Optimal. Leaf size=79 \[ -\frac{\cosh ^3\left (a+b x^2\right )}{18 b^2}-\frac{\cosh \left (a+b x^2\right )}{3 b^2}+\frac{x^2 \sinh \left (a+b x^2\right )}{3 b}+\frac{x^2 \sinh \left (a+b x^2\right ) \cosh ^2\left (a+b x^2\right )}{6 b} \]
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Rubi [A] time = 0.0771761, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5321, 3310, 3296, 2638} \[ -\frac{\cosh ^3\left (a+b x^2\right )}{18 b^2}-\frac{\cosh \left (a+b x^2\right )}{3 b^2}+\frac{x^2 \sinh \left (a+b x^2\right )}{3 b}+\frac{x^2 \sinh \left (a+b x^2\right ) \cosh ^2\left (a+b x^2\right )}{6 b} \]
Antiderivative was successfully verified.
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Rule 5321
Rule 3310
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x^3 \cosh ^3\left (a+b x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x \cosh ^3(a+b x) \, dx,x,x^2\right )\\ &=-\frac{\cosh ^3\left (a+b x^2\right )}{18 b^2}+\frac{x^2 \cosh ^2\left (a+b x^2\right ) \sinh \left (a+b x^2\right )}{6 b}+\frac{1}{3} \operatorname{Subst}\left (\int x \cosh (a+b x) \, dx,x,x^2\right )\\ &=-\frac{\cosh ^3\left (a+b x^2\right )}{18 b^2}+\frac{x^2 \sinh \left (a+b x^2\right )}{3 b}+\frac{x^2 \cosh ^2\left (a+b x^2\right ) \sinh \left (a+b x^2\right )}{6 b}-\frac{\operatorname{Subst}\left (\int \sinh (a+b x) \, dx,x,x^2\right )}{3 b}\\ &=-\frac{\cosh \left (a+b x^2\right )}{3 b^2}-\frac{\cosh ^3\left (a+b x^2\right )}{18 b^2}+\frac{x^2 \sinh \left (a+b x^2\right )}{3 b}+\frac{x^2 \cosh ^2\left (a+b x^2\right ) \sinh \left (a+b x^2\right )}{6 b}\\ \end{align*}
Mathematica [A] time = 0.163931, size = 55, normalized size = 0.7 \[ -\frac{-3 b x^2 \left (9 \sinh \left (a+b x^2\right )+\sinh \left (3 \left (a+b x^2\right )\right )\right )+27 \cosh \left (a+b x^2\right )+\cosh \left (3 \left (a+b x^2\right )\right )}{72 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 93, normalized size = 1.2 \begin{align*}{\frac{ \left ( 3\,b{x}^{2}-1 \right ){{\rm e}^{3\,b{x}^{2}+3\,a}}}{144\,{b}^{2}}}+{\frac{ \left ( 3\,b{x}^{2}-3 \right ){{\rm e}^{b{x}^{2}+a}}}{16\,{b}^{2}}}-{\frac{ \left ( 3\,b{x}^{2}+3 \right ){{\rm e}^{-b{x}^{2}-a}}}{16\,{b}^{2}}}-{\frac{ \left ( 3\,b{x}^{2}+1 \right ){{\rm e}^{-3\,b{x}^{2}-3\,a}}}{144\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16948, size = 135, normalized size = 1.71 \begin{align*} \frac{{\left (3 \, b x^{2} e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x^{2}\right )}}{144 \, b^{2}} + \frac{3 \,{\left (b x^{2} e^{a} - e^{a}\right )} e^{\left (b x^{2}\right )}}{16 \, b^{2}} - \frac{3 \,{\left (b x^{2} + 1\right )} e^{\left (-b x^{2} - a\right )}}{16 \, b^{2}} - \frac{{\left (3 \, b x^{2} + 1\right )} e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{144 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83854, size = 234, normalized size = 2.96 \begin{align*} \frac{3 \, b x^{2} \sinh \left (b x^{2} + a\right )^{3} - \cosh \left (b x^{2} + a\right )^{3} - 3 \, \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right )^{2} + 9 \,{\left (b x^{2} \cosh \left (b x^{2} + a\right )^{2} + 3 \, b x^{2}\right )} \sinh \left (b x^{2} + a\right ) - 27 \, \cosh \left (b x^{2} + a\right )}{72 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.04589, size = 92, normalized size = 1.16 \begin{align*} \begin{cases} - \frac{x^{2} \sinh ^{3}{\left (a + b x^{2} \right )}}{3 b} + \frac{x^{2} \sinh{\left (a + b x^{2} \right )} \cosh ^{2}{\left (a + b x^{2} \right )}}{2 b} + \frac{\sinh ^{2}{\left (a + b x^{2} \right )} \cosh{\left (a + b x^{2} \right )}}{3 b^{2}} - \frac{7 \cosh ^{3}{\left (a + b x^{2} \right )}}{18 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{4} \cosh ^{3}{\left (a \right )}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.37563, size = 250, normalized size = 3.16 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )} e^{\left (3 \, b x^{2} + 3 \, a\right )} - 3 \, a e^{\left (3 \, b x^{2} + 3 \, a\right )} + 27 \,{\left (b x^{2} + a\right )} e^{\left (b x^{2} + a\right )} - 27 \, a e^{\left (b x^{2} + a\right )} - 27 \,{\left (b x^{2} + a\right )} e^{\left (-b x^{2} - a\right )} + 27 \, a e^{\left (-b x^{2} - a\right )} - 3 \,{\left (b x^{2} + a\right )} e^{\left (-3 \, b x^{2} - 3 \, a\right )} + 3 \, a e^{\left (-3 \, b x^{2} - 3 \, a\right )} - e^{\left (3 \, b x^{2} + 3 \, a\right )} - 27 \, e^{\left (b x^{2} + a\right )} - 27 \, e^{\left (-b x^{2} - a\right )} - e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{144 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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