Integrand size = 12, antiderivative size = 140 \[ \int \frac {x}{(a+a \cosh (x))^{3/2}} \, dx=\frac {1}{a \sqrt {a+a \cosh (x)}}+\frac {x \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}} \]
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Time = 0.10 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3400, 4270, 4265, 2317, 2438} \[ \int \frac {x}{(a+a \cosh (x))^{3/2}} \, dx=\frac {x \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {i \operatorname {PolyLog}\left (2,-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {i \operatorname {PolyLog}\left (2,i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {1}{a \sqrt {a \cosh (x)+a}}+\frac {x \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a \cosh (x)+a}} \]
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Rule 2317
Rule 2438
Rule 3400
Rule 4265
Rule 4270
Rubi steps \begin{align*} \text {integral}& = \frac {\cosh \left (\frac {x}{2}\right ) \int x \text {sech}^3\left (\frac {x}{2}\right ) \, dx}{2 a \sqrt {a+a \cosh (x)}} \\ & = \frac {1}{a \sqrt {a+a \cosh (x)}}+\frac {x \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}}+\frac {\cosh \left (\frac {x}{2}\right ) \int x \text {sech}\left (\frac {x}{2}\right ) \, dx}{4 a \sqrt {a+a \cosh (x)}} \\ & = \frac {1}{a \sqrt {a+a \cosh (x)}}+\frac {x \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}}-\frac {\left (i \cosh \left (\frac {x}{2}\right )\right ) \int \log \left (1-i e^{x/2}\right ) \, dx}{2 a \sqrt {a+a \cosh (x)}}+\frac {\left (i \cosh \left (\frac {x}{2}\right )\right ) \int \log \left (1+i e^{x/2}\right ) \, dx}{2 a \sqrt {a+a \cosh (x)}} \\ & = \frac {1}{a \sqrt {a+a \cosh (x)}}+\frac {x \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}}-\frac {\left (i \cosh \left (\frac {x}{2}\right )\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {\left (i \cosh \left (\frac {x}{2}\right )\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}} \\ & = \frac {1}{a \sqrt {a+a \cosh (x)}}+\frac {x \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.79 \[ \int \frac {x}{(a+a \cosh (x))^{3/2}} \, dx=\frac {\cosh \left (\frac {x}{2}\right ) \left (2 \cosh \left (\frac {x}{2}\right )+i \cosh ^2\left (\frac {x}{2}\right ) \left (x \left (\log \left (1-i e^{x/2}\right )-\log \left (1+i e^{x/2}\right )\right )-2 \operatorname {PolyLog}\left (2,-i e^{x/2}\right )+2 \operatorname {PolyLog}\left (2,i e^{x/2}\right )\right )+x \sinh \left (\frac {x}{2}\right )\right )}{(a (1+\cosh (x)))^{3/2}} \]
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\[\int \frac {x}{\left (a +a \cosh \left (x \right )\right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {x}{(a+a \cosh (x))^{3/2}} \, dx=\int { \frac {x}{{\left (a \cosh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x}{(a+a \cosh (x))^{3/2}} \, dx=\int \frac {x}{\left (a \left (\cosh {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x}{(a+a \cosh (x))^{3/2}} \, dx=\int { \frac {x}{{\left (a \cosh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x}{(a+a \cosh (x))^{3/2}} \, dx=\int { \frac {x}{{\left (a \cosh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x}{(a+a \cosh (x))^{3/2}} \, dx=\int \frac {x}{{\left (a+a\,\mathrm {cosh}\left (x\right )\right )}^{3/2}} \,d x \]
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