Optimal. Leaf size=91 \[ -\frac{3}{8} b \cosh (a) \text{Chi}\left (b x^2\right )+\frac{3}{8} b \cosh (3 a) \text{Chi}\left (3 b x^2\right )-\frac{3}{8} b \sinh (a) \text{Shi}\left (b x^2\right )+\frac{3}{8} b \sinh (3 a) \text{Shi}\left (3 b x^2\right )+\frac{3 \sinh \left (a+b x^2\right )}{8 x^2}-\frac{\sinh \left (3 \left (a+b x^2\right )\right )}{8 x^2} \]
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Rubi [A] time = 0.213875, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5340, 5320, 3297, 3303, 3298, 3301} \[ -\frac{3}{8} b \cosh (a) \text{Chi}\left (b x^2\right )+\frac{3}{8} b \cosh (3 a) \text{Chi}\left (3 b x^2\right )-\frac{3}{8} b \sinh (a) \text{Shi}\left (b x^2\right )+\frac{3}{8} b \sinh (3 a) \text{Shi}\left (3 b x^2\right )+\frac{3 \sinh \left (a+b x^2\right )}{8 x^2}-\frac{\sinh \left (3 \left (a+b x^2\right )\right )}{8 x^2} \]
Antiderivative was successfully verified.
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Rule 5340
Rule 5320
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\sinh ^3\left (a+b x^2\right )}{x^3} \, dx &=\int \left (-\frac{3 \sinh \left (a+b x^2\right )}{4 x^3}+\frac{\sinh \left (3 a+3 b x^2\right )}{4 x^3}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\sinh \left (3 a+3 b x^2\right )}{x^3} \, dx-\frac{3}{4} \int \frac{\sinh \left (a+b x^2\right )}{x^3} \, dx\\ &=\frac{1}{8} \operatorname{Subst}\left (\int \frac{\sinh (3 a+3 b x)}{x^2} \, dx,x,x^2\right )-\frac{3}{8} \operatorname{Subst}\left (\int \frac{\sinh (a+b x)}{x^2} \, dx,x,x^2\right )\\ &=\frac{3 \sinh \left (a+b x^2\right )}{8 x^2}-\frac{\sinh \left (3 \left (a+b x^2\right )\right )}{8 x^2}-\frac{1}{8} (3 b) \operatorname{Subst}\left (\int \frac{\cosh (a+b x)}{x} \, dx,x,x^2\right )+\frac{1}{8} (3 b) \operatorname{Subst}\left (\int \frac{\cosh (3 a+3 b x)}{x} \, dx,x,x^2\right )\\ &=\frac{3 \sinh \left (a+b x^2\right )}{8 x^2}-\frac{\sinh \left (3 \left (a+b x^2\right )\right )}{8 x^2}-\frac{1}{8} (3 b \cosh (a)) \operatorname{Subst}\left (\int \frac{\cosh (b x)}{x} \, dx,x,x^2\right )+\frac{1}{8} (3 b \cosh (3 a)) \operatorname{Subst}\left (\int \frac{\cosh (3 b x)}{x} \, dx,x,x^2\right )-\frac{1}{8} (3 b \sinh (a)) \operatorname{Subst}\left (\int \frac{\sinh (b x)}{x} \, dx,x,x^2\right )+\frac{1}{8} (3 b \sinh (3 a)) \operatorname{Subst}\left (\int \frac{\sinh (3 b x)}{x} \, dx,x,x^2\right )\\ &=-\frac{3}{8} b \cosh (a) \text{Chi}\left (b x^2\right )+\frac{3}{8} b \cosh (3 a) \text{Chi}\left (3 b x^2\right )+\frac{3 \sinh \left (a+b x^2\right )}{8 x^2}-\frac{\sinh \left (3 \left (a+b x^2\right )\right )}{8 x^2}-\frac{3}{8} b \sinh (a) \text{Shi}\left (b x^2\right )+\frac{3}{8} b \sinh (3 a) \text{Shi}\left (3 b x^2\right )\\ \end{align*}
