Optimal. Leaf size=51 \[ -\frac {\cosh ^2\left (a+b x^2\right )}{8 b^2}+\frac {x^2 \sinh \left (a+b x^2\right ) \cosh \left (a+b x^2\right )}{4 b}+\frac {x^4}{8} \]
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Rubi [A] time = 0.05, 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 = {5321, 3310, 30} \[ -\frac {\cosh ^2\left (a+b x^2\right )}{8 b^2}+\frac {x^2 \sinh \left (a+b x^2\right ) \cosh \left (a+b x^2\right )}{4 b}+\frac {x^4}{8} \]
Antiderivative was successfully verified.
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Rule 30
Rule 3310
Rule 5321
Rubi steps
\begin {align*} \int x^3 \cosh ^2\left (a+b x^2\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x \cosh ^2(a+b x) \, dx,x,x^2\right )\\ &=-\frac {\cosh ^2\left (a+b x^2\right )}{8 b^2}+\frac {x^2 \cosh \left (a+b x^2\right ) \sinh \left (a+b x^2\right )}{4 b}+\frac {1}{4} \operatorname {Subst}\left (\int x \, dx,x,x^2\right )\\ &=\frac {x^4}{8}-\frac {\cosh ^2\left (a+b x^2\right )}{8 b^2}+\frac {x^2 \cosh \left (a+b x^2\right ) \sinh \left (a+b x^2\right )}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 40, normalized size = 0.78 \[ -\frac {\cosh \left (2 \left (a+b x^2\right )\right )-2 b x^2 \left (\sinh \left (2 \left (a+b x^2\right )\right )+b x^2\right )}{16 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 60, normalized size = 1.18 \[ \frac {2 \, b^{2} x^{4} + 4 \, b x^{2} \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right ) - \cosh \left (b x^{2} + a\right )^{2} - \sinh \left (b x^{2} + a\right )^{2}}{16 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 121, normalized size = 2.37 \[ \frac {4 \, {\left (b x^{2} + a\right )}^{2} - 8 \, {\left (b x^{2} + a\right )} a + 2 \, {\left (b x^{2} + a\right )} e^{\left (2 \, b x^{2} + 2 \, a\right )} - 2 \, a e^{\left (2 \, b x^{2} + 2 \, a\right )} - 2 \, {\left (b x^{2} + a\right )} e^{\left (-2 \, b x^{2} - 2 \, a\right )} + 2 \, a e^{\left (-2 \, b x^{2} - 2 \, a\right )} - e^{\left (2 \, b x^{2} + 2 \, a\right )} - e^{\left (-2 \, b x^{2} - 2 \, a\right )}}{32 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 55, normalized size = 1.08 \[ \frac {x^{4}}{8}+\frac {\left (2 b \,x^{2}-1\right ) {\mathrm e}^{2 b \,x^{2}+2 a}}{32 b^{2}}-\frac {\left (2 b \,x^{2}+1\right ) {\mathrm e}^{-2 b \,x^{2}-2 a}}{32 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 59, normalized size = 1.16 \[ \frac {1}{8} \, x^{4} + \frac {{\left (2 \, b x^{2} e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x^{2}\right )}}{32 \, b^{2}} - \frac {{\left (2 \, b x^{2} + 1\right )} e^{\left (-2 \, b x^{2} - 2 \, a\right )}}{32 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 42, normalized size = 0.82 \[ \frac {x^4}{8}-\frac {\frac {\mathrm {cosh}\left (2\,b\,x^2+2\,a\right )}{16}-\frac {b\,x^2\,\mathrm {sinh}\left (2\,b\,x^2+2\,a\right )}{8}}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.67, size = 78, normalized size = 1.53 \[ \begin {cases} - \frac {x^{4} \sinh ^{2}{\left (a + b x^{2} \right )}}{8} + \frac {x^{4} \cosh ^{2}{\left (a + b x^{2} \right )}}{8} + \frac {x^{2} \sinh {\left (a + b x^{2} \right )} \cosh {\left (a + b x^{2} \right )}}{4 b} - \frac {\cosh ^{2}{\left (a + b x^{2} \right )}}{8 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{4} \cosh ^{2}{\relax (a )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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