Optimal. Leaf size=402 \[ -\frac {3 i x^2 \text {Li}_2\left (-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {3 i x^2 \text {Li}_2\left (i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {12 i x \text {Li}_3\left (-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {12 i x \text {Li}_3\left (i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {24 i \text {Li}_2\left (-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {24 i \text {Li}_2\left (i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {24 i \text {Li}_4\left (-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {24 i \text {Li}_4\left (i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {x^3 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {x^3 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a \cosh (x)+a}}+\frac {3 x^2}{a \sqrt {a \cosh (x)+a}}-\frac {24 x \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}} \]
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Rubi [A] time = 0.26, antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {3319, 4186, 4180, 2279, 2391, 2531, 6609, 2282, 6589} \[ -\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {12 i x \cosh \left (\frac {x}{2}\right ) \text {PolyLog}\left (3,-i e^{x/2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {12 i x \cosh \left (\frac {x}{2}\right ) \text {PolyLog}\left (3,i e^{x/2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {24 i \cosh \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {24 i \cosh \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {24 i \cosh \left (\frac {x}{2}\right ) \text {PolyLog}\left (4,-i e^{x/2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {24 i \cosh \left (\frac {x}{2}\right ) \text {PolyLog}\left (4,i e^{x/2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {3 x^2}{a \sqrt {a \cosh (x)+a}}+\frac {x^3 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {x^3 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a \cosh (x)+a}}-\frac {24 x \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 3319
Rule 4180
Rule 4186
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {x^3}{(a+a \cosh (x))^{3/2}} \, dx &=\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 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \tan ^{-1}\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 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \tan ^{-1}\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 ) \text {Li}_2\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (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 \text {Li}_2\left (-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 \text {Li}_2\left (i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}}+\frac {\left (24 i \cosh \left (\frac {x}{2}\right )\right ) \operatorname {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 ) \operatorname {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 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \tan ^{-1}\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 ) \text {Li}_2\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {24 i \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {12 i x \cosh \left (\frac {x}{2}\right ) \text {Li}_3\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {12 i x \cosh \left (\frac {x}{2}\right ) \text {Li}_3\left (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 \text {Li}_3\left (-i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}}+\frac {\left (12 i \cosh \left (\frac {x}{2}\right )\right ) \int \text {Li}_3\left (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 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \tan ^{-1}\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 ) \text {Li}_2\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {24 i \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {12 i x \cosh \left (\frac {x}{2}\right ) \text {Li}_3\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {12 i x \cosh \left (\frac {x}{2}\right ) \text {Li}_3\left (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 ) \operatorname {Subst}\left (\int \frac {\text {Li}_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 ) \operatorname {Subst}\left (\int \frac {\text {Li}_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 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^3 \tan ^{-1}\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 ) \text {Li}_2\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {24 i \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {3 i x^2 \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {12 i x \cosh \left (\frac {x}{2}\right ) \text {Li}_3\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {12 i x \cosh \left (\frac {x}{2}\right ) \text {Li}_3\left (i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {24 i \cosh \left (\frac {x}{2}\right ) \text {Li}_4\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {24 i \cosh \left (\frac {x}{2}\right ) \text {Li}_4\left (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*}
