Integrand size = 19, antiderivative size = 175 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^5} \, dx=-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a b \cosh (c+d x)}{x^2}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}+b^2 \cosh (c) \text {Chi}(d x)+a b d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{24} a^2 d^4 \cosh (c) \text {Chi}(d x)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a b d \sinh (c+d x)}{x}-\frac {a^2 d^3 \sinh (c+d x)}{24 x}+b^2 \sinh (c) \text {Shi}(d x)+a b d^2 \sinh (c) \text {Shi}(d x)+\frac {1}{24} a^2 d^4 \sinh (c) \text {Shi}(d x) \]
b^2*Chi(d*x)*cosh(c)+a*b*d^2*Chi(d*x)*cosh(c)+1/24*a^2*d^4*Chi(d*x)*cosh(c )-1/4*a^2*cosh(d*x+c)/x^4-a*b*cosh(d*x+c)/x^2-1/24*a^2*d^2*cosh(d*x+c)/x^2 +b^2*Shi(d*x)*sinh(c)+a*b*d^2*Shi(d*x)*sinh(c)+1/24*a^2*d^4*Shi(d*x)*sinh( c)-1/12*a^2*d*sinh(d*x+c)/x^3-a*b*d*sinh(d*x+c)/x-1/24*a^2*d^3*sinh(d*x+c) /x
Time = 0.34 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^5} \, dx=\frac {\left (24 b^2+24 a b d^2+a^2 d^4\right ) x^4 \cosh (c) \text {Chi}(d x)-a \left (\left (6 a+24 b x^2+a d^2 x^2\right ) \cosh (c+d x)+d x \left (2 a+24 b x^2+a d^2 x^2\right ) \sinh (c+d x)\right )+\left (24 b^2+24 a b d^2+a^2 d^4\right ) x^4 \sinh (c) \text {Shi}(d x)}{24 x^4} \]
((24*b^2 + 24*a*b*d^2 + a^2*d^4)*x^4*Cosh[c]*CoshIntegral[d*x] - a*((6*a + 24*b*x^2 + a*d^2*x^2)*Cosh[c + d*x] + d*x*(2*a + 24*b*x^2 + a*d^2*x^2)*Si nh[c + d*x]) + (24*b^2 + 24*a*b*d^2 + a^2*d^4)*x^4*Sinh[c]*SinhIntegral[d* x])/(24*x^4)
Time = 0.55 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5810, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^5} \, dx\) |
| \(\Big \downarrow \) 5810 |
| \(\displaystyle \int \left (\frac {a^2 \cosh (c+d x)}{x^5}+\frac {2 a b \cosh (c+d x)}{x^3}+\frac {b^2 \cosh (c+d x)}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{24} a^2 d^4 \cosh (c) \text {Chi}(d x)+\frac {1}{24} a^2 d^4 \sinh (c) \text {Shi}(d x)-\frac {a^2 d^3 \sinh (c+d x)}{24 x}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a^2 d \sinh (c+d x)}{12 x^3}+a b d^2 \cosh (c) \text {Chi}(d x)+a b d^2 \sinh (c) \text {Shi}(d x)-\frac {a b \cosh (c+d x)}{x^2}-\frac {a b d \sinh (c+d x)}{x}+b^2 \cosh (c) \text {Chi}(d x)+b^2 \sinh (c) \text {Shi}(d x)\) |
-1/4*(a^2*Cosh[c + d*x])/x^4 - (a*b*Cosh[c + d*x])/x^2 - (a^2*d^2*Cosh[c + d*x])/(24*x^2) + b^2*Cosh[c]*CoshIntegral[d*x] + a*b*d^2*Cosh[c]*CoshInte gral[d*x] + (a^2*d^4*Cosh[c]*CoshIntegral[d*x])/24 - (a^2*d*Sinh[c + d*x]) /(12*x^3) - (a*b*d*Sinh[c + d*x])/x - (a^2*d^3*Sinh[c + d*x])/(24*x) + b^2 *Sinh[c]*SinhIntegral[d*x] + a*b*d^2*Sinh[c]*SinhIntegral[d*x] + (a^2*d^4* Sinh[c]*SinhIntegral[d*x])/24
3.1.56.3.1 Defintions of rubi rules used
Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p _.), x_Symbol] :> Int[ExpandIntegrand[Cosh[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Time = 0.24 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.71
| method | result | size |
