Optimal. Leaf size=57 \[ \frac{1}{2} b \sinh (2 a) \text{Chi}\left (2 b x^2\right )+\frac{1}{2} b \cosh (2 a) \text{Shi}\left (2 b x^2\right )-\frac{\cosh \left (2 \left (a+b x^2\right )\right )}{4 x^2}-\frac{1}{4 x^2} \]
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Rubi [A] time = 0.125971, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5341, 5321, 3297, 3303, 3298, 3301} \[ \frac{1}{2} b \sinh (2 a) \text{Chi}\left (2 b x^2\right )+\frac{1}{2} b \cosh (2 a) \text{Shi}\left (2 b x^2\right )-\frac{\cosh \left (2 \left (a+b x^2\right )\right )}{4 x^2}-\frac{1}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 5341
Rule 5321
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\cosh ^2\left (a+b x^2\right )}{x^3} \, dx &=\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} \operatorname{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 \operatorname{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)) \operatorname{Subst}\left (\int \frac{\sinh (2 b x)}{x} \, dx,x,x^2\right )+\frac{1}{2} (b \sinh (2 a)) \operatorname{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*}
Mathematica [A] time = 0.088959, size = 46, normalized size = 0.81 \[ \frac{1}{2} \left (b \sinh (2 a) \text{Chi}\left (2 b x^2\right )+b \cosh (2 a) \text{Shi}\left (2 b x^2\right )-\frac{\cosh ^2\left (a+b x^2\right )}{x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 69, normalized size = 1.2 \begin{align*} -{\frac{1}{4\,{x}^{2}}}-{\frac{{{\rm e}^{-2\,a}}{{\rm e}^{-2\,b{x}^{2}}}}{8\,{x}^{2}}}+{\frac{{{\rm e}^{-2\,a}}b{\it Ei} \left ( 1,2\,b{x}^{2} \right ) }{4}}-{\frac{{{\rm e}^{2\,a}}{{\rm e}^{2\,b{x}^{2}}}}{8\,{x}^{2}}}-{\frac{{{\rm e}^{2\,a}}b{\it Ei} \left ( 1,-2\,b{x}^{2} \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10873, size = 49, normalized size = 0.86 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00239, size = 216, normalized size = 3.79 \begin{align*} -\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}} \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{\cosh ^{2}{\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 = 1.29606, size = 170, normalized size = 2.98 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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