Integrand size = 19, antiderivative size = 133 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^4} \, dx=-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {2 a b \cosh (c+d x)}{x}-\frac {a^2 d^2 \cosh (c+d x)}{6 x}+2 a b d \text {Chi}(d x) \sinh (c)+\frac {1}{6} a^2 d^3 \text {Chi}(d x) \sinh (c)+\frac {b^2 \sinh (c+d x)}{d}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+2 a b d \cosh (c) \text {Shi}(d x)+\frac {1}{6} a^2 d^3 \cosh (c) \text {Shi}(d x) \]
-1/3*a^2*cosh(d*x+c)/x^3-2*a*b*cosh(d*x+c)/x-1/6*a^2*d^2*cosh(d*x+c)/x+2*a *b*d*cosh(c)*Shi(d*x)+1/6*a^2*d^3*cosh(c)*Shi(d*x)+2*a*b*d*Chi(d*x)*sinh(c )+1/6*a^2*d^3*Chi(d*x)*sinh(c)+b^2*sinh(d*x+c)/d-1/6*a^2*d*sinh(d*x+c)/x^2
Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^4} \, dx=\frac {1}{6} \left (-\frac {2 a^2 \cosh (c+d x)}{x^3}-\frac {12 a b \cosh (c+d x)}{x}-\frac {a^2 d^2 \cosh (c+d x)}{x}+a d \left (12 b+a d^2\right ) \text {Chi}(d x) \sinh (c)+\frac {6 b^2 \sinh (c+d x)}{d}-\frac {a^2 d \sinh (c+d x)}{x^2}+a d \left (12 b+a d^2\right ) \cosh (c) \text {Shi}(d x)\right ) \]
((-2*a^2*Cosh[c + d*x])/x^3 - (12*a*b*Cosh[c + d*x])/x - (a^2*d^2*Cosh[c + d*x])/x + a*d*(12*b + a*d^2)*CoshIntegral[d*x]*Sinh[c] + (6*b^2*Sinh[c + d*x])/d - (a^2*d*Sinh[c + d*x])/x^2 + a*d*(12*b + a*d^2)*Cosh[c]*SinhInteg ral[d*x])/6
Time = 0.46 (sec) , antiderivative size = 133, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^4} \, dx\) |
| \(\Big \downarrow \) 5810 |
| \(\displaystyle \int \left (\frac {a^2 \cosh (c+d x)}{x^4}+\frac {2 a b \cosh (c+d x)}{x^2}+b^2 \cosh (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} a^2 d^3 \sinh (c) \text {Chi}(d x)+\frac {1}{6} a^2 d^3 \cosh (c) \text {Shi}(d x)-\frac {a^2 d^2 \cosh (c+d x)}{6 x}-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+2 a b d \sinh (c) \text {Chi}(d x)+2 a b d \cosh (c) \text {Shi}(d x)-\frac {2 a b \cosh (c+d x)}{x}+\frac {b^2 \sinh (c+d x)}{d}\) |
-1/3*(a^2*Cosh[c + d*x])/x^3 - (2*a*b*Cosh[c + d*x])/x - (a^2*d^2*Cosh[c + d*x])/(6*x) + 2*a*b*d*CoshIntegral[d*x]*Sinh[c] + (a^2*d^3*CoshIntegral[d *x]*Sinh[c])/6 + (b^2*Sinh[c + d*x])/d - (a^2*d*Sinh[c + d*x])/(6*x^2) + 2 *a*b*d*Cosh[c]*SinhIntegral[d*x] + (a^2*d^3*Cosh[c]*SinhIntegral[d*x])/6
3.1.55.3.1 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 = 0.26 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.81
| method | result | size |
| risch | \(-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a^{2} d^{4} x^{3}-{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a^{2} d^{4} x^{3}+12 \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a b \,d^{2} x^{3}-12 \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a b \,d^{2} x^{3}+d^{3} {\mathrm e}^{-d x -c} a^{2} x^{2}+d^{3} {\mathrm e}^{d x +c} a^{2} x^{2}-d^{2} {\mathrm e}^{-d x -c} a^{2} x +12 \,{\mathrm e}^{-d x -c} a b d \,x^{2}+6 \,{\mathrm e}^{-d x -c} b^{2} x^{3}+d^{2} {\mathrm e}^{d x +c} a^{2} x +12 \,{\mathrm e}^{d x +c} a b d \,x^{2}-6 \,{\mathrm e}^{d x +c} b^{2} x^{3}+2 \,{\mathrm e}^{-d x -c} a^{2} d +2 \,{\mathrm e}^{d x +c} a^{2} d}{12 d \,x^{3}}\) | \(241\) |
