Integrand size = 17, antiderivative size = 105 \[ \int \frac {\left (a+b x^2\right ) \cosh (c+d x)}{x^4} \, dx=-\frac {a \cosh (c+d x)}{3 x^3}-\frac {b \cosh (c+d x)}{x}-\frac {a d^2 \cosh (c+d x)}{6 x}+b d \text {Chi}(d x) \sinh (c)+\frac {1}{6} a d^3 \text {Chi}(d x) \sinh (c)-\frac {a d \sinh (c+d x)}{6 x^2}+b d \cosh (c) \text {Shi}(d x)+\frac {1}{6} a d^3 \cosh (c) \text {Shi}(d x) \]
-1/3*a*cosh(d*x+c)/x^3-b*cosh(d*x+c)/x-1/6*a*d^2*cosh(d*x+c)/x+b*d*cosh(c) *Shi(d*x)+1/6*a*d^3*cosh(c)*Shi(d*x)+b*d*Chi(d*x)*sinh(c)+1/6*a*d^3*Chi(d* x)*sinh(c)-1/6*a*d*sinh(d*x+c)/x^2
Time = 0.14 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^2\right ) \cosh (c+d x)}{x^4} \, dx=-\frac {2 a \cosh (c+d x)+6 b x^2 \cosh (c+d x)+a d^2 x^2 \cosh (c+d x)-d \left (6 b+a d^2\right ) x^3 \text {Chi}(d x) \sinh (c)+a d x \sinh (c+d x)-d \left (6 b+a d^2\right ) x^3 \cosh (c) \text {Shi}(d x)}{6 x^3} \]
-1/6*(2*a*Cosh[c + d*x] + 6*b*x^2*Cosh[c + d*x] + a*d^2*x^2*Cosh[c + d*x] - d*(6*b + a*d^2)*x^3*CoshIntegral[d*x]*Sinh[c] + a*d*x*Sinh[c + d*x] - d* (6*b + a*d^2)*x^3*Cosh[c]*SinhIntegral[d*x])/x^3
Time = 0.43 (sec) , antiderivative size = 105, 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 = {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.
\(\displaystyle \int \frac {\left (a+b x^2\right ) \cosh (c+d x)}{x^4} \, dx\) |
\(\Big \downarrow \) 5810 |
\(\displaystyle \int \left (\frac {a \cosh (c+d x)}{x^4}+\frac {b \cosh (c+d x)}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{6} a d^3 \sinh (c) \text {Chi}(d x)+\frac {1}{6} a d^3 \cosh (c) \text {Shi}(d x)-\frac {a d^2 \cosh (c+d x)}{6 x}-\frac {a \cosh (c+d x)}{3 x^3}-\frac {a d \sinh (c+d x)}{6 x^2}+b d \sinh (c) \text {Chi}(d x)+b d \cosh (c) \text {Shi}(d x)-\frac {b \cosh (c+d x)}{x}\) |
-1/3*(a*Cosh[c + d*x])/x^3 - (b*Cosh[c + d*x])/x - (a*d^2*Cosh[c + d*x])/( 6*x) + b*d*CoshIntegral[d*x]*Sinh[c] + (a*d^3*CoshIntegral[d*x]*Sinh[c])/6 - (a*d*Sinh[c + d*x])/(6*x^2) + b*d*Cosh[c]*SinhIntegral[d*x] + (a*d^3*Co sh[c]*SinhIntegral[d*x])/6
3.1.47.3.1 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 = 0.10 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.67
method | result | size |
risch | \(-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a \,d^{3} x^{3}-{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a \,d^{3} x^{3}+6 \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) b d \,x^{3}-6 \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) b d \,x^{3}+{\mathrm e}^{-d x -c} a \,d^{2} x^{2}+{\mathrm e}^{d x +c} a \,d^{2} x^{2}-{\mathrm e}^{-d x -c} a d x +6 \,{\mathrm e}^{-d x -c} b \,x^{2}+{\mathrm e}^{d x +c} a d x +6 \,{\mathrm e}^{d x +c} b \,x^{2}+2 \,{\mathrm e}^{-d x -c} a +2 a \,{\mathrm e}^{d x +c}}{12 x^{3}}\) | \(175\) |
meijerg | \(\frac {i d b \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{4}+\frac {d b \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {4 \gamma -4+4 \ln \left (x \right )+4 \ln \left (i d \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}+\frac {4 \,\operatorname {Chi}\left (d x \right )-4 \ln \left (d x \right )-4 \gamma }{\sqrt {\pi }}\right )}{4}-\frac {i a \cosh \left (c \right ) \sqrt {\pi }\, d^{3} \left (-\frac {8 i \left (x^{2} d^{2}+2\right ) \cosh \left (d x \right )}{3 d^{3} x^{3} \sqrt {\pi }}-\frac {8 i \sinh \left (d x \right )}{3 x^{2} d^{2} \sqrt {\pi }}+\frac {8 i \operatorname {Shi}\left (d x \right )}{3 \sqrt {\pi }}\right )}{16}-\frac {a \sinh \left (c \right ) \sqrt {\pi }\, d^{3} \left (\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {4 \left (2 \gamma -\frac {11}{3}+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{3 \sqrt {\pi }}-\frac {8 \left (\frac {55 x^{2} d^{2}}{2}+45\right )}{45 \sqrt {\pi }\, x^{2} d^{2}}+\frac {8 \cosh \left (d x \right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {16 \left (\frac {5 x^{2} d^{2}}{2}+5\right ) \sinh \left (d x \right )}{15 \sqrt {\pi }\, x^{3} d^{3}}-\frac {8 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{3 \sqrt {\pi }}\right )}{16}\) | \(297\) |
