Integrand size = 12, antiderivative size = 62 \[ \int x^3 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {1}{4} b x^2 \cosh \left (a+\frac {b}{x^2}\right )-\frac {1}{4} b^2 \text {Chi}\left (\frac {b}{x^2}\right ) \sinh (a)+\frac {1}{4} x^4 \sinh \left (a+\frac {b}{x^2}\right )-\frac {1}{4} b^2 \cosh (a) \text {Shi}\left (\frac {b}{x^2}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5428, 3378, 3384, 3379, 3382} \[ \int x^3 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=-\frac {1}{4} b^2 \sinh (a) \text {Chi}\left (\frac {b}{x^2}\right )-\frac {1}{4} b^2 \cosh (a) \text {Shi}\left (\frac {b}{x^2}\right )+\frac {1}{4} b x^2 \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{4} x^4 \sinh \left (a+\frac {b}{x^2}\right ) \]
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5428
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {\sinh (a+b x)}{x^3} \, dx,x,\frac {1}{x^2}\right )\right ) \\ & = \frac {1}{4} x^4 \sinh \left (a+\frac {b}{x^2}\right )-\frac {1}{4} b \text {Subst}\left (\int \frac {\cosh (a+b x)}{x^2} \, dx,x,\frac {1}{x^2}\right ) \\ & = \frac {1}{4} b x^2 \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{4} x^4 \sinh \left (a+\frac {b}{x^2}\right )-\frac {1}{4} b^2 \text {Subst}\left (\int \frac {\sinh (a+b x)}{x} \, dx,x,\frac {1}{x^2}\right ) \\ & = \frac {1}{4} b x^2 \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{4} x^4 \sinh \left (a+\frac {b}{x^2}\right )-\frac {1}{4} \left (b^2 \cosh (a)\right ) \text {Subst}\left (\int \frac {\sinh (b x)}{x} \, dx,x,\frac {1}{x^2}\right )-\frac {1}{4} \left (b^2 \sinh (a)\right ) \text {Subst}\left (\int \frac {\cosh (b x)}{x} \, dx,x,\frac {1}{x^2}\right ) \\ & = \frac {1}{4} b x^2 \cosh \left (a+\frac {b}{x^2}\right )-\frac {1}{4} b^2 \text {Chi}\left (\frac {b}{x^2}\right ) \sinh (a)+\frac {1}{4} x^4 \sinh \left (a+\frac {b}{x^2}\right )-\frac {1}{4} b^2 \cosh (a) \text {Shi}\left (\frac {b}{x^2}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.90 \[ \int x^3 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {1}{4} \left (b x^2 \cosh \left (a+\frac {b}{x^2}\right )-b^2 \text {Chi}\left (\frac {b}{x^2}\right ) \sinh (a)+x^4 \sinh \left (a+\frac {b}{x^2}\right )-b^2 \cosh (a) \text {Shi}\left (\frac {b}{x^2}\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(116\) vs. \(2(54)=108\).
Time = 0.65 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.89
method | result | size |
risch | \(\frac {{\mathrm e}^{\frac {2 a \,x^{2}+b}{x^{2}}} {\mathrm e}^{-a} x^{4}}{8}-\frac {{\mathrm e}^{-a} x^{4} {\mathrm e}^{-\frac {b}{x^{2}}}}{8}+\frac {{\mathrm e}^{\frac {2 a \,x^{2}+b}{x^{2}}} {\mathrm e}^{-a} b \,x^{2}}{8}+\frac {{\mathrm e}^{2 a} {\mathrm e}^{-a} \operatorname {Ei}_{1}\left (-\frac {b}{x^{2}}\right ) b^{2}}{8}+\frac {{\mathrm e}^{-a} b \,x^{2} {\mathrm e}^{-\frac {b}{x^{2}}}}{8}-\frac {{\mathrm e}^{-a} b^{2} \operatorname {Ei}_{1}\left (\frac {b}{x^{2}}\right )}{8}\) | \(117\) |
meijerg | \(-\frac {i b^{2} \sqrt {\pi }\, \cosh \left (a \right ) \left (\frac {4 i x^{2} \cosh \left (\frac {b}{x^{2}}\right )}{b \sqrt {\pi }}+\frac {4 i x^{4} \sinh \left (\frac {b}{x^{2}}\right )}{b^{2} \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (\frac {b}{x^{2}}\right )}{\sqrt {\pi }}\right )}{16}+\frac {b^{2} \sqrt {\pi }\, \sinh \left (a \right ) \left (-\frac {4 x^{4} \left (\frac {9 b^{2}}{2 x^{4}}+3\right )}{3 \sqrt {\pi }\, b^{2}}+\frac {4 x^{4} \cosh \left (\frac {b}{x^{2}}\right )}{\sqrt {\pi }\, b^{2}}+\frac {4 x^{2} \sinh \left (\frac {b}{x^{2}}\right )}{\sqrt {\pi }\, b}-\frac {4 \left (\operatorname {Chi}\left (\frac {b}{x^{2}}\right )-\ln \left (\frac {b}{x^{2}}\right )-\gamma \right )}{\sqrt {\pi }}-\frac {2 \left (2 \gamma -3-4 \ln \left (x \right )+2 \ln \left (i b \right )\right )}{\sqrt {\pi }}+\frac {4 x^{4}}{\sqrt {\pi }\, b^{2}}\right )}{16}\) | \(183\) |
