Optimal. Leaf size=42 \[ \frac{1}{2} b \sinh (a) \text{Chi}\left (b x^2\right )+\frac{1}{2} b \cosh (a) \text{Shi}\left (b x^2\right )-\frac{\cosh \left (a+b x^2\right )}{2 x^2} \]
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Rubi [A] time = 0.0905614, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5321, 3297, 3303, 3298, 3301} \[ \frac{1}{2} b \sinh (a) \text{Chi}\left (b x^2\right )+\frac{1}{2} b \cosh (a) \text{Shi}\left (b x^2\right )-\frac{\cosh \left (a+b x^2\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 5321
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\cosh \left (a+b x^2\right )}{x^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\cosh (a+b x)}{x^2} \, dx,x,x^2\right )\\ &=-\frac{\cosh \left (a+b x^2\right )}{2 x^2}+\frac{1}{2} b \operatorname{Subst}\left (\int \frac{\sinh (a+b x)}{x} \, dx,x,x^2\right )\\ &=-\frac{\cosh \left (a+b x^2\right )}{2 x^2}+\frac{1}{2} (b \cosh (a)) \operatorname{Subst}\left (\int \frac{\sinh (b x)}{x} \, dx,x,x^2\right )+\frac{1}{2} (b \sinh (a)) \operatorname{Subst}\left (\int \frac{\cosh (b x)}{x} \, dx,x,x^2\right )\\ &=-\frac{\cosh \left (a+b x^2\right )}{2 x^2}+\frac{1}{2} b \text{Chi}\left (b x^2\right ) \sinh (a)+\frac{1}{2} b \cosh (a) \text{Shi}\left (b x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0415767, size = 38, normalized size = 0.9 \[ \frac{1}{2} \left (b \sinh (a) \text{Chi}\left (b x^2\right )+b \cosh (a) \text{Shi}\left (b x^2\right )-\frac{\cosh \left (a+b x^2\right )}{x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 58, normalized size = 1.4 \begin{align*} -{\frac{{{\rm e}^{-a}}{{\rm e}^{-b{x}^{2}}}}{4\,{x}^{2}}}+{\frac{{{\rm e}^{-a}}b{\it Ei} \left ( 1,b{x}^{2} \right ) }{4}}-{\frac{{{\rm e}^{a}}{{\rm e}^{b{x}^{2}}}}{4\,{x}^{2}}}-{\frac{{{\rm e}^{a}}b{\it Ei} \left ( 1,-b{x}^{2} \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24887, size = 54, normalized size = 1.29 \begin{align*} -\frac{1}{4} \,{\left ({\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} -{\rm Ei}\left (b x^{2}\right ) e^{a}\right )} b - \frac{\cosh \left (b x^{2} + a\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64399, size = 166, normalized size = 3.95 \begin{align*} \frac{{\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 ) +{\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 ) - 2 \, \cosh \left (b x^{2} + a\right )}{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{\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 = 1.32038, size = 146, normalized size = 3.48 \begin{align*} -\frac{{\left (b x^{2} + a\right )} b^{2}{\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} - a b^{2}{\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} -{\left (b x^{2} + a\right )} b^{2}{\rm Ei}\left (b x^{2}\right ) e^{a} + a b^{2}{\rm Ei}\left (b x^{2}\right ) e^{a} + b^{2} e^{\left (b x^{2} + a\right )} + b^{2} e^{\left (-b x^{2} - a\right )}}{4 \, b^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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