| meijerg | \(\frac {b^{2} \cosh \left (c \right ) \sinh \left (d x \right )}{d}-\frac {b^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (d x \right )}{\sqrt {\pi }}\right )}{d}+\frac {i d a b \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{2}+\frac {d b a \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {4 \gamma -4+4 \ln \left (x \right )+4 \ln \left (i d \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}+\frac {4 \,\operatorname {Chi}\left (d x \right )-4 \ln \left (d x \right )-4 \gamma }{\sqrt {\pi }}\right )}{2}-\frac {i a^{2} \cosh \left (c \right ) \sqrt {\pi }\, d^{3} \left (-\frac {8 i \left (x^{2} d^{2}+2\right ) \cosh \left (d x \right )}{3 d^{3} x^{3} \sqrt {\pi }}-\frac {8 i \sinh \left (d x \right )}{3 x^{2} d^{2} \sqrt {\pi }}+\frac {8 i \operatorname {Shi}\left (d x \right )}{3 \sqrt {\pi }}\right )}{16}-\frac {a^{2} \sinh \left (c \right ) \sqrt {\pi }\, d^{3} \left (\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {4 \left (2 \gamma -\frac {11}{3}+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{3 \sqrt {\pi }}-\frac {8 \left (\frac {55 x^{2} d^{2}}{2}+45\right )}{45 \sqrt {\pi }\, x^{2} d^{2}}+\frac {8 \cosh \left (d x \right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {16 \left (\frac {5 x^{2} d^{2}}{2}+5\right ) \sinh \left (d x \right )}{15 \sqrt {\pi }\, x^{3} d^{3}}-\frac {8 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{3 \sqrt {\pi }}\right )}{16}\) | \(342\) |
-1/12/d*(exp(c)*Ei(1,-d*x)*a^2*d^4*x^3-exp(-c)*Ei(1,d*x)*a^2*d^4*x^3+12*ex p(c)*Ei(1,-d*x)*a*b*d^2*x^3-12*exp(-c)*Ei(1,d*x)*a*b*d^2*x^3+d^3*exp(-d*x- c)*a^2*x^2+d^3*exp(d*x+c)*a^2*x^2-d^2*exp(-d*x-c)*a^2*x+12*exp(-d*x-c)*a*b *d*x^2+6*exp(-d*x-c)*b^2*x^3+d^2*exp(d*x+c)*a^2*x+12*exp(d*x+c)*a*b*d*x^2- 6*exp(d*x+c)*b^2*x^3+2*exp(-d*x-c)*a^2*d+2*exp(d*x+c)*a^2*d)/x^3
Time = 0.25 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^4} \, dx=-\frac {2 \, {\left (2 \, a^{2} d + {\left (a^{2} d^{3} + 12 \, a b d\right )} x^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{4} + 12 \, a b d^{2}\right )} x^{3} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{4} + 12 \, a b d^{2}\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \, {\left (a^{2} d^{2} x - 6 \, b^{2} x^{3}\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{2} d^{4} + 12 \, a b d^{2}\right )} x^{3} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{4} + 12 \, a b d^{2}\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{12 \, d x^{3}} \]
