Optimal. Leaf size=140 \[ -\frac {i \text {Li}_2\left (-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {i \text {Li}_2\left (i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {1}{a \sqrt {a \cosh (x)+a}}+\frac {x \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {x \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a \cosh (x)+a}} \]
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Rubi [A] time = 0.10, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3319, 4185, 4180, 2279, 2391} \[ -\frac {i \cosh \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {i \cosh \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {1}{a \sqrt {a \cosh (x)+a}}+\frac {x \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {x \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a \cosh (x)+a}} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 3319
Rule 4180
Rule 4185
Rubi steps
\begin {align*} \int \frac {x}{(a+a \cosh (x))^{3/2}} \, dx &=\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 \tan ^{-1}\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 \tan ^{-1}\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 ) \operatorname {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 ) \operatorname {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 \tan ^{-1}\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 ) \text {Li}_2\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {i \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (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*}
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Mathematica [A] time = 0.11, size = 137, normalized size = 0.98 \[ \frac {2 \cosh ^3\left (\frac {x}{2}\right ) \left (-i \left (\text {Li}_2\left (-i e^{-x/2}\right )-\text {Li}_2\left (i e^{-x/2}\right )\right )-\frac {1}{2} i x \left (\log \left (1-i e^{-x/2}\right )-\log \left (1+i e^{-x/2}\right )\right )\right )}{(a (\cosh (x)+1))^{3/2}}+\frac {2 \cosh ^2\left (\frac {x}{2}\right )}{(a (\cosh (x)+1))^{3/2}}+\frac {x \sinh \left (\frac {x}{2}\right ) \cosh \left (\frac {x}{2}\right )}{(a (\cosh (x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \cosh \relax (x) + a} x}{a^{2} \cosh \relax (x)^{2} + 2 \, a^{2} \cosh \relax (x) + a^{2}}, x\right ) \]
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 \[ \int \frac {x}{{\left (a \cosh \relax (x) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x}{\left (a +a \cosh \relax (x )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \frac {1}{9} \, \sqrt {2} {\left (\frac {3 \, e^{\left (\frac {5}{2} \, x\right )} + 8 \, e^{\left (\frac {3}{2} \, x\right )} - 3 \, e^{\left (\frac {1}{2} \, x\right )}}{a^{\frac {3}{2}} e^{\left (3 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{x} + a^{\frac {3}{2}}} + \frac {3 \, \arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{a^{\frac {3}{2}}}\right )} + 12 \, \sqrt {2} \int \frac {x e^{\left (\frac {3}{2} \, x\right )}}{3 \, {\left (a^{\frac {3}{2}} e^{\left (4 \, x\right )} + 4 \, a^{\frac {3}{2}} e^{\left (3 \, x\right )} + 6 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + 4 \, a^{\frac {3}{2}} e^{x} + a^{\frac {3}{2}}\right )}}\,{d x} - \frac {4 \, {\left (3 \, \sqrt {2} \sqrt {a} x + 2 \, \sqrt {2} \sqrt {a}\right )} e^{\left (\frac {3}{2} \, x\right )}}{9 \, {\left (a^{2} e^{\left (3 \, x\right )} + 3 \, a^{2} e^{\left (2 \, x\right )} + 3 \, a^{2} e^{x} + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{{\left (a+a\,\mathrm {cosh}\relax (x)\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x}{\left (a \left (\cosh {\relax (x )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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