Integrand size = 19, antiderivative size = 167 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^5} \, dx=-\frac {b^2 \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {2 a b \cosh (c+d x)}{x}+\frac {1}{24} a^2 d^4 \cosh (c) \text {Chi}(d x)+2 a b d \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a^2 d^3 \sinh (c+d x)}{24 x}+\frac {b^2 x \sinh (c+d x)}{d}+2 a b d \cosh (c) \text {Shi}(d x)+\frac {1}{24} a^2 d^4 \sinh (c) \text {Shi}(d x) \]
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Time = 0.23 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5395, 3378, 3384, 3379, 3382, 3377, 2718} \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^5} \, dx=\frac {1}{24} a^2 d^4 \cosh (c) \text {Chi}(d x)+\frac {1}{24} a^2 d^4 \sinh (c) \text {Shi}(d x)-\frac {a^2 d^3 \sinh (c+d x)}{24 x}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a^2 d \sinh (c+d x)}{12 x^3}+2 a b d \sinh (c) \text {Chi}(d x)+2 a b d \cosh (c) \text {Shi}(d x)-\frac {2 a b \cosh (c+d x)}{x}-\frac {b^2 \cosh (c+d x)}{d^2}+\frac {b^2 x \sinh (c+d x)}{d} \]
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Rule 2718
Rule 3377
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^2}+b^2 x \cosh (c+d x)\right ) \, dx \\ & = a^2 \int \frac {\cosh (c+d x)}{x^5} \, dx+(2 a b) \int \frac {\cosh (c+d x)}{x^2} \, dx+b^2 \int x \cosh (c+d x) \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {2 a b \cosh (c+d x)}{x}+\frac {b^2 x \sinh (c+d x)}{d}-\frac {b^2 \int \sinh (c+d x) \, dx}{d}+\frac {1}{4} \left (a^2 d\right ) \int \frac {\sinh (c+d x)}{x^4} \, dx+(2 a b d) \int \frac {\sinh (c+d x)}{x} \, dx \\ & = -\frac {b^2 \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {2 a b \cosh (c+d x)}{x}-\frac {a^2 d \sinh (c+d x)}{12 x^3}+\frac {b^2 x \sinh (c+d x)}{d}+\frac {1}{12} \left (a^2 d^2\right ) \int \frac {\cosh (c+d x)}{x^3} \, dx+(2 a b d \cosh (c)) \int \frac {\sinh (d x)}{x} \, dx+(2 a b d \sinh (c)) \int \frac {\cosh (d x)}{x} \, dx \\ & = -\frac {b^2 \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {2 a b \cosh (c+d x)}{x}+2 a b d \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{12 x^3}+\frac {b^2 x \sinh (c+d x)}{d}+2 a b d \cosh (c) \text {Shi}(d x)+\frac {1}{24} \left (a^2 d^3\right ) \int \frac {\sinh (c+d x)}{x^2} \, dx \\ & = -\frac {b^2 \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {2 a b \cosh (c+d x)}{x}+2 a b d \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a^2 d^3 \sinh (c+d x)}{24 x}+\frac {b^2 x \sinh (c+d x)}{d}+2 a b d \cosh (c) \text {Shi}(d x)+\frac {1}{24} \left (a^2 d^4\right ) \int \frac {\cosh (c+d x)}{x} \, dx \\ & = -\frac {b^2 \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {2 a b \cosh (c+d x)}{x}+2 a b d \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a^2 d^3 \sinh (c+d x)}{24 x}+\frac {b^2 x \sinh (c+d x)}{d}+2 a b d \cosh (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 {b^2 \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {2 a b \cosh (c+d x)}{x}+\frac {1}{24} a^2 d^4 \cosh (c) \text {Chi}(d x)+2 a b d \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a^2 d^3 \sinh (c+d x)}{24 x}+\frac {b^2 x \sinh (c+d x)}{d}+2 a b d \cosh (c) \text {Shi}(d x)+\frac {1}{24} a^2 d^4 \sinh (c) \text {Shi}(d x) \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^5} \, dx=\frac {1}{24} \left (-\frac {24 b^2 \cosh (c+d x)}{d^2}-\frac {6 a^2 \cosh (c+d x)}{x^4}-\frac {a^2 d^2 \cosh (c+d x)}{x^2}-\frac {48 a b \cosh (c+d x)}{x}+a d \text {Chi}(d x) \left (a d^3 \cosh (c)+48 b \sinh (c)\right )-\frac {2 a^2 d \sinh (c+d x)}{x^3}-\frac {a^2 d^3 \sinh (c+d x)}{x}+\frac {24 b^2 x \sinh (c+d x)}{d}+a d \left (48 b \cosh (c)+a d^3 \sinh (c)\right ) \text {Shi}(d x)\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(321\) vs. \(2(155)=310\).
