Integrand size = 19, antiderivative size = 150 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^4} \, dx=-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {a^2 d^2 \cosh (c+d x)}{6 x}-\frac {2 b^2 x \cosh (c+d x)}{d^2}+2 a b \cosh (c) \text {Chi}(d x)+\frac {1}{6} a^2 d^3 \text {Chi}(d x) \sinh (c)+\frac {2 b^2 \sinh (c+d x)}{d^3}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+\frac {b^2 x^2 \sinh (c+d x)}{d}+\frac {1}{6} a^2 d^3 \cosh (c) \text {Shi}(d x)+2 a b \sinh (c) \text {Shi}(d x) \]
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Time = 0.19 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5395, 3378, 3384, 3379, 3382, 3377, 2717} \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^4} \, dx=\frac {1}{6} a^2 d^3 \sinh (c) \text {Chi}(d x)+\frac {1}{6} a^2 d^3 \cosh (c) \text {Shi}(d x)-\frac {a^2 d^2 \cosh (c+d x)}{6 x}-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+2 a b \cosh (c) \text {Chi}(d x)+2 a b \sinh (c) \text {Shi}(d x)+\frac {2 b^2 \sinh (c+d x)}{d^3}-\frac {2 b^2 x \cosh (c+d x)}{d^2}+\frac {b^2 x^2 \sinh (c+d x)}{d} \]
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Rule 2717
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^4}+\frac {2 a b \cosh (c+d x)}{x}+b^2 x^2 \cosh (c+d x)\right ) \, dx \\ & = a^2 \int \frac {\cosh (c+d x)}{x^4} \, dx+(2 a b) \int \frac {\cosh (c+d x)}{x} \, dx+b^2 \int x^2 \cosh (c+d x) \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{3 x^3}+\frac {b^2 x^2 \sinh (c+d x)}{d}-\frac {\left (2 b^2\right ) \int x \sinh (c+d x) \, dx}{d}+\frac {1}{3} \left (a^2 d\right ) \int \frac {\sinh (c+d x)}{x^3} \, dx+(2 a b \cosh (c)) \int \frac {\cosh (d x)}{x} \, dx+(2 a b \sinh (c)) \int \frac {\sinh (d x)}{x} \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {2 b^2 x \cosh (c+d x)}{d^2}+2 a b \cosh (c) \text {Chi}(d x)-\frac {a^2 d \sinh (c+d x)}{6 x^2}+\frac {b^2 x^2 \sinh (c+d x)}{d}+2 a b \sinh (c) \text {Shi}(d x)+\frac {\left (2 b^2\right ) \int \cosh (c+d x) \, dx}{d^2}+\frac {1}{6} \left (a^2 d^2\right ) \int \frac {\cosh (c+d x)}{x^2} \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {a^2 d^2 \cosh (c+d x)}{6 x}-\frac {2 b^2 x \cosh (c+d x)}{d^2}+2 a b \cosh (c) \text {Chi}(d x)+\frac {2 b^2 \sinh (c+d x)}{d^3}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+\frac {b^2 x^2 \sinh (c+d x)}{d}+2 a b \sinh (c) \text {Shi}(d x)+\frac {1}{6} \left (a^2 d^3\right ) \int \frac {\sinh (c+d x)}{x} \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {a^2 d^2 \cosh (c+d x)}{6 x}-\frac {2 b^2 x \cosh (c+d x)}{d^2}+2 a b \cosh (c) \text {Chi}(d x)+\frac {2 b^2 \sinh (c+d x)}{d^3}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+\frac {b^2 x^2 \sinh (c+d x)}{d}+2 a b \sinh (c) \text {Shi}(d x)+\frac {1}{6} \left (a^2 d^3 \cosh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx+\frac {1}{6} \left (a^2 d^3 \sinh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {a^2 d^2 \cosh (c+d x)}{6 x}-\frac {2 b^2 x \cosh (c+d x)}{d^2}+2 a b \cosh (c) \text {Chi}(d x)+\frac {1}{6} a^2 d^3 \text {Chi}(d x) \sinh (c)+\frac {2 b^2 \sinh (c+d x)}{d^3}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+\frac {b^2 x^2 \sinh (c+d x)}{d}+\frac {1}{6} a^2 d^3 \cosh (c) \text {Shi}(d x)+2 a b \sinh (c) \text {Shi}(d x) \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^4} \, dx=\frac {1}{6} \left (-\frac {2 a^2 \cosh (c+d x)}{x^3}-\frac {a^2 d^2 \cosh (c+d x)}{x}-\frac {12 b^2 x \cosh (c+d x)}{d^2}+a \text {Chi}(d x) \left (12 b \cosh (c)+a d^3 \sinh (c)\right )+\frac {12 b^2 \sinh (c+d x)}{d^3}-\frac {a^2 d \sinh (c+d x)}{x^2}+\frac {6 b^2 x^2 \sinh (c+d x)}{d}+a \left (a d^3 \cosh (c)+12 b \sinh (c)\right ) \text {Shi}(d x)\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(283\) vs. \(2(140)=280\).
