Integrand size = 17, antiderivative size = 69 \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x^3} \, dx=-\frac {a \cosh (c+d x)}{2 x^2}+\frac {1}{2} a d^2 \cosh (c) \text {Chi}(d x)+\frac {b \sinh (c+d x)}{d}-\frac {a d \sinh (c+d x)}{2 x}+\frac {1}{2} a d^2 \sinh (c) \text {Shi}(d x) \]
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Time = 0.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {5395, 2717, 3378, 3384, 3379, 3382} \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x^3} \, dx=\frac {1}{2} a d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} a d^2 \sinh (c) \text {Shi}(d x)-\frac {a \cosh (c+d x)}{2 x^2}-\frac {a d \sinh (c+d x)}{2 x}+\frac {b \sinh (c+d x)}{d} \]
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Rule 2717
Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5395
Rubi steps \begin{align*} \text {integral}& = \int \left (b \cosh (c+d x)+\frac {a \cosh (c+d x)}{x^3}\right ) \, dx \\ & = a \int \frac {\cosh (c+d x)}{x^3} \, dx+b \int \cosh (c+d x) \, dx \\ & = -\frac {a \cosh (c+d x)}{2 x^2}+\frac {b \sinh (c+d x)}{d}+\frac {1}{2} (a d) \int \frac {\sinh (c+d x)}{x^2} \, dx \\ & = -\frac {a \cosh (c+d x)}{2 x^2}+\frac {b \sinh (c+d x)}{d}-\frac {a d \sinh (c+d x)}{2 x}+\frac {1}{2} \left (a d^2\right ) \int \frac {\cosh (c+d x)}{x} \, dx \\ & = -\frac {a \cosh (c+d x)}{2 x^2}+\frac {b \sinh (c+d x)}{d}-\frac {a d \sinh (c+d x)}{2 x}+\frac {1}{2} \left (a d^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\frac {1}{2} \left (a d^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx \\ & = -\frac {a \cosh (c+d x)}{2 x^2}+\frac {1}{2} a d^2 \cosh (c) \text {Chi}(d x)+\frac {b \sinh (c+d x)}{d}-\frac {a d \sinh (c+d x)}{2 x}+\frac {1}{2} a d^2 \sinh (c) \text {Shi}(d x) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x^3} \, dx=\frac {b \cosh (d x) \sinh (c)}{d}-\frac {a \cosh (d x) (\cosh (c)+d x \sinh (c))}{2 x^2}+\frac {b \cosh (c) \sinh (d x)}{d}-\frac {a (d x \cosh (c)+\sinh (c)) \sinh (d x)}{2 x^2}+\frac {1}{2} a d^2 (\cosh (c) \text {Chi}(d x)+\sinh (c) \text {Shi}(d x)) \]
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Time = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.72
method | result | size |
risch | \(-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a \,d^{3} x^{2}+{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a \,d^{3} x^{2}-d^{2} {\mathrm e}^{-d x -c} a x +d^{2} {\mathrm e}^{d x +c} a x +2 \,{\mathrm e}^{-d x -c} b \,x^{2}-2 \,{\mathrm e}^{d x +c} b \,x^{2}+d \,{\mathrm e}^{-d x -c} a +a d \,{\mathrm e}^{d x +c}}{4 d \,x^{2}}\) | \(119\) |
meijerg | \(\frac {b \cosh \left (c \right ) \sinh \left (d x \right )}{d}-\frac {b \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (d x \right )}{\sqrt {\pi }}\right )}{d}-\frac {a \cosh \left (c \right ) \sqrt {\pi }\, d^{2} \left (\frac {4}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {2 \left (2 \gamma -3+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{\sqrt {\pi }}-\frac {4 \left (\frac {9 x^{2} d^{2}}{2}+3\right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \cosh \left (d x \right )}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}-\frac {4 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{\sqrt {\pi }}\right )}{8}+\frac {i a \sinh \left (c \right ) \sqrt {\pi }\, d^{2} \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}+\frac {4 i \sinh \left (d x \right )}{x^{2} d^{2} \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{8}\) | \(206\) |
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Time = 0.26 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.46 \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x^3} \, dx=-\frac {2 \, a d \cosh \left (d x + c\right ) - {\left (a d^{3} x^{2} {\rm Ei}\left (d x\right ) + a d^{3} x^{2} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \, {\left (a d^{2} x - 2 \, b x^{2}\right )} \sinh \left (d x + c\right ) - {\left (a d^{3} x^{2} {\rm Ei}\left (d x\right ) - a d^{3} x^{2} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{4 \, d x^{2}} \]
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\[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x^3} \, dx=\int \frac {\left (a + b x^{3}\right ) \cosh {\left (c + d x \right )}}{x^{3}}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.26 \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x^3} \, dx=\frac {1}{4} \, {\left (a d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + a d e^{c} \Gamma \left (-1, -d x\right ) - \frac {2 \, {\left (d x e^{c} - e^{c}\right )} b e^{\left (d x\right )}}{d^{2}} - \frac {2 \, {\left (d x + 1\right )} b e^{\left (-d x - c\right )}}{d^{2}}\right )} d + \frac {1}{2} \, {\left (2 \, b x - \frac {a}{x^{2}}\right )} \cosh \left (d x + c\right ) \]
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Time = 0.26 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.71 \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x^3} \, dx=\frac {a d^{3} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a d^{3} x^{2} {\rm Ei}\left (d x\right ) e^{c} - a d^{2} x e^{\left (d x + c\right )} + a d^{2} x e^{\left (-d x - c\right )} + 2 \, b x^{2} e^{\left (d x + c\right )} - 2 \, b x^{2} e^{\left (-d x - c\right )} - a d e^{\left (d x + c\right )} - a d e^{\left (-d x - c\right )}}{4 \, d x^{2}} \]
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Timed out. \[ \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x^3} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (b\,x^3+a\right )}{x^3} \,d x \]
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