Integrand size = 10, antiderivative size = 60 \[ \int x \sinh \left (a+\frac {b}{x}\right ) \, dx=\frac {1}{2} b x \cosh \left (a+\frac {b}{x}\right )-\frac {1}{2} b^2 \text {Chi}\left (\frac {b}{x}\right ) \sinh (a)+\frac {1}{2} x^2 \sinh \left (a+\frac {b}{x}\right )-\frac {1}{2} b^2 \cosh (a) \text {Shi}\left (\frac {b}{x}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5428, 3378, 3384, 3379, 3382} \[ \int x \sinh \left (a+\frac {b}{x}\right ) \, dx=-\frac {1}{2} b^2 \sinh (a) \text {Chi}\left (\frac {b}{x}\right )-\frac {1}{2} b^2 \cosh (a) \text {Shi}\left (\frac {b}{x}\right )+\frac {1}{2} x^2 \sinh \left (a+\frac {b}{x}\right )+\frac {1}{2} b x \cosh \left (a+\frac {b}{x}\right ) \]
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5428
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\sinh (a+b x)}{x^3} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{2} x^2 \sinh \left (a+\frac {b}{x}\right )-\frac {1}{2} b \text {Subst}\left (\int \frac {\cosh (a+b x)}{x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{2} b x \cosh \left (a+\frac {b}{x}\right )+\frac {1}{2} x^2 \sinh \left (a+\frac {b}{x}\right )-\frac {1}{2} b^2 \text {Subst}\left (\int \frac {\sinh (a+b x)}{x} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{2} b x \cosh \left (a+\frac {b}{x}\right )+\frac {1}{2} x^2 \sinh \left (a+\frac {b}{x}\right )-\frac {1}{2} \left (b^2 \cosh (a)\right ) \text {Subst}\left (\int \frac {\sinh (b x)}{x} \, dx,x,\frac {1}{x}\right )-\frac {1}{2} \left (b^2 \sinh (a)\right ) \text {Subst}\left (\int \frac {\cosh (b x)}{x} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{2} b x \cosh \left (a+\frac {b}{x}\right )-\frac {1}{2} b^2 \text {Chi}\left (\frac {b}{x}\right ) \sinh (a)+\frac {1}{2} x^2 \sinh \left (a+\frac {b}{x}\right )-\frac {1}{2} b^2 \cosh (a) \text {Shi}\left (\frac {b}{x}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.90 \[ \int x \sinh \left (a+\frac {b}{x}\right ) \, dx=\frac {1}{2} \left (-b^2 \text {Chi}\left (\frac {b}{x}\right ) \sinh (a)+x \left (b \cosh \left (a+\frac {b}{x}\right )+x \sinh \left (a+\frac {b}{x}\right )\right )-b^2 \cosh (a) \text {Shi}\left (\frac {b}{x}\right )\right ) \]
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Time = 1.03 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.55
method | result | size |
risch | \(-\frac {{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (\frac {b}{x}\right ) b^{2}}{4}+\frac {{\mathrm e}^{-\frac {a x +b}{x}} b x}{4}-\frac {{\mathrm e}^{-\frac {a x +b}{x}} x^{2}}{4}+\frac {{\mathrm e}^{a} \operatorname {Ei}_{1}\left (-\frac {b}{x}\right ) b^{2}}{4}+\frac {{\mathrm e}^{\frac {a x +b}{x}} x b}{4}+\frac {{\mathrm e}^{\frac {a x +b}{x}} x^{2}}{4}\) | \(93\) |
