Optimal. Leaf size=33 \[ \frac {\sinh ^3\left (a+b x^2\right )}{6 b}+\frac {\sinh \left (a+b x^2\right )}{2 b} \]
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Rubi [A] time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 = {5321, 2633} \[ \frac {\sinh ^3\left (a+b x^2\right )}{6 b}+\frac {\sinh \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 2633
Rule 5321
Rubi steps
\begin {align*} \int x \cosh ^3\left (a+b x^2\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \cosh ^3(a+b x) \, dx,x,x^2\right )\\ &=\frac {i \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh \left (a+b x^2\right )\right )}{2 b}\\ &=\frac {\sinh \left (a+b x^2\right )}{2 b}+\frac {\sinh ^3\left (a+b x^2\right )}{6 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 1.00 \[ \frac {\sinh ^3\left (a+b x^2\right )}{6 b}+\frac {\sinh \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 38, normalized size = 1.15 \[ \frac {\sinh \left (b x^{2} + a\right )^{3} + 3 \, {\left (\cosh \left (b x^{2} + a\right )^{2} + 3\right )} \sinh \left (b x^{2} + a\right )}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 56, normalized size = 1.70 \[ -\frac {{\left (9 \, e^{\left (2 \, b x^{2} + 2 \, a\right )} + 1\right )} e^{\left (-3 \, b x^{2} - 3 \, a\right )} - e^{\left (3 \, b x^{2} + 3 \, a\right )} - 9 \, e^{\left (b x^{2} + a\right )}}{48 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 28, normalized size = 0.85 \[ \frac {\left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (b \,x^{2}+a \right )\right )}{3}\right ) \sinh \left (b \,x^{2}+a \right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.30, size = 62, normalized size = 1.88 \[ \frac {e^{\left (3 \, b x^{2} + 3 \, a\right )}}{48 \, b} + \frac {3 \, e^{\left (b x^{2} + a\right )}}{16 \, b} - \frac {3 \, e^{\left (-b x^{2} - a\right )}}{16 \, b} - \frac {e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{48 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 26, normalized size = 0.79 \[ \frac {{\mathrm {sinh}\left (b\,x^2+a\right )}^3+3\,\mathrm {sinh}\left (b\,x^2+a\right )}{6\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.82, size = 44, normalized size = 1.33 \[ \begin {cases} - \frac {\sinh ^{3}{\left (a + b x^{2} \right )}}{3 b} + \frac {\sinh {\left (a + b x^{2} \right )} \cosh ^{2}{\left (a + b x^{2} \right )}}{2 b} & \text {for}\: b \neq 0 \\\frac {x^{2} \cosh ^{3}{\relax (a )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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