Mathematica [A] time = 0.116758, size = 90, normalized size = 0.99 \[ -\frac{3 b x^2 \cosh (a) \text{Chi}\left (b x^2\right )-3 b x^2 \cosh (3 a) \text{Chi}\left (3 b x^2\right )+3 b x^2 \sinh (a) \text{Shi}\left (b x^2\right )-3 b x^2 \sinh (3 a) \text{Shi}\left (3 b x^2\right )-3 \sinh \left (a+b x^2\right )+\sinh \left (3 \left (a+b x^2\right )\right )}{8 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 120, normalized size = 1.3 \begin{align*}{\frac{{{\rm e}^{-3\,a}}{{\rm e}^{-3\,b{x}^{2}}}}{16\,{x}^{2}}}-{\frac{3\,{{\rm e}^{-3\,a}}b{\it Ei} \left ( 1,3\,b{x}^{2} \right ) }{16}}-{\frac{3\,{{\rm e}^{-a}}{{\rm e}^{-b{x}^{2}}}}{16\,{x}^{2}}}+{\frac{3\,{{\rm e}^{-a}}b{\it Ei} \left ( 1,b{x}^{2} \right ) }{16}}-{\frac{{{\rm e}^{3\,a}}{{\rm e}^{3\,b{x}^{2}}}}{16\,{x}^{2}}}-{\frac{3\,{{\rm e}^{3\,a}}b{\it Ei} \left ( 1,-3\,b{x}^{2} \right ) }{16}}+{\frac{3\,{{\rm e}^{a}}{{\rm e}^{b{x}^{2}}}}{16\,{x}^{2}}}+{\frac{3\,{{\rm e}^{a}}b{\it Ei} \left ( 1,-b{x}^{2} \right ) }{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20758, size = 78, normalized size = 0.86 \begin{align*} \frac{3}{16} \, b e^{\left (-3 \, a\right )} \Gamma \left (-1, 3 \, b x^{2}\right ) - \frac{3}{16} \, b e^{\left (-a\right )} \Gamma \left (-1, b x^{2}\right ) - \frac{3}{16} \, b e^{a} \Gamma \left (-1, -b x^{2}\right ) + \frac{3}{16} \, b e^{\left (3 \, a\right )} \Gamma \left (-1, -3 \, b x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7403, size = 385, normalized size = 4.23 \begin{align*} -\frac{2 \, \sinh \left (b x^{2} + a\right )^{3} - 3 \,{\left (b x^{2}{\rm Ei}\left (3 \, b x^{2}\right ) + b x^{2}{\rm Ei}\left (-3 \, b x^{2}\right )\right )} \cosh \left (3 \, a\right ) + 3 \,{\left (b x^{2}{\rm Ei}\left (b x^{2}\right ) + b x^{2}{\rm Ei}\left (-b x^{2}\right )\right )} \cosh \left (a\right ) + 6 \,{\left (\cosh \left (b x^{2} + a\right )^{2} - 1\right )} \sinh \left (b x^{2} + a\right ) - 3 \,{\left (b x^{2}{\rm Ei}\left (3 \, b x^{2}\right ) - b x^{2}{\rm Ei}\left (-3 \, b x^{2}\right )\right )} \sinh \left (3 \, a\right ) + 3 \,{\left (b x^{2}{\rm Ei}\left (b x^{2}\right ) - b x^{2}{\rm Ei}\left (-b x^{2}\right )\right )} \sinh \left (a\right )}{16 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{3}{\left (a + b x^{2} \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28424, size = 301, normalized size = 3.31 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )} b^{2}{\rm Ei}\left (3 \, b x^{2}\right ) e^{\left (3 \, a\right )} - 3 \, a b^{2}{\rm Ei}\left (3 \, b x^{2}\right ) e^{\left (3 \, a\right )} - 3 \,{\left (b x^{2} + a\right )} b^{2}{\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + 3 \, a b^{2}{\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + 3 \,{\left (b x^{2} + a\right )} b^{2}{\rm Ei}\left (-3 \, b x^{2}\right ) e^{\left (-3 \, a\right )} - 3 \, a b^{2}{\rm Ei}\left (-3 \, b x^{2}\right ) e^{\left (-3 \, a\right )} - 3 \,{\left (b x^{2} + a\right )} b^{2}{\rm Ei}\left (b x^{2}\right ) e^{a} + 3 \, a b^{2}{\rm Ei}\left (b x^{2}\right ) e^{a} - b^{2} e^{\left (3 \, b x^{2} + 3 \, a\right )} + 3 \, b^{2} e^{\left (b x^{2} + a\right )} - 3 \, b^{2} e^{\left (-b x^{2} - a\right )} + b^{2} e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{16 \, b^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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