Time = 0.35 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.93
method | result | size |
risch | \(\frac {-{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a^{2} d^{6} x^{4}-{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a^{2} d^{6} x^{4}+{\mathrm e}^{-d x -c} a^{2} d^{5} x^{3}+48 \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a b \,d^{3} x^{4}-{\mathrm e}^{d x +c} a^{2} d^{5} x^{3}-48 \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a b \,d^{3} x^{4}-d^{4} {\mathrm e}^{-d x -c} a^{2} x^{2}-24 \,{\mathrm e}^{-d x -c} b^{2} d \,x^{5}-d^{4} {\mathrm e}^{d x +c} a^{2} x^{2}+24 \,{\mathrm e}^{d x +c} b^{2} d \,x^{5}-48 \,{\mathrm e}^{-d x -c} a b \,d^{2} x^{3}-48 \,{\mathrm e}^{d x +c} a b \,d^{2} x^{3}+2 \,{\mathrm e}^{-d x -c} a^{2} d^{3} x -24 \,{\mathrm e}^{-d x -c} b^{2} x^{4}-2 \,{\mathrm e}^{d x +c} a^{2} d^{3} x -24 \,{\mathrm e}^{d x +c} b^{2} x^{4}-6 d^{2} {\mathrm e}^{-d x -c} a^{2}-6 d^{2} {\mathrm e}^{d x +c} a^{2}}{48 d^{2} x^{4}}\) | \(322\) |
meijerg | \(-\frac {2 b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {d x \sinh \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{2}}+\frac {b^{2} \sinh \left (c \right ) \left (\cosh \left (d x \right ) x d -\sinh \left (d x \right )\right )}{d^{2}}+\frac {i d a b \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{2}+\frac {d b a \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {4 \gamma -4+4 \ln \left (x \right )+4 \ln \left (i d \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}+\frac {4 \,\operatorname {Chi}\left (d x \right )-4 \ln \left (d x \right )-4 \gamma }{\sqrt {\pi }}\right )}{2}+\frac {a^{2} \cosh \left (c \right ) \sqrt {\pi }\, d^{4} \left (-\frac {8}{\sqrt {\pi }\, x^{4} d^{4}}-\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {\frac {4 \gamma }{3}-\frac {25}{9}+\frac {4 \ln \left (x \right )}{3}+\frac {4 \ln \left (i d \right )}{3}}{\sqrt {\pi }}+\frac {\frac {25}{9} d^{4} x^{4}+8 x^{2} d^{2}+8}{\sqrt {\pi }\, x^{4} d^{4}}-\frac {8 \left (\frac {15 x^{2} d^{2}}{2}+45\right ) \cosh \left (d x \right )}{45 \sqrt {\pi }\, x^{4} d^{4}}-\frac {8 \left (\frac {15 x^{2} d^{2}}{2}+15\right ) \sinh \left (d x \right )}{45 \sqrt {\pi }\, x^{3} d^{3}}+\frac {\frac {4 \,\operatorname {Chi}\left (d x \right )}{3}-\frac {4 \ln \left (d x \right )}{3}-\frac {4 \gamma }{3}}{\sqrt {\pi }}\right )}{32}-\frac {i a^{2} \sinh \left (c \right ) \sqrt {\pi }\, d^{4} \left (-\frac {8 i \left (\frac {x^{2} d^{2}}{2}+1\right ) \cosh \left (d x \right )}{3 d^{3} x^{3} \sqrt {\pi }}-\frac {8 i \left (\frac {x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{3 d^{4} x^{4} \sqrt {\pi }}+\frac {4 i \operatorname {Shi}\left (d x \right )}{3 \sqrt {\pi }}\right )}{32}\) | \(405\) |
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Time = 0.26 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^5} \, dx=-\frac {2 \, {\left (a^{2} d^{4} x^{2} + 48 \, a b d^{2} x^{3} + 24 \, b^{2} x^{4} + 6 \, a^{2} d^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{6} + 48 \, a b d^{3}\right )} x^{4} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{6} - 48 \, a b d^{3}\right )} x^{4} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \, {\left (a^{2} d^{5} x^{3} - 24 \, b^{2} d x^{5} + 2 \, a^{2} d^{3} x\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{2} d^{6} + 48 \, a b d^{3}\right )} x^{4} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{6} - 48 \, a b d^{3}\right )} x^{4} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{48 \, d^{2} x^{4}} \]
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\[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^5} \, dx=\int \frac {\left (a + b x^{3}\right )^{2} \cosh {\left (c + d x \right )}}{x^{5}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^5} \, dx=\frac {1}{8} \, {\left (a^{2} d^{3} e^{\left (-c\right )} \Gamma \left (-3, d x\right ) + a^{2} d^{3} e^{c} \Gamma \left (-3, -d x\right ) - 8 \, a b {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 8 \, a b {\rm Ei}\left (d x\right ) e^{c} - \frac {2 \, {\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{3}} - \frac {2 \, {\left (d^{2} x^{2} + 2 \, d x + 2\right )} b^{2} e^{\left (-d x - c\right )}}{d^{3}}\right )} d + \frac {1}{4} \, {\left (2 \, b^{2} x^{2} - \frac {8 \, a b x^{3} + a^{2}}{x^{4}}\right )} \cosh \left (d x + c\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (155) = 310\).
Time = 0.26 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.89 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^5} \, dx=\frac {a^{2} d^{6} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{6} x^{4} {\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{5} x^{3} e^{\left (d x + c\right )} + a^{2} d^{5} x^{3} e^{\left (-d x - c\right )} - 48 \, a b d^{3} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 48 \, a b d^{3} x^{4} {\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{4} x^{2} e^{\left (d x + c\right )} + 24 \, b^{2} d x^{5} e^{\left (d x + c\right )} - a^{2} d^{4} x^{2} e^{\left (-d x - c\right )} - 24 \, b^{2} d x^{5} e^{\left (-d x - c\right )} - 48 \, a b d^{2} x^{3} e^{\left (d x + c\right )} - 48 \, a b d^{2} x^{3} e^{\left (-d x - c\right )} - 2 \, a^{2} d^{3} x e^{\left (d x + c\right )} - 24 \, b^{2} x^{4} e^{\left (d x + c\right )} + 2 \, a^{2} d^{3} x e^{\left (-d x - c\right )} - 24 \, b^{2} x^{4} e^{\left (-d x - c\right )} - 6 \, a^{2} d^{2} e^{\left (d x + c\right )} - 6 \, a^{2} d^{2} e^{\left (-d x - c\right )}}{48 \, d^{2} x^{4}} \]
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Timed out. \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^5} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (b\,x^3+a\right )}^2}{x^5} \,d x \]
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