Optimal. Leaf size=34 \[ \frac{x^2 \sinh \left (a+b x^2\right )}{2 b}-\frac{\cosh \left (a+b x^2\right )}{2 b^2} \]
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Rubi [A] time = 0.036081, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5321, 3296, 2638} \[ \frac{x^2 \sinh \left (a+b x^2\right )}{2 b}-\frac{\cosh \left (a+b x^2\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 5321
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x^3 \cosh \left (a+b x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x \cosh (a+b x) \, dx,x,x^2\right )\\ &=\frac{x^2 \sinh \left (a+b x^2\right )}{2 b}-\frac{\operatorname{Subst}\left (\int \sinh (a+b x) \, dx,x,x^2\right )}{2 b}\\ &=-\frac{\cosh \left (a+b x^2\right )}{2 b^2}+\frac{x^2 \sinh \left (a+b x^2\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0330678, size = 31, normalized size = 0.91 \[ \frac{b x^2 \sinh \left (a+b x^2\right )-\cosh \left (a+b x^2\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 45, normalized size = 1.3 \begin{align*}{\frac{ \left ( b{x}^{2}-1 \right ){{\rm e}^{b{x}^{2}+a}}}{4\,{b}^{2}}}-{\frac{ \left ( b{x}^{2}+1 \right ){{\rm e}^{-b{x}^{2}-a}}}{4\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.985214, size = 108, normalized size = 3.18 \begin{align*} \frac{1}{4} \, x^{4} \cosh \left (b x^{2} + a\right ) - \frac{1}{8} \, b{\left (\frac{{\left (b^{2} x^{4} e^{a} - 2 \, b x^{2} e^{a} + 2 \, e^{a}\right )} e^{\left (b x^{2}\right )}}{b^{3}} + \frac{{\left (b^{2} x^{4} + 2 \, b x^{2} + 2\right )} e^{\left (-b x^{2} - a\right )}}{b^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79538, size = 69, normalized size = 2.03 \begin{align*} \frac{b x^{2} \sinh \left (b x^{2} + a\right ) - \cosh \left (b x^{2} + a\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.03247, size = 36, normalized size = 1.06 \begin{align*} \begin{cases} \frac{x^{2} \sinh{\left (a + b x^{2} \right )}}{2 b} - \frac{\cosh{\left (a + b x^{2} \right )}}{2 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{4} \cosh{\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 [A] time = 1.29896, size = 65, normalized size = 1.91 \begin{align*} \frac{\frac{{\left (b x^{2} - 1\right )} e^{\left (b x^{2} + a\right )}}{b} - \frac{{\left (b x^{2} + 1\right )} e^{\left (-b x^{2} - a\right )}}{b}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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