Optimal. Leaf size=68 \[ 2 x^3 \tanh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-12 x^2 \sqrt {a \cosh (x)+a}-96 \sqrt {a \cosh (x)+a}+48 x \tanh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \]
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Rubi [A] time = 0.12, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3319, 3296, 2638} \[ -12 x^2 \sqrt {a \cosh (x)+a}+2 x^3 \tanh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-96 \sqrt {a \cosh (x)+a}+48 x \tanh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3319
Rubi steps
\begin {align*} \int x^3 \sqrt {a+a \cosh (x)} \, dx &=\left (\sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int x^3 \cosh \left (\frac {x}{2}\right ) \, dx\\ &=2 x^3 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )-\left (6 \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int x^2 \sinh \left (\frac {x}{2}\right ) \, dx\\ &=-12 x^2 \sqrt {a+a \cosh (x)}+2 x^3 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+\left (24 \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int x \cosh \left (\frac {x}{2}\right ) \, dx\\ &=-12 x^2 \sqrt {a+a \cosh (x)}+48 x \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+2 x^3 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )-\left (48 \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \sinh \left (\frac {x}{2}\right ) \, dx\\ &=-96 \sqrt {a+a \cosh (x)}-12 x^2 \sqrt {a+a \cosh (x)}+48 x \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+2 x^3 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 33, normalized size = 0.49 \[ 2 \left (x \left (x^2+24\right ) \tanh \left (\frac {x}{2}\right )-6 \left (x^2+8\right )\right ) \sqrt {a (\cosh (x)+1)} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \cosh \relax (x) + a} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 62, normalized size = 0.91 \[ \frac {\sqrt {2}\, \sqrt {a \left ({\mathrm e}^{x}+1\right )^{2} {\mathrm e}^{-x}}\, \left (x^{3} {\mathrm e}^{x}-x^{3}-6 x^{2} {\mathrm e}^{x}-6 x^{2}+24 x \,{\mathrm e}^{x}-24 x -48 \,{\mathrm e}^{x}-48\right )}{{\mathrm e}^{x}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 88, normalized size = 1.29 \[ -{\left (\sqrt {2} \sqrt {a} x^{3} + 6 \, \sqrt {2} \sqrt {a} x^{2} + 24 \, \sqrt {2} \sqrt {a} x - {\left (\sqrt {2} \sqrt {a} x^{3} - 6 \, \sqrt {2} \sqrt {a} x^{2} + 24 \, \sqrt {2} \sqrt {a} x - 48 \, \sqrt {2} \sqrt {a}\right )} e^{x} + 48 \, \sqrt {2} \sqrt {a}\right )} e^{\left (-\frac {1}{2} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 63, normalized size = 0.93 \[ -\frac {\sqrt {a+a\,\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}\,\left (48\,x+96\,{\mathrm {e}}^x+12\,x^2\,{\mathrm {e}}^x-2\,x^3\,{\mathrm {e}}^x-48\,x\,{\mathrm {e}}^x+12\,x^2+2\,x^3+96\right )}{{\mathrm {e}}^x+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \sqrt {a \left (\cosh {\relax (x )} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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