-1/12*(exp(c)*Ei(1,-d*x)*a*d^3*x^3-exp(-c)*Ei(1,d*x)*a*d^3*x^3+6*exp(c)*Ei (1,-d*x)*b*d*x^3-6*exp(-c)*Ei(1,d*x)*b*d*x^3+exp(-d*x-c)*a*d^2*x^2+exp(d*x +c)*a*d^2*x^2-exp(-d*x-c)*a*d*x+6*exp(-d*x-c)*b*x^2+exp(d*x+c)*a*d*x+6*exp (d*x+c)*b*x^2+2*exp(-d*x-c)*a+2*a*exp(d*x+c))/x^3
Time = 0.25 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.21 \[ \int \frac {\left (a+b x^2\right ) \cosh (c+d x)}{x^4} \, dx=-\frac {2 \, a d x \sinh \left (d x + c\right ) + 2 \, {\left ({\left (a d^{2} + 6 \, b\right )} x^{2} + 2 \, a\right )} \cosh \left (d x + c\right ) - {\left ({\left (a d^{3} + 6 \, b d\right )} x^{3} {\rm Ei}\left (d x\right ) - {\left (a d^{3} + 6 \, b d\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - {\left ({\left (a d^{3} + 6 \, b d\right )} x^{3} {\rm Ei}\left (d x\right ) + {\left (a d^{3} + 6 \, b d\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{12 \, x^{3}} \]
-1/12*(2*a*d*x*sinh(d*x + c) + 2*((a*d^2 + 6*b)*x^2 + 2*a)*cosh(d*x + c) - ((a*d^3 + 6*b*d)*x^3*Ei(d*x) - (a*d^3 + 6*b*d)*x^3*Ei(-d*x))*cosh(c) - (( a*d^3 + 6*b*d)*x^3*Ei(d*x) + (a*d^3 + 6*b*d)*x^3*Ei(-d*x))*sinh(c))/x^3
\[ \int \frac {\left (a+b x^2\right ) \cosh (c+d x)}{x^4} \, dx=\int \frac {\left (a + b x^{2}\right ) \cosh {\left (c + d x \right )}}{x^{4}}\, dx \]
Time = 0.25 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \frac {\left (a+b x^2\right ) \cosh (c+d x)}{x^4} \, dx=\frac {1}{6} \, {\left (a d^{2} e^{\left (-c\right )} \Gamma \left (-2, d x\right ) - a d^{2} e^{c} \Gamma \left (-2, -d x\right ) - 3 \, b {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 3 \, b {\rm Ei}\left (d x\right ) e^{c}\right )} d - \frac {{\left (3 \, b x^{2} + a\right )} \cosh \left (d x + c\right )}{3 \, x^{3}} \]
1/6*(a*d^2*e^(-c)*gamma(-2, d*x) - a*d^2*e^c*gamma(-2, -d*x) - 3*b*Ei(-d*x )*e^(-c) + 3*b*Ei(d*x)*e^c)*d - 1/3*(3*b*x^2 + a)*cosh(d*x + c)/x^3
Time = 0.27 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.62 \[ \int \frac {\left (a+b x^2\right ) \cosh (c+d x)}{x^4} \, dx=-\frac {a d^{3} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a d^{3} x^{3} {\rm Ei}\left (d x\right ) e^{c} + 6 \, b d x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 6 \, b d x^{3} {\rm Ei}\left (d x\right ) e^{c} + a d^{2} x^{2} e^{\left (d x + c\right )} + a d^{2} x^{2} e^{\left (-d x - c\right )} + a d x e^{\left (d x + c\right )} + 6 \, b x^{2} e^{\left (d x + c\right )} - a d x e^{\left (-d x - c\right )} + 6 \, b x^{2} e^{\left (-d x - c\right )} + 2 \, a e^{\left (d x + c\right )} + 2 \, a e^{\left (-d x - c\right )}}{12 \, x^{3}} \]
-1/12*(a*d^3*x^3*Ei(-d*x)*e^(-c) - a*d^3*x^3*Ei(d*x)*e^c + 6*b*d*x^3*Ei(-d *x)*e^(-c) - 6*b*d*x^3*Ei(d*x)*e^c + a*d^2*x^2*e^(d*x + c) + a*d^2*x^2*e^( -d*x - c) + a*d*x*e^(d*x + c) + 6*b*x^2*e^(d*x + c) - a*d*x*e^(-d*x - c) + 6*b*x^2*e^(-d*x - c) + 2*a*e^(d*x + c) + 2*a*e^(-d*x - c))/x^3
Timed out. \[ \int \frac {\left (a+b x^2\right ) \cosh (c+d x)}{x^4} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (b\,x^2+a\right )}{x^4} \,d x \]