Integrand size = 17, antiderivative size = 186 \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)^2} \, dx=-\frac {\cosh (c+d x)}{a^2 x}-\frac {b \cosh (c+d x)}{a^2 (a+b x)}-\frac {2 b \cosh (c) \text {Chi}(d x)}{a^3}+\frac {2 b \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a^3}+\frac {d \text {Chi}(d x) \sinh (c)}{a^2}+\frac {d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{a^2}+\frac {d \cosh (c) \text {Shi}(d x)}{a^2}-\frac {2 b \sinh (c) \text {Shi}(d x)}{a^3}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^2}+\frac {2 b \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^3} \]
-2*b*Chi(d*x)*cosh(c)/a^3+2*b*Chi(a*d/b+d*x)*cosh(-c+a*d/b)/a^3-cosh(d*x+c )/a^2/x-b*cosh(d*x+c)/a^2/(b*x+a)+d*cosh(c)*Shi(d*x)/a^2+d*cosh(-c+a*d/b)* Shi(a*d/b+d*x)/a^2+d*Chi(d*x)*sinh(c)/a^2-2*b*Shi(d*x)*sinh(c)/a^3-d*Chi(a *d/b+d*x)*sinh(-c+a*d/b)/a^2-2*b*Shi(a*d/b+d*x)*sinh(-c+a*d/b)/a^3
Time = 0.92 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.98 \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)^2} \, dx=\frac {-\frac {a (a+2 b x) \cosh (c) \cosh (d x)}{x (a+b x)}-2 b \cosh (c) \text {Chi}(d x)+2 b \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right )+a d \text {Chi}(d x) \sinh (c)+a d \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \sinh \left (c-\frac {a d}{b}\right )-\frac {a (a+2 b x) \sinh (c) \sinh (d x)}{x (a+b x)}+a d \cosh (c) \text {Shi}(d x)-2 b \sinh (c) \text {Shi}(d x)+a d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+2 b \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{a^3} \]
(-((a*(a + 2*b*x)*Cosh[c]*Cosh[d*x])/(x*(a + b*x))) - 2*b*Cosh[c]*CoshInte gral[d*x] + 2*b*Cosh[c - (a*d)/b]*CoshIntegral[d*(a/b + x)] + a*d*CoshInte gral[d*x]*Sinh[c] + a*d*CoshIntegral[d*(a/b + x)]*Sinh[c - (a*d)/b] - (a*( a + 2*b*x)*Sinh[c]*Sinh[d*x])/(x*(a + b*x)) + a*d*Cosh[c]*SinhIntegral[d*x ] - 2*b*Sinh[c]*SinhIntegral[d*x] + a*d*Cosh[c - (a*d)/b]*SinhIntegral[d*( a/b + x)] + 2*b*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)])/a^3
Time = 0.75 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cosh (c+d x)}{x^2 (a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 b^2 \cosh (c+d x)}{a^3 (a+b x)}-\frac {2 b \cosh (c+d x)}{a^3 x}+\frac {b^2 \cosh (c+d x)}{a^2 (a+b x)^2}+\frac {\cosh (c+d x)}{a^2 x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b \cosh (c) \text {Chi}(d x)}{a^3}+\frac {2 b \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a^3}-\frac {2 b \sinh (c) \text {Shi}(d x)}{a^3}+\frac {2 b \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a^3}+\frac {d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a^2}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a^2}-\frac {b \cosh (c+d x)}{a^2 (a+b x)}+\frac {d \sinh (c) \text {Chi}(d x)}{a^2}+\frac {d \cosh (c) \text {Shi}(d x)}{a^2}-\frac {\cosh (c+d x)}{a^2 x}\) |
-(Cosh[c + d*x]/(a^2*x)) - (b*Cosh[c + d*x])/(a^2*(a + b*x)) - (2*b*Cosh[c ]*CoshIntegral[d*x])/a^3 + (2*b*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d *x])/a^3 + (d*CoshIntegral[d*x]*Sinh[c])/a^2 + (d*CoshIntegral[(a*d)/b + d *x]*Sinh[c - (a*d)/b])/a^2 + (d*Cosh[c]*SinhIntegral[d*x])/a^2 - (2*b*Sinh [c]*SinhIntegral[d*x])/a^3 + (d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d *x])/a^2 + (2*b*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/a^3
3.1.32.3.1 Defintions of rubi rules used
Time = 0.24 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.68
| method | result | size |
| risch | \(-\frac {d \,{\mathrm e}^{-d x -c} b}{a^{2} \left (d x b +d a \right )}-\frac {d \,{\mathrm e}^{-d x -c}}{2 a x \left (d x b +d a \right )}+\frac {d \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right )}{2 a^{2}}+\frac {{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) b}{a^{3}}+\frac {d \,{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right )}{2 a^{2}}-\frac {{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) b}{a^{3}}-\frac {{\mathrm e}^{d x +c}}{2 a^{2} x}-\frac {d \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2 a^{2}}+\frac {b \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{a^{3}}-\frac {d \,{\mathrm e}^{d x +c}}{2 a^{2} \left (\frac {d a}{b}+d x \right )}-\frac {d \,{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{2 a^{2}}-\frac {b \,{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{a^{3}}\) | \(312\) |
