Integrand size = 14, antiderivative size = 53 \[ \int x^2 \sqrt {a+a \cosh (x)} \, dx=-8 x \sqrt {a+a \cosh (x)}+16 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+2 x^2 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3400, 3377, 2717} \[ \int x^2 \sqrt {a+a \cosh (x)} \, dx=2 x^2 \tanh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-8 x \sqrt {a \cosh (x)+a}+16 \tanh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \]
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Rule 2717
Rule 3377
Rule 3400
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int x^2 \cosh \left (\frac {x}{2}\right ) \, dx \\ & = 2 x^2 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )-\left (4 \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int x \sinh \left (\frac {x}{2}\right ) \, dx \\ & = -8 x \sqrt {a+a \cosh (x)}+2 x^2 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+\left (8 \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \cosh \left (\frac {x}{2}\right ) \, dx \\ & = -8 x \sqrt {a+a \cosh (x)}+16 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )+2 x^2 \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.58 \[ \int x^2 \sqrt {a+a \cosh (x)} \, dx=8 \sqrt {a (1+\cosh (x))} \left (-x+\frac {1}{4} \left (8+x^2\right ) \tanh \left (\frac {x}{2}\right )\right ) \]
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Time = 0.05 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.94
method | result | size |
risch | \(\frac {\sqrt {2}\, \sqrt {a \left ({\mathrm e}^{x}+1\right )^{2} {\mathrm e}^{-x}}\, \left (x^{2} {\mathrm e}^{x}-x^{2}-4 x \,{\mathrm e}^{x}-4 x +8 \,{\mathrm e}^{x}-8\right )}{{\mathrm e}^{x}+1}\) | \(50\) |
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Exception generated. \[ \int x^2 \sqrt {a+a \cosh (x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x^2 \sqrt {a+a \cosh (x)} \, dx=\int x^{2} \sqrt {a \left (\cosh {\left (x \right )} + 1\right )}\, dx \]
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none
Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.25 \[ \int x^2 \sqrt {a+a \cosh (x)} \, dx=-{\left (\sqrt {2} \sqrt {a} x^{2} + 4 \, \sqrt {2} \sqrt {a} x - {\left (\sqrt {2} \sqrt {a} x^{2} - 4 \, \sqrt {2} \sqrt {a} x + 8 \, \sqrt {2} \sqrt {a}\right )} e^{x} + 8 \, \sqrt {2} \sqrt {a}\right )} e^{\left (-\frac {1}{2} \, x\right )} \]
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\[ \int x^2 \sqrt {a+a \cosh (x)} \, dx=\int { \sqrt {a \cosh \left (x\right ) + a} x^{2} \,d x } \]
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Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.96 \[ \int x^2 \sqrt {a+a \cosh (x)} \, dx=-\frac {\sqrt {a+a\,\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}\,\left (8\,x-16\,{\mathrm {e}}^x-2\,x^2\,{\mathrm {e}}^x+8\,x\,{\mathrm {e}}^x+2\,x^2+16\right )}{{\mathrm {e}}^x+1} \]
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