Optimal. Leaf size=145 \[ \frac {4}{3} a x^2 \sinh \left (\frac {x}{2}\right ) \cosh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}+\frac {8}{3} a x^2 \tanh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-\frac {16}{9} a x \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-\frac {32}{3} a x \sqrt {a \cosh (x)+a}+\frac {224}{9} a \tanh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}+\frac {32}{27} a \sinh ^2\left (\frac {x}{2}\right ) \tanh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \]
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Rubi [A] time = 0.15, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3319, 3311, 3296, 2637, 2633} \[ \frac {4}{3} a x^2 \sinh \left (\frac {x}{2}\right ) \cosh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}+\frac {8}{3} a x^2 \tanh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-\frac {16}{9} a x \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-\frac {32}{3} a x \sqrt {a \cosh (x)+a}+\frac {224}{9} a \tanh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}+\frac {32}{27} a \sinh ^2\left (\frac {x}{2}\right ) \tanh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \]
Antiderivative was successfully verified.
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Rule 2633
Rule 2637
Rule 3296
Rule 3311
Rule 3319
Rubi steps
\begin {align*} \int x^2 (a+a \cosh (x))^{3/2} \, dx &=\left (2 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int x^2 \cosh ^3\left (\frac {x}{2}\right ) \, dx\\ &=-\frac {16}{9} a x \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}+\frac {4}{3} a x^2 \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {1}{3} \left (4 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int x^2 \cosh \left (\frac {x}{2}\right ) \, dx+\frac {1}{9} \left (16 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \cosh ^3\left (\frac {x}{2}\right ) \, dx\\ &=-\frac {16}{9} a x \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}+\frac {4}{3} a x^2 \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {8}{3} a x^2 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+\frac {1}{9} \left (32 i a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh \left (\frac {x}{2}\right )\right )-\frac {1}{3} \left (16 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int x \sinh \left (\frac {x}{2}\right ) \, dx\\ &=-\frac {32}{3} a x \sqrt {a+a \cosh (x)}-\frac {16}{9} a x \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}+\frac {4}{3} a x^2 \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {32}{9} a \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+\frac {8}{3} a x^2 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+\frac {32}{27} a \sqrt {a+a \cosh (x)} \sinh ^2\left (\frac {x}{2}\right ) \tanh \left (\frac {x}{2}\right )+\frac {1}{3} \left (32 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \cosh \left (\frac {x}{2}\right ) \, dx\\ &=-\frac {32}{3} a x \sqrt {a+a \cosh (x)}-\frac {16}{9} a x \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}+\frac {4}{3} a x^2 \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {224}{9} a \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+\frac {8}{3} a x^2 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+\frac {32}{27} a \sqrt {a+a \cosh (x)} \sinh ^2\left (\frac {x}{2}\right ) \tanh \left (\frac {x}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.22, size = 54, normalized size = 0.37 \[ \frac {2}{27} a \sqrt {a (\cosh (x)+1)} \left (\left (45 x^2+328\right ) \tanh \left (\frac {x}{2}\right )+\cosh (x) \left (\left (9 x^2+8\right ) \tanh \left (\frac {x}{2}\right )-12 x\right )-156 x\right ) \]
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 [A] time = 0.13, size = 144, normalized size = 0.99 \[ -\frac {1}{54} \, \sqrt {2} {\left (54 \, a^{\frac {3}{2}} x^{2} e^{\left (-\frac {1}{2} \, x\right )} + 9 \, a^{\frac {3}{2}} x^{2} e^{\left (-\frac {3}{2} \, x\right )} + 216 \, a^{\frac {3}{2}} x e^{\left (-\frac {1}{2} \, x\right )} + 12 \, a^{\frac {3}{2}} x e^{\left (-\frac {3}{2} \, x\right )} + 432 \, a^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, x\right )} + 8 \, a^{\frac {3}{2}} e^{\left (-\frac {3}{2} \, x\right )} - {\left (9 \, a^{\frac {3}{2}} x^{2} - 12 \, a^{\frac {3}{2}} x + 8 \, a^{\frac {3}{2}}\right )} e^{\left (\frac {3}{2} \, x\right )} - 81 \, {\left (a^{\frac {3}{2}} x^{2} - 4 \, a^{\frac {3}{2}} x + 8 \, a^{\frac {3}{2}}\right )} e^{\left (\frac {1}{2} \, x\right )} + 27 \, {\left (a^{\frac {3}{2}} x^{2} + 4 \, a^{\frac {3}{2}} x + 8 \, a^{\frac {3}{2}}\right )} e^{\left (-\frac {1}{2} \, x\right )}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a +a \cosh \relax (x )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 136, normalized size = 0.94 \[ -\frac {1}{54} \, {\left (9 \, \sqrt {2} a^{\frac {3}{2}} x^{2} + 12 \, \sqrt {2} a^{\frac {3}{2}} x + 8 \, \sqrt {2} a^{\frac {3}{2}} - {\left (9 \, \sqrt {2} a^{\frac {3}{2}} x^{2} - 12 \, \sqrt {2} a^{\frac {3}{2}} x + 8 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{\left (3 \, x\right )} - 81 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{2} - 4 \, \sqrt {2} a^{\frac {3}{2}} x + 8 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{\left (2 \, x\right )} + 81 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{2} + 4 \, \sqrt {2} a^{\frac {3}{2}} x + 8 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{x}\right )} e^{\left (-\frac {3}{2} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\left (a+a\,\mathrm {cosh}\relax (x)\right )}^{3/2} \,d x \]
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 \[ \int x^{2} \left (a \left (\cosh {\relax (x )} + 1\right )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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