Integrand size = 12, antiderivative size = 32 \[ \int x \sqrt {a+a \cosh (x)} \, dx=-4 \sqrt {a+a \cosh (x)}+2 x \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3400, 3377, 2718} \[ \int x \sqrt {a+a \cosh (x)} \, dx=2 x \tanh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-4 \sqrt {a \cosh (x)+a} \]
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Rule 2718
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 \cosh \left (\frac {x}{2}\right ) \, dx \\ & = 2 x \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )-\left (2 \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \sinh \left (\frac {x}{2}\right ) \, dx \\ & = -4 \sqrt {a+a \cosh (x)}+2 x \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.69 \[ \int x \sqrt {a+a \cosh (x)} \, dx=2 \sqrt {a (1+\cosh (x))} \left (-2+x \tanh \left (\frac {x}{2}\right )\right ) \]
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Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19
method | result | size |
risch | \(\frac {\sqrt {2}\, \sqrt {a \left ({\mathrm e}^{x}+1\right )^{2} {\mathrm e}^{-x}}\, \left (x \,{\mathrm e}^{x}-x -2 \,{\mathrm e}^{x}-2\right )}{{\mathrm e}^{x}+1}\) | \(38\) |
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Exception generated. \[ \int x \sqrt {a+a \cosh (x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x \sqrt {a+a \cosh (x)} \, dx=\int x \sqrt {a \left (\cosh {\left (x \right )} + 1\right )}\, dx \]
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none
Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int x \sqrt {a+a \cosh (x)} \, dx=-{\left (\sqrt {2} \sqrt {a} x - {\left (\sqrt {2} \sqrt {a} x - 2 \, \sqrt {2} \sqrt {a}\right )} e^{x} + 2 \, \sqrt {2} \sqrt {a}\right )} e^{\left (-\frac {1}{2} \, x\right )} \]
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\[ \int x \sqrt {a+a \cosh (x)} \, dx=\int { \sqrt {a \cosh \left (x\right ) + a} x \,d x } \]
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Time = 1.70 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int x \sqrt {a+a \cosh (x)} \, dx=-\frac {\sqrt {a+a\,\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}\,\left (2\,x+4\,{\mathrm {e}}^x-2\,x\,{\mathrm {e}}^x+4\right )}{{\mathrm {e}}^x+1} \]
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