| risch | \(-\frac {{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a^{2} d^{4} x^{4}+{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a^{2} d^{4} x^{4}+24 \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a b \,d^{2} x^{4}+24 \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a b \,d^{2} x^{4}+{\mathrm e}^{d x +c} a^{2} d^{3} x^{3}-{\mathrm e}^{-d x -c} a^{2} d^{3} x^{3}+24 \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) b^{2} x^{4}+24 \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) b^{2} x^{4}+{\mathrm e}^{d x +c} a^{2} d^{2} x^{2}+24 \,{\mathrm e}^{d x +c} a b d \,x^{3}+{\mathrm e}^{-d x -c} a^{2} d^{2} x^{2}-24 \,{\mathrm e}^{-d x -c} a b d \,x^{3}+2 \,{\mathrm e}^{d x +c} a^{2} d x +24 \,{\mathrm e}^{d x +c} a b \,x^{2}-2 \,{\mathrm e}^{-d x -c} a^{2} d x +24 \,{\mathrm e}^{-d x -c} a b \,x^{2}+6 \,{\mathrm e}^{d x +c} a^{2}+6 \,{\mathrm e}^{-d x -c} a^{2}}{48 x^{4}}\) | \(299\) |
| meijerg | \(\frac {b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {2 \gamma +2 \ln \left (x \right )+2 \ln \left (i d \right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Chi}\left (d x \right )-2 \ln \left (d x \right )-2 \gamma }{\sqrt {\pi }}\right )}{2}+b^{2} \operatorname {Shi}\left (d x \right ) \sinh \left (c \right )-\frac {d^{2} a b \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {4}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {2 \left (2 \gamma -3+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{\sqrt {\pi }}-\frac {4 \left (\frac {9 x^{2} d^{2}}{2}+3\right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \cosh \left (d x \right )}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}-\frac {4 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{\sqrt {\pi }}\right )}{4}+\frac {i d^{2} b a \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}+\frac {4 i \sinh \left (d x \right )}{x^{2} d^{2} \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{4}+\frac {a^{2} \cosh \left (c \right ) \sqrt {\pi }\, d^{4} \left (-\frac {8}{\sqrt {\pi }\, x^{4} d^{4}}-\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {\frac {4 \gamma }{3}-\frac {25}{9}+\frac {4 \ln \left (x \right )}{3}+\frac {4 \ln \left (i d \right )}{3}}{\sqrt {\pi }}+\frac {\frac {25}{9} d^{4} x^{4}+8 x^{2} d^{2}+8}{\sqrt {\pi }\, x^{4} d^{4}}-\frac {8 \left (\frac {15 x^{2} d^{2}}{2}+45\right ) \cosh \left (d x \right )}{45 \sqrt {\pi }\, x^{4} d^{4}}-\frac {8 \left (\frac {15 x^{2} d^{2}}{2}+15\right ) \sinh \left (d x \right )}{45 \sqrt {\pi }\, x^{3} d^{3}}+\frac {\frac {4 \,\operatorname {Chi}\left (d x \right )}{3}-\frac {4 \ln \left (d x \right )}{3}-\frac {4 \gamma }{3}}{\sqrt {\pi }}\right )}{32}-\frac {i a^{2} \sinh \left (c \right ) \sqrt {\pi }\, d^{4} \left (-\frac {8 i \left (\frac {x^{2} d^{2}}{2}+1\right ) \cosh \left (d x \right )}{3 d^{3} x^{3} \sqrt {\pi }}-\frac {8 i \left (\frac {x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{3 d^{4} x^{4} \sqrt {\pi }}+\frac {4 i \operatorname {Shi}\left (d x \right )}{3 \sqrt {\pi }}\right )}{32}\) | \(464\) |
-1/48*(exp(-c)*Ei(1,d*x)*a^2*d^4*x^4+exp(c)*Ei(1,-d*x)*a^2*d^4*x^4+24*exp( -c)*Ei(1,d*x)*a*b*d^2*x^4+24*exp(c)*Ei(1,-d*x)*a*b*d^2*x^4+exp(d*x+c)*a^2* d^3*x^3-exp(-d*x-c)*a^2*d^3*x^3+24*exp(-c)*Ei(1,d*x)*b^2*x^4+24*exp(c)*Ei( 1,-d*x)*b^2*x^4+exp(d*x+c)*a^2*d^2*x^2+24*exp(d*x+c)*a*b*d*x^3+exp(-d*x-c) *a^2*d^2*x^2-24*exp(-d*x-c)*a*b*d*x^3+2*exp(d*x+c)*a^2*d*x+24*exp(d*x+c)*a *b*x^2-2*exp(-d*x-c)*a^2*d*x+24*exp(-d*x-c)*a*b*x^2+6*exp(d*x+c)*a^2+6*exp (-d*x-c)*a^2)/x^4
