Integrand size = 14, antiderivative size = 185 \[ \int x^3 (a+a \cosh (x))^{3/2} \, dx=-\frac {1280}{9} a \sqrt {a+a \cosh (x)}-16 a x^2 \sqrt {a+a \cosh (x)}-\frac {64}{27} a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}-\frac {8}{3} a x^2 \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}+\frac {32}{9} a x \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {4}{3} a x^3 \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {640}{9} a x \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+\frac {8}{3} a x^3 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right ) \]
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Time = 0.16 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3400, 3392, 3377, 2718, 3391} \[ \int x^3 (a+a \cosh (x))^{3/2} \, dx=\frac {4}{3} a x^3 \sinh \left (\frac {x}{2}\right ) \cosh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}+\frac {8}{3} a x^3 \tanh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-\frac {8}{3} a x^2 \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-16 a x^2 \sqrt {a \cosh (x)+a}-\frac {64}{27} a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-\frac {1280}{9} a \sqrt {a \cosh (x)+a}+\frac {32}{9} a x \sinh \left (\frac {x}{2}\right ) \cosh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}+\frac {640}{9} a x \tanh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \]
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Rule 2718
Rule 3377
Rule 3391
Rule 3392
Rule 3400
Rubi steps \begin{align*} \text {integral}& = \left (2 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int x^3 \cosh ^3\left (\frac {x}{2}\right ) \, dx \\ & = -\frac {8}{3} a x^2 \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}+\frac {4}{3} a x^3 \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^3 \cosh \left (\frac {x}{2}\right ) \, dx+\frac {1}{3} \left (16 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int x \cosh ^3\left (\frac {x}{2}\right ) \, dx \\ & = -\frac {64}{27} a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}-\frac {8}{3} a x^2 \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}+\frac {32}{9} a x \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {4}{3} a x^3 \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {8}{3} a x^3 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+\frac {1}{9} \left (32 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int x \cosh \left (\frac {x}{2}\right ) \, dx-\left (8 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int x^2 \sinh \left (\frac {x}{2}\right ) \, dx \\ & = -16 a x^2 \sqrt {a+a \cosh (x)}-\frac {64}{27} a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}-\frac {8}{3} a x^2 \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}+\frac {32}{9} a x \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {4}{3} a x^3 \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {64}{9} a x \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+\frac {8}{3} a x^3 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )-\frac {1}{9} \left (64 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \sinh \left (\frac {x}{2}\right ) \, dx+\left (32 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int x \cosh \left (\frac {x}{2}\right ) \, dx \\ & = -\frac {128}{9} a \sqrt {a+a \cosh (x)}-16 a x^2 \sqrt {a+a \cosh (x)}-\frac {64}{27} a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}-\frac {8}{3} a x^2 \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}+\frac {32}{9} a x \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {4}{3} a x^3 \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {640}{9} a x \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+\frac {8}{3} a x^3 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )-\left (64 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \sinh \left (\frac {x}{2}\right ) \, dx \\ & = -\frac {1280}{9} a \sqrt {a+a \cosh (x)}-16 a x^2 \sqrt {a+a \cosh (x)}-\frac {64}{27} a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}-\frac {8}{3} a x^2 \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}+\frac {32}{9} a x \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {4}{3} a x^3 \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {640}{9} a x \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+\frac {8}{3} a x^3 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right ) \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.38 \[ \int x^3 (a+a \cosh (x))^{3/2} \, dx=\frac {2}{27} a \sqrt {a (1+\cosh (x))} \left (-2 \left (968+117 x^2\right )+3 x \left (328+15 x^2\right ) \tanh \left (\frac {x}{2}\right )+\cosh (x) \left (-2 \left (8+9 x^2\right )+3 x \left (8+3 x^2\right ) \tanh \left (\frac {x}{2}\right )\right )\right ) \]
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\[\int x^{3} \left (a +a \cosh \left (x \right )\right )^{\frac {3}{2}}d x\]
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Exception generated. \[ \int x^3 (a+a \cosh (x))^{3/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x^3 (a+a \cosh (x))^{3/2} \, dx=\int x^{3} \left (a \left (\cosh {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.97 \[ \int x^3 (a+a \cosh (x))^{3/2} \, dx=-\frac {1}{54} \, {\left (9 \, \sqrt {2} a^{\frac {3}{2}} x^{3} + 18 \, \sqrt {2} a^{\frac {3}{2}} x^{2} + 24 \, \sqrt {2} a^{\frac {3}{2}} x + 16 \, \sqrt {2} a^{\frac {3}{2}} - {\left (9 \, \sqrt {2} a^{\frac {3}{2}} x^{3} - 18 \, \sqrt {2} a^{\frac {3}{2}} x^{2} + 24 \, \sqrt {2} a^{\frac {3}{2}} x - 16 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{\left (3 \, x\right )} - 81 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} - 6 \, \sqrt {2} a^{\frac {3}{2}} x^{2} + 24 \, \sqrt {2} a^{\frac {3}{2}} x - 48 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{\left (2 \, x\right )} + 81 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} + 6 \, \sqrt {2} a^{\frac {3}{2}} x^{2} + 24 \, \sqrt {2} a^{\frac {3}{2}} x + 48 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{x}\right )} e^{\left (-\frac {3}{2} \, x\right )} \]
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Time = 0.29 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.04 \[ \int x^3 (a+a \cosh (x))^{3/2} \, dx=-\frac {1}{54} \, \sqrt {2} {\left (54 \, a^{\frac {3}{2}} x^{3} e^{\left (-\frac {1}{2} \, x\right )} + 9 \, a^{\frac {3}{2}} x^{3} e^{\left (-\frac {3}{2} \, x\right )} + 324 \, a^{\frac {3}{2}} x^{2} e^{\left (-\frac {1}{2} \, x\right )} + 18 \, a^{\frac {3}{2}} x^{2} e^{\left (-\frac {3}{2} \, x\right )} + 1296 \, a^{\frac {3}{2}} x e^{\left (-\frac {1}{2} \, x\right )} + 24 \, a^{\frac {3}{2}} x e^{\left (-\frac {3}{2} \, x\right )} + 2592 \, a^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, x\right )} + 16 \, a^{\frac {3}{2}} e^{\left (-\frac {3}{2} \, x\right )} - {\left (9 \, a^{\frac {3}{2}} x^{3} - 18 \, a^{\frac {3}{2}} x^{2} + 24 \, a^{\frac {3}{2}} x - 16 \, a^{\frac {3}{2}}\right )} e^{\left (\frac {3}{2} \, x\right )} - 81 \, {\left (a^{\frac {3}{2}} x^{3} - 6 \, a^{\frac {3}{2}} x^{2} + 24 \, a^{\frac {3}{2}} x - 48 \, a^{\frac {3}{2}}\right )} e^{\left (\frac {1}{2} \, x\right )} + 27 \, {\left (a^{\frac {3}{2}} x^{3} + 6 \, a^{\frac {3}{2}} x^{2} + 24 \, a^{\frac {3}{2}} x + 48 \, a^{\frac {3}{2}}\right )} e^{\left (-\frac {1}{2} \, x\right )}\right )} \]
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Timed out. \[ \int x^3 (a+a \cosh (x))^{3/2} \, dx=\int x^3\,{\left (a+a\,\mathrm {cosh}\left (x\right )\right )}^{3/2} \,d x \]
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