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Mathematica [A] time = 2.97, size = 716, normalized size = 1.78 \[ -\frac {i \cosh \left (\frac {x}{2}\right ) \left (48 x^2 \text {Li}_2\left (-i e^{x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )-48 \left (-x^2-2 i \pi x+\pi ^2+8\right ) \text {Li}_2\left (-i e^{-x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )+96 i \pi x \text {Li}_2\left (i e^{x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )+192 x \text {Li}_3\left (-i e^{-x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )-192 x \text {Li}_3\left (-i e^{x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )+384 \text {Li}_2\left (i e^{-x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )-48 \pi ^2 \text {Li}_2\left (i e^{x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )+192 i \pi \text {Li}_3\left (-i e^{-x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )-192 i \pi \text {Li}_3\left (i e^{x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )+384 \text {Li}_4\left (-i e^{-x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )+384 \text {Li}_4\left (-i e^{x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )+x^4 \left (-\cosh ^2\left (\frac {x}{2}\right )\right )+8 i x^3 \sinh \left (\frac {x}{2}\right )-4 i \pi x^3 \cosh ^2\left (\frac {x}{2}\right )-8 x^3 \log \left (1+i e^{-x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )+8 x^3 \log \left (1+i e^{x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )+6 \pi ^2 x^2 \cosh ^2\left (\frac {x}{2}\right )+48 i x^2 \cosh \left (\frac {x}{2}\right )-24 i \pi x^2 \log \left (1+i e^{-x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )+24 i \pi x^2 \log \left (1-i e^{x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )+4 i \pi ^3 x \cosh ^2\left (\frac {x}{2}\right )+7 \pi ^4 \cosh ^2\left (\frac {x}{2}\right )-192 x \log \left (1-i e^{-x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )+24 \pi ^2 x \log \left (1+i e^{-x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )+192 x \log \left (1+i e^{-x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )-24 \pi ^2 x \log \left (1-i e^{x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )+8 i \pi ^3 \log \left (1+i e^{-x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )-8 i \pi ^3 \log \left (1+i e^{x/2}\right ) \cosh ^2\left (\frac {x}{2}\right )+8 i \pi ^3 \cosh ^2\left (\frac {x}{2}\right ) \log \left (\tan \left (\frac {1}{4} (\pi +i x)\right )\right )\right )}{8 (a (\cosh (x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \cosh \relax (x) + a} x^{3}}{a^{2} \cosh \relax (x)^{2} + 2 \, a^{2} \cosh \relax (x) + a^{2}}, x\right ) \]
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 \[ \int \frac {x^{3}}{{\left (a \cosh \relax (x) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^{3}}{\left (a +a \cosh \relax (x )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \frac {8}{27} \, \sqrt {2} {\left (\frac {3 \, e^{\left (\frac {5}{2} \, x\right )} + 8 \, e^{\left (\frac {3}{2} \, x\right )} - 3 \, e^{\left (\frac {1}{2} \, x\right )}}{a^{\frac {3}{2}} e^{\left (3 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{x} + a^{\frac {3}{2}}} + \frac {3 \, \arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{a^{\frac {3}{2}}}\right )} + 36 \, \sqrt {2} \int \frac {x^{3} e^{\left (\frac {3}{2} \, x\right )}}{9 \, {\left (a^{\frac {3}{2}} e^{\left (4 \, x\right )} + 4 \, a^{\frac {3}{2}} e^{\left (3 \, x\right )} + 6 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + 4 \, a^{\frac {3}{2}} e^{x} + a^{\frac {3}{2}}\right )}}\,{d x} + 72 \, \sqrt {2} \int \frac {x^{2} e^{\left (\frac {3}{2} \, x\right )}}{9 \, {\left (a^{\frac {3}{2}} e^{\left (4 \, x\right )} + 4 \, a^{\frac {3}{2}} e^{\left (3 \, x\right )} + 6 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + 4 \, a^{\frac {3}{2}} e^{x} + a^{\frac {3}{2}}\right )}}\,{d x} + 96 \, \sqrt {2} \int \frac {x e^{\left (\frac {3}{2} \, x\right )}}{9 \, {\left (a^{\frac {3}{2}} e^{\left (4 \, x\right )} + 4 \, a^{\frac {3}{2}} e^{\left (3 \, x\right )} + 6 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + 4 \, a^{\frac {3}{2}} e^{x} + a^{\frac {3}{2}}\right )}}\,{d x} - \frac {4 \, {\left (9 \, \sqrt {2} \sqrt {a} x^{3} + 18 \, \sqrt {2} \sqrt {a} x^{2} + 24 \, \sqrt {2} \sqrt {a} x + 16 \, \sqrt {2} \sqrt {a}\right )} e^{\left (\frac {3}{2} \, x\right )}}{27 \, {\left (a^{2} e^{\left (3 \, x\right )} + 3 \, a^{2} e^{\left (2 \, x\right )} + 3 \, a^{2} e^{x} + a^{2}\right )}} \]
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 \[ \int \frac {x^3}{{\left (a+a\,\mathrm {cosh}\relax (x)\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{3}}{\left (a \left (\cosh {\relax (x )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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