Integrand size = 14, antiderivative size = 248 \[ \int \frac {x^2}{(a+a \cosh (x))^{3/2}} \, dx=\frac {2 x}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 \arctan \left (\sinh \left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {2 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {2 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {4 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}} \]
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Time = 0.16 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3400, 4271, 3855, 4265, 2611, 2320, 6724} \[ \int \frac {x^2}{(a+a \cosh (x))^{3/2}} \, dx=\frac {x^2 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {4 \cosh \left (\frac {x}{2}\right ) \arctan \left (\sinh \left (\frac {x}{2}\right )\right )}{a \sqrt {a \cosh (x)+a}}-\frac {2 i x \operatorname {PolyLog}\left (2,-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {2 i x \operatorname {PolyLog}\left (2,i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {4 i \operatorname {PolyLog}\left (3,-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {4 i \operatorname {PolyLog}\left (3,i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {x^2 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a \cosh (x)+a}}+\frac {2 x}{a \sqrt {a \cosh (x)+a}} \]
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Rule 2320
Rule 2611
Rule 3400
Rule 3855
Rule 4265
Rule 4271
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\cosh \left (\frac {x}{2}\right ) \int x^2 \text {sech}^3\left (\frac {x}{2}\right ) \, dx}{2 a \sqrt {a+a \cosh (x)}} \\ & = \frac {2 x}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}}+\frac {\cosh \left (\frac {x}{2}\right ) \int x^2 \text {sech}\left (\frac {x}{2}\right ) \, dx}{4 a \sqrt {a+a \cosh (x)}}-\frac {\left (2 \cosh \left (\frac {x}{2}\right )\right ) \int \text {sech}\left (\frac {x}{2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}} \\ & = \frac {2 x}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 \arctan \left (\sinh \left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}}-\frac {\left (i \cosh \left (\frac {x}{2}\right )\right ) \int x \log \left (1-i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}}+\frac {\left (i \cosh \left (\frac {x}{2}\right )\right ) \int x \log \left (1+i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}} \\ & = \frac {2 x}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 \arctan \left (\sinh \left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {2 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {2 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}}+\frac {\left (2 i \cosh \left (\frac {x}{2}\right )\right ) \int \operatorname {PolyLog}\left (2,-i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}}-\frac {\left (2 i \cosh \left (\frac {x}{2}\right )\right ) \int \operatorname {PolyLog}\left (2,i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}} \\ & = \frac {2 x}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 \arctan \left (\sinh \left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {2 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {2 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}}+\frac {\left (4 i \cosh \left (\frac {x}{2}\right )\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {\left (4 i \cosh \left (\frac {x}{2}\right )\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}} \\ & = \frac {2 x}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 \arctan \left (\sinh \left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {2 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {2 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {4 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.65 \[ \int \frac {x^2}{(a+a \cosh (x))^{3/2}} \, dx=\frac {\cosh \left (\frac {x}{2}\right ) \left (4 x \cosh \left (\frac {x}{2}\right )+i \cosh ^2\left (\frac {x}{2}\right ) \left (16 i \arctan \left (e^{x/2}\right )+x^2 \log \left (1-i e^{x/2}\right )-x^2 \log \left (1+i e^{x/2}\right )-4 x \operatorname {PolyLog}\left (2,-i e^{x/2}\right )+4 x \operatorname {PolyLog}\left (2,i e^{x/2}\right )+8 \operatorname {PolyLog}\left (3,-i e^{x/2}\right )-8 \operatorname {PolyLog}\left (3,i e^{x/2}\right )\right )+x^2 \sinh \left (\frac {x}{2}\right )\right )}{(a (1+\cosh (x)))^{3/2}} \]
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\[\int \frac {x^{2}}{\left (a +a \cosh \left (x \right )\right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {x^2}{(a+a \cosh (x))^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (a \cosh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^2}{(a+a \cosh (x))^{3/2}} \, dx=\int \frac {x^{2}}{\left (a \left (\cosh {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^2}{(a+a \cosh (x))^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (a \cosh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^2}{(a+a \cosh (x))^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (a \cosh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{(a+a \cosh (x))^{3/2}} \, dx=\int \frac {x^2}{{\left (a+a\,\mathrm {cosh}\left (x\right )\right )}^{3/2}} \,d x \]
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