Optimal. Leaf size=34 \[ \frac {x^2 \cosh \left (a+b x^2\right )}{2 b}-\frac {\sinh \left (a+b x^2\right )}{2 b^2} \]
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Rubi [A] time = 0.04, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5320, 3296, 2637} \[ \frac {x^2 \cosh \left (a+b x^2\right )}{2 b}-\frac {\sinh \left (a+b x^2\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 5320
Rubi steps
\begin {align*} \int x^3 \sinh \left (a+b x^2\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x \sinh (a+b x) \, dx,x,x^2\right )\\ &=\frac {x^2 \cosh \left (a+b x^2\right )}{2 b}-\frac {\operatorname {Subst}\left (\int \cosh (a+b x) \, dx,x,x^2\right )}{2 b}\\ &=\frac {x^2 \cosh \left (a+b x^2\right )}{2 b}-\frac {\sinh \left (a+b x^2\right )}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 31, normalized size = 0.91 \[ \frac {b x^2 \cosh \left (a+b x^2\right )-\sinh \left (a+b x^2\right )}{2 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 29, normalized size = 0.85 \[ \frac {b x^{2} \cosh \left (b x^{2} + a\right ) - \sinh \left (b x^{2} + a\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 47, normalized size = 1.38 \[ \frac {\frac {{\left (b x^{2} - 1\right )} e^{\left (b x^{2} + a\right )}}{b} + \frac {{\left (b x^{2} + 1\right )} e^{\left (-b x^{2} - a\right )}}{b}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 45, normalized size = 1.32 \[ \frac {\left (b \,x^{2}-1\right ) {\mathrm e}^{b \,x^{2}+a}}{4 b^{2}}+\frac {\left (b \,x^{2}+1\right ) {\mathrm e}^{-b \,x^{2}-a}}{4 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 81, normalized size = 2.38 \[ \frac {1}{4} \, x^{4} \sinh \left (b x^{2} + a\right ) - \frac {1}{8} \, b {\left (\frac {{\left (b^{2} x^{4} e^{a} - 2 \, b x^{2} e^{a} + 2 \, e^{a}\right )} e^{\left (b x^{2}\right )}}{b^{3}} - \frac {{\left (b^{2} x^{4} + 2 \, b x^{2} + 2\right )} e^{\left (-b x^{2} - a\right )}}{b^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 28, normalized size = 0.82 \[ -\frac {\mathrm {sinh}\left (b\,x^2+a\right )-b\,x^2\,\mathrm {cosh}\left (b\,x^2+a\right )}{2\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.80, size = 36, normalized size = 1.06 \[ \begin {cases} \frac {x^{2} \cosh {\left (a + b x^{2} \right )}}{2 b} - \frac {\sinh {\left (a + b x^{2} \right )}}{2 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{4} \sinh {\relax (a )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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