Integrand size = 14, antiderivative size = 402 \[ \int \frac {x^3}{(a+a \cosh (x))^{3/2}} \, dx=\frac {3 x^2}{a \sqrt {a+a \cosh (x)}}-\frac {24 x \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {12 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {12 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (4,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (4,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}} \]
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Time = 0.23 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {3400, 4271, 4265, 2317, 2438, 2611, 6744, 2320, 6724} \[ \int \frac {x^3}{(a+a \cosh (x))^{3/2}} \, dx=\frac {x^3 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {24 x \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {3 i x^2 \operatorname {PolyLog}\left (2,i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {12 i x \operatorname {PolyLog}\left (3,-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {12 i x \operatorname {PolyLog}\left (3,i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {24 i \operatorname {PolyLog}\left (2,-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {24 i \operatorname {PolyLog}\left (2,i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {24 i \operatorname {PolyLog}\left (4,-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {24 i \operatorname {PolyLog}\left (4,i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {x^3 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a \cosh (x)+a}}+\frac {3 x^2}{a \sqrt {a \cosh (x)+a}} \]
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Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3400
Rule 4265
Rule 4271
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \frac {\cosh \left (\frac {x}{2}\right ) \int x^3 \text {sech}^3\left (\frac {x}{2}\right ) \, dx}{2 a \sqrt {a+a \cosh (x)}} \\ & = \frac {3 x^2}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}}+\frac {\cosh \left (\frac {x}{2}\right ) \int x^3 \text {sech}\left (\frac {x}{2}\right ) \, dx}{4 a \sqrt {a+a \cosh (x)}}-\frac {\left (6 \cosh \left (\frac {x}{2}\right )\right ) \int x \text {sech}\left (\frac {x}{2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}} \\ & = \frac {3 x^2}{a \sqrt {a+a \cosh (x)}}-\frac {24 x \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}}-\frac {\left (3 i \cosh \left (\frac {x}{2}\right )\right ) \int x^2 \log \left (1-i e^{x/2}\right ) \, dx}{2 a \sqrt {a+a \cosh (x)}}+\frac {\left (3 i \cosh \left (\frac {x}{2}\right )\right ) \int x^2 \log \left (1+i e^{x/2}\right ) \, dx}{2 a \sqrt {a+a \cosh (x)}}+\frac {\left (12 i \cosh \left (\frac {x}{2}\right )\right ) \int \log \left (1-i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}}-\frac {\left (12 i \cosh \left (\frac {x}{2}\right )\right ) \int \log \left (1+i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}} \\ & = \frac {3 x^2}{a \sqrt {a+a \cosh (x)}}-\frac {24 x \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}}+\frac {\left (6 i \cosh \left (\frac {x}{2}\right )\right ) \int x \operatorname {PolyLog}\left (2,-i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}}-\frac {\left (6 i \cosh \left (\frac {x}{2}\right )\right ) \int x \operatorname {PolyLog}\left (2,i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}}+\frac {\left (24 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 (24 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 {3 x^2}{a \sqrt {a+a \cosh (x)}}-\frac {24 x \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {12 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {12 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}}-\frac {\left (12 i \cosh \left (\frac {x}{2}\right )\right ) \int \operatorname {PolyLog}\left (3,-i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}}+\frac {\left (12 i \cosh \left (\frac {x}{2}\right )\right ) \int \operatorname {PolyLog}\left (3,i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}} \\ & = \frac {3 x^2}{a \sqrt {a+a \cosh (x)}}-\frac {24 x \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {12 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {12 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}}-\frac {\left (24 i \cosh \left (\frac {x}{2}\right )\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {\left (24 i \cosh \left (\frac {x}{2}\right )\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}} \\ & = \frac {3 x^2}{a \sqrt {a+a \cosh (x)}}-\frac {24 x \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {12 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {12 i x \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (3,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (4,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {24 i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (4,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}} \\ \end{align*}
Time = 0.68 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.58 \[ \int \frac {x^3}{(a+a \cosh (x))^{3/2}} \, dx=\frac {\cosh \left (\frac {x}{2}\right ) \left (6 x^2 \cosh \left (\frac {x}{2}\right )+8 i \cosh ^2\left (\frac {x}{2}\right ) \left (-3 x \log \left (1-i e^{x/2}\right )+\frac {1}{8} x^3 \log \left (1-i e^{x/2}\right )+3 x \log \left (1+i e^{x/2}\right )-\frac {1}{8} x^3 \log \left (1+i e^{x/2}\right )-\frac {3}{4} \left (-8+x^2\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )+\frac {3}{4} \left (-8+x^2\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )+3 x \operatorname {PolyLog}\left (3,-i e^{x/2}\right )-3 x \operatorname {PolyLog}\left (3,i e^{x/2}\right )-6 \operatorname {PolyLog}\left (4,-i e^{x/2}\right )+6 \operatorname {PolyLog}\left (4,i e^{x/2}\right )\right )+x^3 \sinh \left (\frac {x}{2}\right )\right )}{(a (1+\cosh (x)))^{3/2}} \]
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\[\int \frac {x^{3}}{\left (a +a \cosh \left (x \right )\right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {x^3}{(a+a \cosh (x))^{3/2}} \, dx=\int { \frac {x^{3}}{{\left (a \cosh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^3}{(a+a \cosh (x))^{3/2}} \, dx=\int \frac {x^{3}}{\left (a \left (\cosh {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^3}{(a+a \cosh (x))^{3/2}} \, dx=\int { \frac {x^{3}}{{\left (a \cosh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^3}{(a+a \cosh (x))^{3/2}} \, dx=\int { \frac {x^{3}}{{\left (a \cosh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^3}{(a+a \cosh (x))^{3/2}} \, dx=\int \frac {x^3}{{\left (a+a\,\mathrm {cosh}\left (x\right )\right )}^{3/2}} \,d x \]
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