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 = 0.04, 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 = {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 2638
Rule 3296
Rule 5321
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.46, size = 29, normalized size = 0.85 \[ \frac {b x^{2} \sinh \left (b x^{2} + a\right ) - \cosh \left (b x^{2} + a\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 48, normalized size = 1.41 \[ \frac {\frac {{\left (b x^{2} - 1\right )} e^{\left (b x^{2} + a\right )}}{b} - \frac {{\left (b x^{2} + 1\right )} e^{\left (-b x^{2} - a\right )}}{b}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, 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 = 0.32, size = 80, normalized size = 2.35 \[ \frac {1}{4} \, x^{4} \cosh \left (b x^{2} + a\right ) - \frac {1}{8} \, b {\left (\frac {{\left (b^{2} x^{4} e^{a} - 2 \, b x^{2} e^{a} + 2 \, e^{a}\right )} e^{\left (b x^{2}\right )}}{b^{3}} + \frac {{\left (b^{2} x^{4} + 2 \, b x^{2} + 2\right )} e^{\left (-b x^{2} - a\right )}}{b^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 28, normalized size = 0.82 \[ -\frac {\mathrm {cosh}\left (b\,x^2+a\right )-b\,x^2\,\mathrm {sinh}\left (b\,x^2+a\right )}{2\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.82, size = 36, normalized size = 1.06 \[ \begin {cases} \frac {x^{2} \sinh {\left (a + b x^{2} \right )}}{2 b} - \frac {\cosh {\left (a + b x^{2} \right )}}{2 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{4} \cosh {\relax (a )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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