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Time = 0.24 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.44 \[ \int x^3 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {1}{4} \, x^{4} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right ) + \frac {1}{4} \, b x^{2} \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) - \frac {1}{8} \, {\left (b^{2} {\rm Ei}\left (\frac {b}{x^{2}}\right ) - b^{2} {\rm Ei}\left (-\frac {b}{x^{2}}\right )\right )} \cosh \left (a\right ) - \frac {1}{8} \, {\left (b^{2} {\rm Ei}\left (\frac {b}{x^{2}}\right ) + b^{2} {\rm Ei}\left (-\frac {b}{x^{2}}\right )\right )} \sinh \left (a\right ) \]
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\[ \int x^3 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\int x^{3} \sinh {\left (a + \frac {b}{x^{2}} \right )}\, dx \]
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Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.71 \[ \int x^3 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {1}{4} \, x^{4} \sinh \left (a + \frac {b}{x^{2}}\right ) + \frac {1}{8} \, {\left (b e^{\left (-a\right )} \Gamma \left (-1, \frac {b}{x^{2}}\right ) - b e^{a} \Gamma \left (-1, -\frac {b}{x^{2}}\right )\right )} b \]
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Leaf count of result is larger than twice the leaf count of optimal. 353 vs. \(2 (54) = 108\).
Time = 0.25 (sec) , antiderivative size = 353, normalized size of antiderivative = 5.69 \[ \int x^3 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {a^{2} b^{3} {\rm Ei}\left (a - \frac {a x^{2} + b}{x^{2}}\right ) e^{\left (-a\right )} - a^{2} b^{3} {\rm Ei}\left (-a + \frac {a x^{2} + b}{x^{2}}\right ) e^{a} - \frac {2 \, {\left (a x^{2} + b\right )} a b^{3} {\rm Ei}\left (a - \frac {a x^{2} + b}{x^{2}}\right ) e^{\left (-a\right )}}{x^{2}} + \frac {2 \, {\left (a x^{2} + b\right )} a b^{3} {\rm Ei}\left (-a + \frac {a x^{2} + b}{x^{2}}\right ) e^{a}}{x^{2}} - a b^{3} e^{\left (\frac {a x^{2} + b}{x^{2}}\right )} - a b^{3} e^{\left (-\frac {a x^{2} + b}{x^{2}}\right )} + b^{3} e^{\left (\frac {a x^{2} + b}{x^{2}}\right )} - b^{3} e^{\left (-\frac {a x^{2} + b}{x^{2}}\right )} + \frac {{\left (a x^{2} + b\right )}^{2} b^{3} {\rm Ei}\left (a - \frac {a x^{2} + b}{x^{2}}\right ) e^{\left (-a\right )}}{x^{4}} - \frac {{\left (a x^{2} + b\right )}^{2} b^{3} {\rm Ei}\left (-a + \frac {a x^{2} + b}{x^{2}}\right ) e^{a}}{x^{4}} + \frac {{\left (a x^{2} + b\right )} b^{3} e^{\left (\frac {a x^{2} + b}{x^{2}}\right )}}{x^{2}} + \frac {{\left (a x^{2} + b\right )} b^{3} e^{\left (-\frac {a x^{2} + b}{x^{2}}\right )}}{x^{2}}}{8 \, {\left (a^{2} - \frac {2 \, {\left (a x^{2} + b\right )} a}{x^{2}} + \frac {{\left (a x^{2} + b\right )}^{2}}{x^{4}}\right )} b} \]
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Timed out. \[ \int x^3 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\int x^3\,\mathrm {sinh}\left (a+\frac {b}{x^2}\right ) \,d x \]
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