Optimal. Leaf size=248 \[ -\frac {2 i x \text {Li}_2\left (-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {2 i x \text {Li}_2\left (i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {4 i \text {Li}_3\left (-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {4 i \text {Li}_3\left (i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {x^2 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {x^2 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a \cosh (x)+a}}+\frac {2 x}{a \sqrt {a \cosh (x)+a}}-\frac {4 \cosh \left (\frac {x}{2}\right ) \tan ^{-1}\left (\sinh \left (\frac {x}{2}\right )\right )}{a \sqrt {a \cosh (x)+a}} \]
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Rubi [A] time = 0.19, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3319, 4186, 3770, 4180, 2531, 2282, 6589} \[ -\frac {2 i x \cosh \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {2 i x \cosh \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {4 i \cosh \left (\frac {x}{2}\right ) \text {PolyLog}\left (3,-i e^{x/2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {4 i \cosh \left (\frac {x}{2}\right ) \text {PolyLog}\left (3,i e^{x/2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {x^2 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {x^2 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a \cosh (x)+a}}+\frac {2 x}{a \sqrt {a \cosh (x)+a}}-\frac {4 \cosh \left (\frac {x}{2}\right ) \tan ^{-1}\left (\sinh \left (\frac {x}{2}\right )\right )}{a \sqrt {a \cosh (x)+a}} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 3319
Rule 3770
Rule 4180
Rule 4186
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^2}{(a+a \cosh (x))^{3/2}} \, dx &=\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 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 \tan ^{-1}\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 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 \tan ^{-1}\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 ) \text {Li}_2\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {2 i x \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (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 \text {Li}_2\left (-i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}}-\frac {\left (2 i \cosh \left (\frac {x}{2}\right )\right ) \int \text {Li}_2\left (i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}}\\ &=\frac {2 x}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 \tan ^{-1}\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 ) \text {Li}_2\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {2 i x \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (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 ) \operatorname {Subst}\left (\int \frac {\text {Li}_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 ) \operatorname {Subst}\left (\int \frac {\text {Li}_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 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 \tan ^{-1}\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 ) \text {Li}_2\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {2 i x \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {4 i \cosh \left (\frac {x}{2}\right ) \text {Li}_3\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 i \cosh \left (\frac {x}{2}\right ) \text {Li}_3\left (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*}
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Mathematica [A] time = 0.91, size = 214, normalized size = 0.86 \[ \frac {\cosh \left (\frac {x}{2}\right ) \left (-4 i x \cosh ^2\left (\frac {x}{2}\right ) \text {Li}_2\left (-i \left (\cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )\right )\right )+4 i x \cosh ^2\left (\frac {x}{2}\right ) \text {Li}_2\left (i \left (\cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )\right )\right )+8 i \cosh ^2\left (\frac {x}{2}\right ) \text {Li}_3\left (-i \left (\cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )\right )\right )-8 i \cosh ^2\left (\frac {x}{2}\right ) \text {Li}_3\left (i \left (\cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )\right )\right )+x^2 \sinh \left (\frac {x}{2}\right )+2 x^2 \cosh ^2\left (\frac {x}{2}\right ) \tan ^{-1}\left (\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right )\right )+4 x \cosh \left (\frac {x}{2}\right )-16 \cosh ^2\left (\frac {x}{2}\right ) \tan ^{-1}\left (\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right )\right )\right )}{(a (\cosh (x)+1))^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \cosh \relax (x) + a} x^{2}}{a^{2} \cosh \relax (x)^{2} + 2 \, a^{2} \cosh \relax (x) + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (a \cosh \relax (x) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (a +a \cosh \relax (x )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {4}{27} \, \sqrt {2} {\left (\frac {3 \, e^{\left (\frac {5}{2} \, x\right )} + 8 \, e^{\left (\frac {3}{2} \, x\right )} - 3 \, e^{\left (\frac {1}{2} \, x\right )}}{a^{\frac {3}{2}} e^{\left (3 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{x} + a^{\frac {3}{2}}} + \frac {3 \, \arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{a^{\frac {3}{2}}}\right )} + 36 \, \sqrt {2} \int \frac {x^{2} e^{\left (\frac {3}{2} \, x\right )}}{9 \, {\left (a^{\frac {3}{2}} e^{\left (4 \, x\right )} + 4 \, a^{\frac {3}{2}} e^{\left (3 \, x\right )} + 6 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + 4 \, a^{\frac {3}{2}} e^{x} + a^{\frac {3}{2}}\right )}}\,{d x} + 48 \, \sqrt {2} \int \frac {x e^{\left (\frac {3}{2} \, x\right )}}{9 \, {\left (a^{\frac {3}{2}} e^{\left (4 \, x\right )} + 4 \, a^{\frac {3}{2}} e^{\left (3 \, x\right )} + 6 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + 4 \, a^{\frac {3}{2}} e^{x} + a^{\frac {3}{2}}\right )}}\,{d x} - \frac {4 \, {\left (9 \, \sqrt {2} x^{2} + 12 \, \sqrt {2} x + 8 \, \sqrt {2}\right )} e^{\left (\frac {3}{2} \, x\right )}}{27 \, {\left (a^{\frac {3}{2}} e^{\left (3 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{x} + a^{\frac {3}{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2}{{\left (a+a\,\mathrm {cosh}\relax (x)\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (a \left (\cosh {\relax (x )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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