Optimal. Leaf size=157 \[ -\frac {4 i \text {Li}_2\left (-i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^2 \sqrt {a \cosh (c+d x)+a}}+\frac {4 i \text {Li}_2\left (i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^2 \sqrt {a \cosh (c+d x)+a}}+\frac {4 x \tan ^{-1}\left (e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a \cosh (c+d x)+a}} \]
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Rubi [A] time = 0.08, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3319, 4180, 2279, 2391} \[ -\frac {4 i \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \text {PolyLog}\left (2,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a \cosh (c+d x)+a}}+\frac {4 i \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \text {PolyLog}\left (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a \cosh (c+d x)+a}}+\frac {4 x \tan ^{-1}\left (e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a \cosh (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 3319
Rule 4180
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a+a \cosh (c+d x)}} \, dx &=\frac {\sin \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right ) \int x \csc \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right ) \, dx}{\sqrt {a+a \cosh (c+d x)}}\\ &=\frac {4 x \tan ^{-1}\left (e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cosh (c+d x)}}-\frac {\left (2 i \sin \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \int \log \left (1-i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \, dx}{d \sqrt {a+a \cosh (c+d x)}}+\frac {\left (2 i \sin \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \int \log \left (1+i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \, dx}{d \sqrt {a+a \cosh (c+d x)}}\\ &=\frac {4 x \tan ^{-1}\left (e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cosh (c+d x)}}-\frac {\left (4 i \sin \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}}+\frac {\left (4 i \sin \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}}\\ &=\frac {4 x \tan ^{-1}\left (e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cosh (c+d x)}}-\frac {4 i \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \text {Li}_2\left (-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}}+\frac {4 i \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \text {Li}_2\left (i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.71, size = 117, normalized size = 0.75 \[ \frac {4 \cosh \left (\frac {1}{2} (c+d x)\right ) \left (-i \text {Li}_2\left (-i \left (\cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )+i \text {Li}_2\left (i \left (\cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )+d x \tan ^{-1}\left (\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right )\right )\right )}{d^2 \sqrt {a (\cosh (c+d x)+1)}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{\sqrt {a \cosh \left (d x + c\right ) + a}}, x\right ) \]
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 \[ \int \frac {x}{\sqrt {a \cosh \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {a +a \cosh \left (d x +c \right )}}\, dx \]
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 \[ 2 \, \sqrt {2} d \int \frac {x e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{\sqrt {a} d e^{\left (2 \, d x + 2 \, c\right )} + 2 \, \sqrt {a} d e^{\left (d x + c\right )} + \sqrt {a} d}\,{d x} + 4 \, \sqrt {2} {\left (\frac {e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{{\left (\sqrt {a} d e^{\left (d x + c\right )} + \sqrt {a} d\right )} d} + \frac {\arctan \left (e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}\right )}{\sqrt {a} d^{2}}\right )} - \frac {2 \, {\left (\sqrt {2} \sqrt {a} d x e^{\left (\frac {1}{2} \, c\right )} + 2 \, \sqrt {2} \sqrt {a} e^{\left (\frac {1}{2} \, c\right )}\right )} e^{\left (\frac {1}{2} \, d x\right )}}{a d^{2} e^{\left (d x + c\right )} + a d^{2}} \]
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 \[ \int \frac {x}{\sqrt {a+a\,\mathrm {cosh}\left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {x}{\sqrt {a \left (\cosh {\left (c + d x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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