Optimal. Leaf size=33 \[ \frac{\cosh ^3\left (a+b x^2\right )}{6 b}-\frac{\cosh \left (a+b x^2\right )}{2 b} \]
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Rubi [A] time = 0.0330922, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5320, 2633} \[ \frac{\cosh ^3\left (a+b x^2\right )}{6 b}-\frac{\cosh \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 5320
Rule 2633
Rubi steps
\begin{align*} \int x \sinh ^3\left (a+b x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sinh ^3(a+b x) \, dx,x,x^2\right )\\ &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh \left (a+b x^2\right )\right )}{2 b}\\ &=-\frac{\cosh \left (a+b x^2\right )}{2 b}+\frac{\cosh ^3\left (a+b x^2\right )}{6 b}\\ \end{align*}
Mathematica [A] time = 0.013943, size = 33, normalized size = 1. \[ \frac{\cosh \left (3 \left (a+b x^2\right )\right )}{24 b}-\frac{3 \cosh \left (a+b x^2\right )}{8 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 28, normalized size = 0.9 \begin{align*}{\frac{\cosh \left ( b{x}^{2}+a \right ) }{2\,b} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( b{x}^{2}+a \right ) \right ) ^{2}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.05527, size = 84, normalized size = 2.55 \begin{align*} \frac{e^{\left (3 \, b x^{2} + 3 \, a\right )}}{48 \, b} - \frac{3 \, e^{\left (b x^{2} + a\right )}}{16 \, b} - \frac{3 \, e^{\left (-b x^{2} - a\right )}}{16 \, b} + \frac{e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{48 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78307, size = 116, normalized size = 3.52 \begin{align*} \frac{\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 \, \cosh \left (b x^{2} + a\right )}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.40858, size = 44, normalized size = 1.33 \begin{align*} \begin{cases} \frac{\sinh ^{2}{\left (a + b x^{2} \right )} \cosh{\left (a + b x^{2} \right )}}{2 b} - \frac{\cosh ^{3}{\left (a + b x^{2} \right )}}{3 b} & \text{for}\: b \neq 0 \\\frac{x^{2} \sinh ^{3}{\left (a \right )}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22626, size = 76, normalized size = 2.3 \begin{align*} -\frac{{\left (9 \, e^{\left (2 \, b x^{2} + 2 \, a\right )} - 1\right )} e^{\left (-3 \, b x^{2} - 3 \, a\right )} - e^{\left (3 \, b x^{2} + 3 \, a\right )} + 9 \, e^{\left (b x^{2} + a\right )}}{48 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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