Optimal. Leaf size=51 \[ -\frac {x^4}{8}+\frac {x^2 \cosh \left (a+b x^2\right ) \sinh \left (a+b x^2\right )}{4 b}-\frac {\sinh ^2\left (a+b x^2\right )}{8 b^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5428, 3391, 30}
\begin {gather*} -\frac {\sinh ^2\left (a+b x^2\right )}{8 b^2}+\frac {x^2 \sinh \left (a+b x^2\right ) \cosh \left (a+b x^2\right )}{4 b}-\frac {x^4}{8} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 3391
Rule 5428
Rubi steps
\begin {align*} \int x^3 \sinh ^2\left (a+b x^2\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int x \sinh ^2(a+b x) \, dx,x,x^2\right )\\ &=\frac {x^2 \cosh \left (a+b x^2\right ) \sinh \left (a+b x^2\right )}{4 b}-\frac {\sinh ^2\left (a+b x^2\right )}{8 b^2}-\frac {1}{4} \text {Subst}\left (\int x \, dx,x,x^2\right )\\ &=-\frac {x^4}{8}+\frac {x^2 \cosh \left (a+b x^2\right ) \sinh \left (a+b x^2\right )}{4 b}-\frac {\sinh ^2\left (a+b x^2\right )}{8 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 42, normalized size = 0.82 \begin {gather*} -\frac {\cosh \left (2 \left (a+b x^2\right )\right )+2 b x^2 \left (b x^2-\sinh \left (2 \left (a+b x^2\right )\right )\right )}{16 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.08, size = 55, normalized size = 1.08
method | result | size |
risch | \(-\frac {x^{4}}{8}+\frac {\left (2 x^{2} b -1\right ) {\mathrm e}^{2 x^{2} b +2 a}}{32 b^{2}}-\frac {\left (2 x^{2} b +1\right ) {\mathrm e}^{-2 x^{2} b -2 a}}{32 b^{2}}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 59, normalized size = 1.16 \begin {gather*} -\frac {1}{8} \, x^{4} + \frac {{\left (2 \, b x^{2} e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x^{2}\right )}}{32 \, b^{2}} - \frac {{\left (2 \, b x^{2} + 1\right )} e^{\left (-2 \, b x^{2} - 2 \, a\right )}}{32 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 56, normalized size = 1.10 \begin {gather*} -\frac {2 \, b^{2} x^{4} - 4 \, b x^{2} \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right ) + \cosh \left (b x^{2} + a\right )^{2} + \sinh \left (b x^{2} + a\right )^{2}}{16 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.26, size = 78, normalized size = 1.53 \begin {gather*} \begin {cases} \frac {x^{4} \sinh ^{2}{\left (a + b x^{2} \right )}}{8} - \frac {x^{4} \cosh ^{2}{\left (a + b x^{2} \right )}}{8} + \frac {x^{2} \sinh {\left (a + b x^{2} \right )} \cosh {\left (a + b x^{2} \right )}}{4 b} - \frac {\cosh ^{2}{\left (a + b x^{2} \right )}}{8 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{4} \sinh ^{2}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs.
\(2 (45) = 90\).
time = 0.41, size = 141, normalized size = 2.76 \begin {gather*} -\frac {4 \, {\left (b x^{2} + a\right )}^{2} - 2 \, {\left (b x^{2} + a\right )} e^{\left (2 \, b x^{2} + 2 \, a\right )} + 2 \, {\left (b x^{2} + a\right )} e^{\left (-2 \, b x^{2} - 2 \, a\right )} + e^{\left (2 \, b x^{2} + 2 \, a\right )} + e^{\left (-2 \, b x^{2} - 2 \, a\right )}}{32 \, b^{2}} + \frac {4 \, {\left (b x^{2} + a\right )} a - a e^{\left (2 \, b x^{2} + 2 \, a\right )} - {\left (2 \, a e^{\left (2 \, b x^{2} + 2 \, a\right )} - a\right )} e^{\left (-2 \, b x^{2} - 2 \, a\right )}}{16 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 42, normalized size = 0.82 \begin {gather*} -\frac {\frac {\mathrm {cosh}\left (2\,b\,x^2+2\,a\right )}{16}-\frac {b\,x^2\,\mathrm {sinh}\left (2\,b\,x^2+2\,a\right )}{8}}{b^2}-\frac {x^4}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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