x);

simplify(GammaQ(a_,x_)*Gamma(a_)):=Gamma(a,x);
simplify(Gamma(a_,x_)/GammaQ(a_,x_)):=Gamma(a);
simplify(Gamma(n_,a_,x_)):=d(Gamma(a,x),x,n);
simplify(Gamma(n_,x_,0)):=d(Gamma(x),x,n);

simplify(pow(a_,b_)):=a^b;
simplify(1/pow(a_,b_)):=a^(-b);


simplify(Dawson(x_)):=exp(-x*x)*erfi(x)*sqrt(pi)/2;