meijerg | \(-\frac {i b^{2} \sqrt {\pi }\, \cosh \left (a \right ) \left (\frac {4 i x \cosh \left (\frac {b}{x}\right )}{b \sqrt {\pi }}+\frac {4 i x^{2} \sinh \left (\frac {b}{x}\right )}{b^{2} \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (\frac {b}{x}\right )}{\sqrt {\pi }}\right )}{8}+\frac {b^{2} \sqrt {\pi }\, \sinh \left (a \right ) \left (-\frac {4 x^{2} \left (\frac {9 b^{2}}{2 x^{2}}+3\right )}{3 \sqrt {\pi }\, b^{2}}+\frac {4 x^{2} \cosh \left (\frac {b}{x}\right )}{\sqrt {\pi }\, b^{2}}+\frac {4 x \sinh \left (\frac {b}{x}\right )}{\sqrt {\pi }\, b}-\frac {4 \left (\operatorname {Chi}\left (\frac {b}{x}\right )-\ln \left (\frac {b}{x}\right )-\gamma \right )}{\sqrt {\pi }}-\frac {2 \left (2 \gamma -3-2 \ln \left (x \right )+2 \ln \left (i b \right )\right )}{\sqrt {\pi }}+\frac {4 x^{2}}{\sqrt {\pi }\, b^{2}}\right )}{8}\) | \(179\) |
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Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.38 \[ \int x \sinh \left (a+\frac {b}{x}\right ) \, dx=\frac {1}{2} \, b x \cosh \left (\frac {a x + b}{x}\right ) + \frac {1}{2} \, x^{2} \sinh \left (\frac {a x + b}{x}\right ) - \frac {1}{4} \, {\left (b^{2} {\rm Ei}\left (\frac {b}{x}\right ) - b^{2} {\rm Ei}\left (-\frac {b}{x}\right )\right )} \cosh \left (a\right ) - \frac {1}{4} \, {\left (b^{2} {\rm Ei}\left (\frac {b}{x}\right ) + b^{2} {\rm Ei}\left (-\frac {b}{x}\right )\right )} \sinh \left (a\right ) \]
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\[ \int x \sinh \left (a+\frac {b}{x}\right ) \, dx=\int x \sinh {\left (a + \frac {b}{x} \right )}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.73 \[ \int x \sinh \left (a+\frac {b}{x}\right ) \, dx=\frac {1}{2} \, x^{2} \sinh \left (a + \frac {b}{x}\right ) + \frac {1}{4} \, {\left (b e^{\left (-a\right )} \Gamma \left (-1, \frac {b}{x}\right ) - b e^{a} \Gamma \left (-1, -\frac {b}{x}\right )\right )} b \]
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Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (52) = 104\).
Time = 0.28 (sec) , antiderivative size = 313, normalized size of antiderivative = 5.22 \[ \int x \sinh \left (a+\frac {b}{x}\right ) \, dx=\frac {a^{2} b^{3} {\rm Ei}\left (a - \frac {a x + b}{x}\right ) e^{\left (-a\right )} - a^{2} b^{3} {\rm Ei}\left (-a + \frac {a x + b}{x}\right ) e^{a} - \frac {2 \, {\left (a x + b\right )} a b^{3} {\rm Ei}\left (a - \frac {a x + b}{x}\right ) e^{\left (-a\right )}}{x} + \frac {2 \, {\left (a x + b\right )} a b^{3} {\rm Ei}\left (-a + \frac {a x + b}{x}\right ) e^{a}}{x} + \frac {{\left (a x + b\right )}^{2} b^{3} {\rm Ei}\left (a - \frac {a x + b}{x}\right ) e^{\left (-a\right )}}{x^{2}} - \frac {{\left (a x + b\right )}^{2} b^{3} {\rm Ei}\left (-a + \frac {a x + b}{x}\right ) e^{a}}{x^{2}} - a b^{3} e^{\left (\frac {a x + b}{x}\right )} - a b^{3} e^{\left (-\frac {a x + b}{x}\right )} + b^{3} e^{\left (\frac {a x + b}{x}\right )} + \frac {{\left (a x + b\right )} b^{3} e^{\left (\frac {a x + b}{x}\right )}}{x} - b^{3} e^{\left (-\frac {a x + b}{x}\right )} + \frac {{\left (a x + b\right )} b^{3} e^{\left (-\frac {a x + b}{x}\right )}}{x}}{4 \, {\left (a^{2} - \frac {2 \, {\left (a x + b\right )} a}{x} + \frac {{\left (a x + b\right )}^{2}}{x^{2}}\right )} b} \]
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Timed out. \[ \int x \sinh \left (a+\frac {b}{x}\right ) \, dx=\int x\,\mathrm {sinh}\left (a+\frac {b}{x}\right ) \,d x \]
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