Time = 0.24 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^5} \, dx=-\frac {2 \, {\left ({\left (a^{2} d^{2} + 24 \, a b\right )} x^{2} + 6 \, a^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{4} + 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{4} + 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \, {\left (2 \, a^{2} d x + {\left (a^{2} d^{3} + 24 \, a b d\right )} x^{3}\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{2} d^{4} + 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{4} + 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{48 \, x^{4}} \]
-1/48*(2*((a^2*d^2 + 24*a*b)*x^2 + 6*a^2)*cosh(d*x + c) - ((a^2*d^4 + 24*a *b*d^2 + 24*b^2)*x^4*Ei(d*x) + (a^2*d^4 + 24*a*b*d^2 + 24*b^2)*x^4*Ei(-d*x ))*cosh(c) + 2*(2*a^2*d*x + (a^2*d^3 + 24*a*b*d)*x^3)*sinh(d*x + c) - ((a^ 2*d^4 + 24*a*b*d^2 + 24*b^2)*x^4*Ei(d*x) - (a^2*d^4 + 24*a*b*d^2 + 24*b^2) *x^4*Ei(-d*x))*sinh(c))/x^4
\[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^5} \, dx=\int \frac {\left (a + b x^{2}\right )^{2} \cosh {\left (c + d x \right )}}{x^{5}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^5} \, dx=\frac {1}{8} \, {\left ({\left (d^{3} e^{\left (-c\right )} \Gamma \left (-3, d x\right ) + d^{3} e^{c} \Gamma \left (-3, -d x\right )\right )} a^{2} + 4 \, {\left (d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + d e^{c} \Gamma \left (-1, -d x\right )\right )} a b - \frac {4 \, b^{2} \cosh \left (d x + c\right ) \log \left (x^{2}\right )}{d} + \frac {4 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} b^{2}}{d}\right )} d + \frac {1}{4} \, {\left (2 \, b^{2} \log \left (x^{2}\right ) - \frac {4 \, a b x^{2} + a^{2}}{x^{4}}\right )} \cosh \left (d x + c\right ) \]
1/8*((d^3*e^(-c)*gamma(-3, d*x) + d^3*e^c*gamma(-3, -d*x))*a^2 + 4*(d*e^(- c)*gamma(-1, d*x) + d*e^c*gamma(-1, -d*x))*a*b - 4*b^2*cosh(d*x + c)*log(x ^2)/d + 4*(Ei(-d*x)*e^(-c) + Ei(d*x)*e^c)*b^2/d)*d + 1/4*(2*b^2*log(x^2) - (4*a*b*x^2 + a^2)/x^4)*cosh(d*x + c)
Time = 0.27 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^5} \, dx=\frac {a^{2} d^{4} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{4} x^{4} {\rm Ei}\left (d x\right ) e^{c} + 24 \, a b d^{2} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 24 \, a b d^{2} x^{4} {\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{3} x^{3} e^{\left (d x + c\right )} + a^{2} d^{3} x^{3} e^{\left (-d x - c\right )} + 24 \, b^{2} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 24 \, b^{2} x^{4} {\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{2} x^{2} e^{\left (d x + c\right )} - 24 \, a b d x^{3} e^{\left (d x + c\right )} - a^{2} d^{2} x^{2} e^{\left (-d x - c\right )} + 24 \, a b d x^{3} e^{\left (-d x - c\right )} - 2 \, a^{2} d x e^{\left (d x + c\right )} - 24 \, a b x^{2} e^{\left (d x + c\right )} + 2 \, a^{2} d x e^{\left (-d x - c\right )} - 24 \, a b x^{2} e^{\left (-d x - c\right )} - 6 \, a^{2} e^{\left (d x + c\right )} - 6 \, a^{2} e^{\left (-d x - c\right )}}{48 \, x^{4}} \]
1/48*(a^2*d^4*x^4*Ei(-d*x)*e^(-c) + a^2*d^4*x^4*Ei(d*x)*e^c + 24*a*b*d^2*x ^4*Ei(-d*x)*e^(-c) + 24*a*b*d^2*x^4*Ei(d*x)*e^c - a^2*d^3*x^3*e^(d*x + c) + a^2*d^3*x^3*e^(-d*x - c) + 24*b^2*x^4*Ei(-d*x)*e^(-c) + 24*b^2*x^4*Ei(d* x)*e^c - a^2*d^2*x^2*e^(d*x + c) - 24*a*b*d*x^3*e^(d*x + c) - a^2*d^2*x^2* e^(-d*x - c) + 24*a*b*d*x^3*e^(-d*x - c) - 2*a^2*d*x*e^(d*x + c) - 24*a*b* x^2*e^(d*x + c) + 2*a^2*d*x*e^(-d*x - c) - 24*a*b*x^2*e^(-d*x - c) - 6*a^2 *e^(d*x + c) - 6*a^2*e^(-d*x - c))/x^4
Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^5} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (b\,x^2+a\right )}^2}{x^5} \,d x \]