-1/12*(2*(2*a^2*d + (a^2*d^3 + 12*a*b*d)*x^2)*cosh(d*x + c) - ((a^2*d^4 + 12*a*b*d^2)*x^3*Ei(d*x) - (a^2*d^4 + 12*a*b*d^2)*x^3*Ei(-d*x))*cosh(c) + 2 *(a^2*d^2*x - 6*b^2*x^3)*sinh(d*x + c) - ((a^2*d^4 + 12*a*b*d^2)*x^3*Ei(d* x) + (a^2*d^4 + 12*a*b*d^2)*x^3*Ei(-d*x))*sinh(c))/(d*x^3)
\[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^4} \, dx=\int \frac {\left (a + b x^{2}\right )^{2} \cosh {\left (c + d x \right )}}{x^{4}}\, dx \]
Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^4} \, dx=\frac {1}{6} \, {\left (a^{2} d^{2} e^{\left (-c\right )} \Gamma \left (-2, d x\right ) - a^{2} d^{2} e^{c} \Gamma \left (-2, -d x\right ) - 6 \, a b {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 6 \, a b {\rm Ei}\left (d x\right ) e^{c} - \frac {3 \, {\left (d x e^{c} - e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{2}} - \frac {3 \, {\left (d x + 1\right )} b^{2} e^{\left (-d x - c\right )}}{d^{2}}\right )} d + \frac {1}{3} \, {\left (3 \, b^{2} x - \frac {6 \, a b x^{2} + a^{2}}{x^{3}}\right )} \cosh \left (d x + c\right ) \]
1/6*(a^2*d^2*e^(-c)*gamma(-2, d*x) - a^2*d^2*e^c*gamma(-2, -d*x) - 6*a*b*E i(-d*x)*e^(-c) + 6*a*b*Ei(d*x)*e^c - 3*(d*x*e^c - e^c)*b^2*e^(d*x)/d^2 - 3 *(d*x + 1)*b^2*e^(-d*x - c)/d^2)*d + 1/3*(3*b^2*x - (6*a*b*x^2 + a^2)/x^3) *cosh(d*x + c)
Time = 0.26 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.77 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^4} \, dx=-\frac {a^{2} d^{4} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a^{2} d^{4} x^{3} {\rm Ei}\left (d x\right ) e^{c} + 12 \, a b d^{2} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 12 \, a b d^{2} x^{3} {\rm Ei}\left (d x\right ) e^{c} + a^{2} d^{3} x^{2} e^{\left (d x + c\right )} + a^{2} d^{3} x^{2} e^{\left (-d x - c\right )} + a^{2} d^{2} x e^{\left (d x + c\right )} + 12 \, a b d x^{2} e^{\left (d x + c\right )} - 6 \, b^{2} x^{3} e^{\left (d x + c\right )} - a^{2} d^{2} x e^{\left (-d x - c\right )} + 12 \, a b d x^{2} e^{\left (-d x - c\right )} + 6 \, b^{2} x^{3} e^{\left (-d x - c\right )} + 2 \, a^{2} d e^{\left (d x + c\right )} + 2 \, a^{2} d e^{\left (-d x - c\right )}}{12 \, d x^{3}} \]
-1/12*(a^2*d^4*x^3*Ei(-d*x)*e^(-c) - a^2*d^4*x^3*Ei(d*x)*e^c + 12*a*b*d^2* x^3*Ei(-d*x)*e^(-c) - 12*a*b*d^2*x^3*Ei(d*x)*e^c + a^2*d^3*x^2*e^(d*x + c) + a^2*d^3*x^2*e^(-d*x - c) + a^2*d^2*x*e^(d*x + c) + 12*a*b*d*x^2*e^(d*x + c) - 6*b^2*x^3*e^(d*x + c) - a^2*d^2*x*e^(-d*x - c) + 12*a*b*d*x^2*e^(-d *x - c) + 6*b^2*x^3*e^(-d*x - c) + 2*a^2*d*e^(d*x + c) + 2*a^2*d*e^(-d*x - c))/(d*x^3)
Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^4} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (b\,x^2+a\right )}^2}{x^4} \,d x \]