Integrand size = 14, antiderivative size = 57 \[ \int \frac {\sinh ^2\left (a+b x^2\right )}{x^3} \, dx=\frac {1}{4 x^2}-\frac {\cosh \left (2 \left (a+b x^2\right )\right )}{4 x^2}+\frac {1}{2} b \text {Chi}\left (2 b x^2\right ) \sinh (2 a)+\frac {1}{2} b \cosh (2 a) \text {Shi}\left (2 b x^2\right ) \]
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Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5448, 5429, 3378, 3384, 3379, 3382} \[ \int \frac {\sinh ^2\left (a+b x^2\right )}{x^3} \, dx=\frac {1}{2} b \sinh (2 a) \text {Chi}\left (2 b x^2\right )+\frac {1}{2} b \cosh (2 a) \text {Shi}\left (2 b x^2\right )-\frac {\cosh \left (2 \left (a+b x^2\right )\right )}{4 x^2}+\frac {1}{4 x^2} \]
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5429
Rule 5448
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{2 x^3}+\frac {\cosh \left (2 a+2 b x^2\right )}{2 x^3}\right ) \, dx \\ & = \frac {1}{4 x^2}+\frac {1}{2} \int \frac {\cosh \left (2 a+2 b x^2\right )}{x^3} \, dx \\ & = \frac {1}{4 x^2}+\frac {1}{4} \text {Subst}\left (\int \frac {\cosh (2 a+2 b x)}{x^2} \, dx,x,x^2\right ) \\ & = \frac {1}{4 x^2}-\frac {\cosh \left (2 \left (a+b x^2\right )\right )}{4 x^2}+\frac {1}{2} b \text {Subst}\left (\int \frac {\sinh (2 a+2 b x)}{x} \, dx,x,x^2\right ) \\ & = \frac {1}{4 x^2}-\frac {\cosh \left (2 \left (a+b x^2\right )\right )}{4 x^2}+\frac {1}{2} (b \cosh (2 a)) \text {Subst}\left (\int \frac {\sinh (2 b x)}{x} \, dx,x,x^2\right )+\frac {1}{2} (b \sinh (2 a)) \text {Subst}\left (\int \frac {\cosh (2 b x)}{x} \, dx,x,x^2\right ) \\ & = \frac {1}{4 x^2}-\frac {\cosh \left (2 \left (a+b x^2\right )\right )}{4 x^2}+\frac {1}{2} b \text {Chi}\left (2 b x^2\right ) \sinh (2 a)+\frac {1}{2} b \cosh (2 a) \text {Shi}\left (2 b x^2\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81 \[ \int \frac {\sinh ^2\left (a+b x^2\right )}{x^3} \, dx=\frac {1}{2} \left (b \text {Chi}\left (2 b x^2\right ) \sinh (2 a)-\frac {\sinh ^2\left (a+b x^2\right )}{x^2}+b \cosh (2 a) \text {Shi}\left (2 b x^2\right )\right ) \]
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Time = 0.51 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.16
method | result | size |
risch | \(-\frac {2 \,\operatorname {Ei}_{1}\left (-2 x^{2} b \right ) {\mathrm e}^{2 a} b \,x^{2}-2 \,\operatorname {Ei}_{1}\left (2 x^{2} b \right ) {\mathrm e}^{-2 a} b \,x^{2}+{\mathrm e}^{2 x^{2} b +2 a}+{\mathrm e}^{-2 x^{2} b -2 a}-2}{8 x^{2}}\) | \(66\) |
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Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.58 \[ \int \frac {\sinh ^2\left (a+b x^2\right )}{x^3} \, dx=-\frac {\cosh \left (b x^{2} + a\right )^{2} - {\left (b x^{2} {\rm Ei}\left (2 \, b x^{2}\right ) - b x^{2} {\rm Ei}\left (-2 \, b x^{2}\right )\right )} \cosh \left (2 \, a\right ) + \sinh \left (b x^{2} + a\right )^{2} - {\left (b x^{2} {\rm Ei}\left (2 \, b x^{2}\right ) + b x^{2} {\rm Ei}\left (-2 \, b x^{2}\right )\right )} \sinh \left (2 \, a\right ) - 1}{4 \, x^{2}} \]
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\[ \int \frac {\sinh ^2\left (a+b x^2\right )}{x^3} \, dx=\int \frac {\sinh ^{2}{\left (a + b x^{2} \right )}}{x^{3}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.63 \[ \int \frac {\sinh ^2\left (a+b x^2\right )}{x^3} \, dx=-\frac {1}{4} \, b e^{\left (-2 \, a\right )} \Gamma \left (-1, 2 \, b x^{2}\right ) + \frac {1}{4} \, b e^{\left (2 \, a\right )} \Gamma \left (-1, -2 \, b x^{2}\right ) + \frac {1}{4 \, x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (50) = 100\).
Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.21 \[ \int \frac {\sinh ^2\left (a+b x^2\right )}{x^3} \, dx=\frac {2 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (2 \, b x^{2}\right ) e^{\left (2 \, a\right )} - 2 \, a b^{2} {\rm Ei}\left (2 \, b x^{2}\right ) e^{\left (2 \, a\right )} - 2 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (-2 \, b x^{2}\right ) e^{\left (-2 \, a\right )} + 2 \, a b^{2} {\rm Ei}\left (-2 \, b x^{2}\right ) e^{\left (-2 \, a\right )} - b^{2} e^{\left (2 \, b x^{2} + 2 \, a\right )} - b^{2} e^{\left (-2 \, b x^{2} - 2 \, a\right )} + 2 \, b^{2}}{8 \, b^{2} x^{2}} \]
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Timed out. \[ \int \frac {\sinh ^2\left (a+b x^2\right )}{x^3} \, dx=\int \frac {{\mathrm {sinh}\left (b\,x^2+a\right )}^2}{x^3} \,d x \]
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