-d*exp(-d*x-c)/a^2/(b*d*x+a*d)*b-1/2*d*exp(-d*x-c)/a/x/(b*d*x+a*d)+1/2*d/a ^2*exp(-c)*Ei(1,d*x)+1/a^3*exp(-c)*Ei(1,d*x)*b+1/2*d/a^2*exp((a*d-b*c)/b)* Ei(1,d*x+c+(a*d-b*c)/b)-1/a^3*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*b-1 /2/a^2/x*exp(d*x+c)-1/2*d/a^2*exp(c)*Ei(1,-d*x)+1/a^3*b*exp(c)*Ei(1,-d*x)- 1/2*d/a^2*exp(d*x+c)/(d/b*a+d*x)-1/2*d/a^2*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-( a*d-b*c)/b)-b/a^3*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)
Time = 0.26 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.03 \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)^2} \, dx=-\frac {2 \, {\left (2 \, a b x + a^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left ({\left (a b d - 2 \, b^{2}\right )} x^{2} + {\left (a^{2} d - 2 \, a b\right )} x\right )} {\rm Ei}\left (d x\right ) - {\left ({\left (a b d + 2 \, b^{2}\right )} x^{2} + {\left (a^{2} d + 2 \, a b\right )} x\right )} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - {\left ({\left ({\left (a b d + 2 \, b^{2}\right )} x^{2} + {\left (a^{2} d + 2 \, a b\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left ({\left (a b d - 2 \, b^{2}\right )} x^{2} + {\left (a^{2} d - 2 \, a b\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - {\left ({\left ({\left (a b d - 2 \, b^{2}\right )} x^{2} + {\left (a^{2} d - 2 \, a b\right )} x\right )} {\rm Ei}\left (d x\right ) + {\left ({\left (a b d + 2 \, b^{2}\right )} x^{2} + {\left (a^{2} d + 2 \, a b\right )} x\right )} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) + {\left ({\left ({\left (a b d + 2 \, b^{2}\right )} x^{2} + {\left (a^{2} d + 2 \, a b\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left ({\left (a b d - 2 \, b^{2}\right )} x^{2} + {\left (a^{2} d - 2 \, a b\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (a^{3} b x^{2} + a^{4} x\right )}} \]
-1/2*(2*(2*a*b*x + a^2)*cosh(d*x + c) - (((a*b*d - 2*b^2)*x^2 + (a^2*d - 2 *a*b)*x)*Ei(d*x) - ((a*b*d + 2*b^2)*x^2 + (a^2*d + 2*a*b)*x)*Ei(-d*x))*cos h(c) - (((a*b*d + 2*b^2)*x^2 + (a^2*d + 2*a*b)*x)*Ei((b*d*x + a*d)/b) - (( a*b*d - 2*b^2)*x^2 + (a^2*d - 2*a*b)*x)*Ei(-(b*d*x + a*d)/b))*cosh(-(b*c - a*d)/b) - (((a*b*d - 2*b^2)*x^2 + (a^2*d - 2*a*b)*x)*Ei(d*x) + ((a*b*d + 2*b^2)*x^2 + (a^2*d + 2*a*b)*x)*Ei(-d*x))*sinh(c) + (((a*b*d + 2*b^2)*x^2 + (a^2*d + 2*a*b)*x)*Ei((b*d*x + a*d)/b) + ((a*b*d - 2*b^2)*x^2 + (a^2*d - 2*a*b)*x)*Ei(-(b*d*x + a*d)/b))*sinh(-(b*c - a*d)/b))/(a^3*b*x^2 + a^4*x)
Timed out. \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)^2} \, dx=\text {Timed out} \]
\[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)^2} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b x + a\right )}^{2} x^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 3353 vs. \(2 (191) = 382\).
Time = 0.33 (sec) , antiderivative size = 3353, normalized size of antiderivative = 18.03 \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)^2} \, dx=\text {Too large to display} \]
-1/2*((b*x + a)^2*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d^2*Ei(-(b*x + a )*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b + c)*e^(-c)/b - 2*(b*x + a)*a*(b*c /(b*x + a) - a*d/(b*x + a) + d)*c*d^2*Ei(-(b*x + a)*(b*c/(b*x + a) - a*d/( b*x + a) + d)/b + c)*e^(-c) + a*b*c^2*d^2*Ei(-(b*x + a)*(b*c/(b*x + a) - a *d/(b*x + a) + d)/b + c)*e^(-c) + (b*x + a)*a^2*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^3*Ei(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b + c)*e^( -c)/b - a^2*c*d^3*Ei(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b + c) *e^(-c) - (b*x + a)^2*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d^2*Ei((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c)*e^c/b + 2*(b*x + a)*a*(b*c /(b*x + a) - a*d/(b*x + a) + d)*c*d^2*Ei((b*x + a)*(b*c/(b*x + a) - a*d/(b *x + a) + d)/b - c)*e^c - a*b*c^2*d^2*Ei((b*x + a)*(b*c/(b*x + a) - a*d/(b *x + a) + d)/b - c)*e^c - (b*x + a)*a^2*(b*c/(b*x + a) - a*d/(b*x + a) + d )*d^3*Ei((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c)*e^c/b + a^2* c*d^3*Ei((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c)*e^c - (b*x + a)^2*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d^2*Ei(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^((b*c - a*d)/b)/b + 2*(b*x + a )*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c*d^2*Ei(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^((b*c - a*d)/b) - a*b*c^2*d^2*Ei(( (b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^((b*c - a* d)/b) - (b*x + a)*a^2*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^3*Ei(((b*x ...
Timed out. \[ \int \frac {\cosh (c+d x)}{x^2 (a+b x)^2} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x^2\,{\left (a+b\,x\right )}^2} \,d x \]