Integrand size = 16, antiderivative size = 157 \[ \int \frac {x}{\sqrt {a+a \cosh (c+d x)}} \, dx=\frac {4 x \arctan \left (e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cosh (c+d x)}}-\frac {4 i \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}}+\frac {4 i \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}} \]
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Time = 0.08 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3400, 4265, 2317, 2438} \[ \int \frac {x}{\sqrt {a+a \cosh (c+d x)}} \, dx=\frac {4 x \arctan \left (e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a \cosh (c+d x)+a}}-\frac {4 i \operatorname {PolyLog}\left (2,-i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^2 \sqrt {a \cosh (c+d x)+a}}+\frac {4 i \operatorname {PolyLog}\left (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^2 \sqrt {a \cosh (c+d x)+a}} \]
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Rule 2317
Rule 2438
Rule 3400
Rule 4265
Rubi steps \begin{align*} \text {integral}& = \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 \arctan \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 \arctan \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 ) \text {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 ) \text {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 \arctan \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 ) \operatorname {PolyLog}\left (2,-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 ) \operatorname {PolyLog}\left (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.89 \[ \int \frac {x}{\sqrt {a+a \cosh (c+d x)}} \, dx=\frac {4 i \cosh \left (\frac {1}{2} (c+d x)\right ) \left (i c \arctan \left (e^{\frac {1}{2} (c+d x)}\right )+\frac {1}{2} (c+d x) \log \left (1-i e^{\frac {1}{2} (c+d x)}\right )-\frac {1}{2} (c+d x) \log \left (1+i e^{\frac {1}{2} (c+d x)}\right )-\operatorname {PolyLog}\left (2,-i e^{\frac {1}{2} (c+d x)}\right )+\operatorname {PolyLog}\left (2,i e^{\frac {1}{2} (c+d x)}\right )\right )}{d^2 \sqrt {a (1+\cosh (c+d x))}} \]
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\[\int \frac {x}{\sqrt {a +a \cosh \left (d x +c \right )}}d x\]
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\[ \int \frac {x}{\sqrt {a+a \cosh (c+d x)}} \, dx=\int { \frac {x}{\sqrt {a \cosh \left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {x}{\sqrt {a+a \cosh (c+d x)}} \, dx=\int \frac {x}{\sqrt {a \left (\cosh {\left (c + d x \right )} + 1\right )}}\, dx \]
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\[ \int \frac {x}{\sqrt {a+a \cosh (c+d x)}} \, dx=\int { \frac {x}{\sqrt {a \cosh \left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {x}{\sqrt {a+a \cosh (c+d x)}} \, dx=\int { \frac {x}{\sqrt {a \cosh \left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {x}{\sqrt {a+a \cosh (c+d x)}} \, dx=\int \frac {x}{\sqrt {a+a\,\mathrm {cosh}\left (c+d\,x\right )}} \,d x \]
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