Time = 0.31 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.89
method | result | size |
risch | \(-\frac {-{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a^{2} d^{6} x^{3}+{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a^{2} d^{6} x^{3}+{\mathrm e}^{d x +c} a^{2} d^{5} x^{2}-6 \,{\mathrm e}^{d x +c} b^{2} d^{2} x^{5}+{\mathrm e}^{-d x -c} a^{2} d^{5} x^{2}+6 \,{\mathrm e}^{-d x -c} b^{2} d^{2} x^{5}+12 \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a b \,d^{3} x^{3}+12 \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a b \,d^{3} x^{3}+d^{4} {\mathrm e}^{d x +c} a^{2} x +12 \,{\mathrm e}^{d x +c} b^{2} d \,x^{4}-d^{4} {\mathrm e}^{-d x -c} a^{2} x +12 \,{\mathrm e}^{-d x -c} b^{2} d \,x^{4}+2 \,{\mathrm e}^{d x +c} a^{2} d^{3}-12 \,{\mathrm e}^{d x +c} b^{2} x^{3}+2 \,{\mathrm e}^{-d x -c} a^{2} d^{3}+12 \,{\mathrm e}^{-d x -c} b^{2} x^{3}}{12 d^{3} x^{3}}\) | \(284\) |
meijerg | \(\frac {4 i b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {i \left (\frac {3 x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{3}}+\frac {4 b^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (\frac {x^{2} d^{2}}{2}+1\right ) \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {d x \sinh \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{3}}+a b \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {2 \gamma +2 \ln \left (x \right )+2 \ln \left (i d \right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Chi}\left (d x \right )-2 \ln \left (d x \right )-2 \gamma }{\sqrt {\pi }}\right )+2 a b \,\operatorname {Shi}\left (d x \right ) \sinh \left (c \right )-\frac {i a^{2} \cosh \left (c \right ) \sqrt {\pi }\, d^{3} \left (-\frac {8 i \left (x^{2} d^{2}+2\right ) \cosh \left (d x \right )}{3 d^{3} x^{3} \sqrt {\pi }}-\frac {8 i \sinh \left (d x \right )}{3 x^{2} d^{2} \sqrt {\pi }}+\frac {8 i \operatorname {Shi}\left (d x \right )}{3 \sqrt {\pi }}\right )}{16}-\frac {a^{2} \sinh \left (c \right ) \sqrt {\pi }\, d^{3} \left (\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {4 \left (2 \gamma -\frac {11}{3}+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{3 \sqrt {\pi }}-\frac {8 \left (\frac {55 x^{2} d^{2}}{2}+45\right )}{45 \sqrt {\pi }\, x^{2} d^{2}}+\frac {8 \cosh \left (d x \right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {16 \left (\frac {5 x^{2} d^{2}}{2}+5\right ) \sinh \left (d x \right )}{15 \sqrt {\pi }\, x^{3} d^{3}}-\frac {8 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{3 \sqrt {\pi }}\right )}{16}\) | \(347\) |
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Time = 0.25 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^4} \, dx=-\frac {2 \, {\left (a^{2} d^{5} x^{2} + 12 \, b^{2} d x^{4} + 2 \, a^{2} d^{3}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{6} + 12 \, a b d^{3}\right )} x^{3} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{6} - 12 \, a b d^{3}\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - 2 \, {\left (6 \, b^{2} d^{2} x^{5} - a^{2} d^{4} x + 12 \, b^{2} x^{3}\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{2} d^{6} + 12 \, a b d^{3}\right )} x^{3} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{6} - 12 \, a b d^{3}\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{12 \, d^{3} x^{3}} \]
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\[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^4} \, dx=\int \frac {\left (a + b x^{3}\right )^{2} \cosh {\left (c + d x \right )}}{x^{4}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^4} \, dx=\frac {1}{6} \, {\left ({\left (d^{2} e^{\left (-c\right )} \Gamma \left (-2, d x\right ) - d^{2} e^{c} \Gamma \left (-2, -d x\right )\right )} a^{2} - b^{2} {\left (\frac {{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} e^{\left (d x\right )}}{d^{4}} + \frac {{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} e^{\left (-d x - c\right )}}{d^{4}}\right )} - \frac {4 \, a b \cosh \left (d x + c\right ) \log \left (x^{3}\right )}{d} + \frac {6 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} a b}{d}\right )} d + \frac {1}{3} \, {\left (b^{2} x^{3} + 2 \, a b \log \left (x^{3}\right ) - \frac {a^{2}}{x^{3}}\right )} \cosh \left (d x + c\right ) \]
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Time = 0.25 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.86 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^4} \, dx=-\frac {a^{2} d^{6} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a^{2} d^{6} x^{3} {\rm Ei}\left (d x\right ) e^{c} + a^{2} d^{5} x^{2} e^{\left (d x + c\right )} - 6 \, b^{2} d^{2} x^{5} e^{\left (d x + c\right )} + a^{2} d^{5} x^{2} e^{\left (-d x - c\right )} + 6 \, b^{2} d^{2} x^{5} e^{\left (-d x - c\right )} - 12 \, a b d^{3} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 12 \, a b d^{3} x^{3} {\rm Ei}\left (d x\right ) e^{c} + a^{2} d^{4} x e^{\left (d x + c\right )} + 12 \, b^{2} d x^{4} e^{\left (d x + c\right )} - a^{2} d^{4} x e^{\left (-d x - c\right )} + 12 \, b^{2} d x^{4} e^{\left (-d x - c\right )} + 2 \, a^{2} d^{3} e^{\left (d x + c\right )} - 12 \, b^{2} x^{3} e^{\left (d x + c\right )} + 2 \, a^{2} d^{3} e^{\left (-d x - c\right )} + 12 \, b^{2} x^{3} e^{\left (-d x - c\right )}}{12 \, d^{3} x^{3}} \]
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Timed out. \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^4} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (b\,x^3+a\right )}^2}{x^4} \,d x \]
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