Integrand size = 19, antiderivative size = 175 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^5} \, dx=-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a b \cosh (c+d x)}{x^2}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}+b^2 \cosh (c) \text {Chi}(d x)+a b d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{24} a^2 d^4 \cosh (c) \text {Chi}(d x)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a b d \sinh (c+d x)}{x}-\frac {a^2 d^3 \sinh (c+d x)}{24 x}+b^2 \sinh (c) \text {Shi}(d x)+a b d^2 \sinh (c) \text {Shi}(d x)+\frac {1}{24} a^2 d^4 \sinh (c) \text {Shi}(d x) \]
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Time = 0.25 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5395, 3378, 3384, 3379, 3382} \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^5} \, dx=\frac {1}{24} a^2 d^4 \cosh (c) \text {Chi}(d x)+\frac {1}{24} a^2 d^4 \sinh (c) \text {Shi}(d x)-\frac {a^2 d^3 \sinh (c+d x)}{24 x}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a^2 d \sinh (c+d x)}{12 x^3}+a b d^2 \cosh (c) \text {Chi}(d x)+a b d^2 \sinh (c) \text {Shi}(d x)-\frac {a b \cosh (c+d x)}{x^2}-\frac {a b d \sinh (c+d x)}{x}+b^2 \cosh (c) \text {Chi}(d x)+b^2 \sinh (c) \text {Shi}(d x) \]
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5395
Rubi steps \begin{align*} \text {integral}& = \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 \\ & = a^2 \int \frac {\cosh (c+d x)}{x^5} \, dx+(2 a b) \int \frac {\cosh (c+d x)}{x^3} \, dx+b^2 \int \frac {\cosh (c+d x)}{x} \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a b \cosh (c+d x)}{x^2}+\frac {1}{4} \left (a^2 d\right ) \int \frac {\sinh (c+d x)}{x^4} \, dx+(a b d) \int \frac {\sinh (c+d x)}{x^2} \, dx+\left (b^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\left (b^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a b \cosh (c+d x)}{x^2}+b^2 \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}+b^2 \sinh (c) \text {Shi}(d x)+\frac {1}{12} \left (a^2 d^2\right ) \int \frac {\cosh (c+d x)}{x^3} \, dx+\left (a b d^2\right ) \int \frac {\cosh (c+d x)}{x} \, 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)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a b d \sinh (c+d x)}{x}+b^2 \sinh (c) \text {Shi}(d x)+\frac {1}{24} \left (a^2 d^3\right ) \int \frac {\sinh (c+d x)}{x^2} \, dx+\left (a b d^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\left (a b d^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, 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 {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} \left (a^2 d^4\right ) \int \frac {\cosh (c+d x)}{x} \, 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 {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} \left (a^2 d^4 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\frac {1}{24} \left (a^2 d^4 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, 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) \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^5} \, dx=\frac {\left (24 b^2+24 a b d^2+a^2 d^4\right ) x^4 \cosh (c) \text {Chi}(d x)-a \left (\left (6 a+24 b x^2+a d^2 x^2\right ) \cosh (c+d x)+d x \left (2 a+24 b x^2+a d^2 x^2\right ) \sinh (c+d x)\right )+\left (24 b^2+24 a b d^2+a^2 d^4\right ) x^4 \sinh (c) \text {Shi}(d x)}{24 x^4} \]
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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}\) | \(464\) |
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Time = 0.24 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^5} \, dx=-\frac {2 \, {\left ({\left (a^{2} d^{2} + 24 \, a b\right )} x^{2} + 6 \, a^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{4} + 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{4} + 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \, {\left (2 \, a^{2} d x + {\left (a^{2} d^{3} + 24 \, a b d\right )} x^{3}\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{2} d^{4} + 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{4} + 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{48 \, x^{4}} \]
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\[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^5} \, dx=\int \frac {\left (a + b x^{2}\right )^{2} \cosh {\left (c + d x \right )}}{x^{5}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^5} \, dx=\frac {1}{8} \, {\left ({\left (d^{3} e^{\left (-c\right )} \Gamma \left (-3, d x\right ) + d^{3} e^{c} \Gamma \left (-3, -d x\right )\right )} a^{2} + 4 \, {\left (d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + d e^{c} \Gamma \left (-1, -d x\right )\right )} a b - \frac {4 \, b^{2} \cosh \left (d x + c\right ) \log \left (x^{2}\right )}{d} + \frac {4 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} b^{2}}{d}\right )} d + \frac {1}{4} \, {\left (2 \, b^{2} \log \left (x^{2}\right ) - \frac {4 \, a b x^{2} + a^{2}}{x^{4}}\right )} \cosh \left (d x + c\right ) \]
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Time = 0.27 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^5} \, dx=\frac {a^{2} d^{4} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{4} x^{4} {\rm Ei}\left (d x\right ) e^{c} + 24 \, a b d^{2} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 24 \, a b d^{2} x^{4} {\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{3} x^{3} e^{\left (d x + c\right )} + a^{2} d^{3} x^{3} e^{\left (-d x - c\right )} + 24 \, b^{2} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 24 \, b^{2} x^{4} {\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{2} x^{2} e^{\left (d x + c\right )} - 24 \, a b d x^{3} e^{\left (d x + c\right )} - a^{2} d^{2} x^{2} e^{\left (-d x - c\right )} + 24 \, a b d x^{3} e^{\left (-d x - c\right )} - 2 \, a^{2} d x e^{\left (d x + c\right )} - 24 \, a b x^{2} e^{\left (d x + c\right )} + 2 \, a^{2} d x e^{\left (-d x - c\right )} - 24 \, a b x^{2} e^{\left (-d x - c\right )} - 6 \, a^{2} e^{\left (d x + c\right )} - 6 \, a^{2} e^{\left (-d x - c\right )}}{48 \, x^{4}} \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^5} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (b\,x^2+a\right )}^2}{x^5} \,d x \]
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