Optimal. Leaf size=39 \[ -\frac{1}{2} \text{PolyLog}\left (2,e^{2 x}\right )+x \log (a \sinh (x))+\frac{x^2}{2}-x \log \left (1-e^{2 x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0594839, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {2548, 3716, 2190, 2279, 2391} \[ -\frac{1}{2} \text{PolyLog}\left (2,e^{2 x}\right )+x \log (a \sinh (x))+\frac{x^2}{2}-x \log \left (1-e^{2 x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2548
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \log (a \sinh (x)) \, dx &=x \log (a \sinh (x))-\int x \coth (x) \, dx\\ &=\frac{x^2}{2}+x \log (a \sinh (x))+2 \int \frac{e^{2 x} x}{1-e^{2 x}} \, dx\\ &=\frac{x^2}{2}-x \log \left (1-e^{2 x}\right )+x \log (a \sinh (x))+\int \log \left (1-e^{2 x}\right ) \, dx\\ &=\frac{x^2}{2}-x \log \left (1-e^{2 x}\right )+x \log (a \sinh (x))+\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 x}\right )\\ &=\frac{x^2}{2}-x \log \left (1-e^{2 x}\right )+x \log (a \sinh (x))-\frac{\text{Li}_2\left (e^{2 x}\right )}{2}\\ \end{align*}
Mathematica [A] time = 0.0203845, size = 36, normalized size = 0.92 \[ \frac{1}{2} \left (\text{PolyLog}\left (2,e^{-2 x}\right )-x \left (-2 \log (a \sinh (x))+x+2 \log \left (1-e^{-2 x}\right )\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.126, size = 295, normalized size = 7.6 \begin{align*} -x\ln \left ({{\rm e}^{x}} \right ) +{\frac{i}{2}}\pi \,{\it csgn} \left ( i \left ({{\rm e}^{2\,x}}-1 \right ) \right ) \left ({\it csgn} \left ( i{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}-1 \right ) \right ) \right ) ^{2}x-{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ia \left ({{\rm e}^{2\,x}}-1 \right ){{\rm e}^{-x}} \right ) \right ) ^{3}x-{\frac{i}{2}}\pi \,{\it csgn} \left ( i \left ({{\rm e}^{2\,x}}-1 \right ) \right ){\it csgn} \left ( i{{\rm e}^{-x}} \right ){\it csgn} \left ( i{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}-1 \right ) \right ) x+{\frac{i}{2}}\pi \,{\it csgn} \left ( i{{\rm e}^{-x}} \right ) \left ({\it csgn} \left ( i{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}-1 \right ) \right ) \right ) ^{2}x+{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ia \left ({{\rm e}^{2\,x}}-1 \right ){{\rm e}^{-x}} \right ) \right ) ^{2}{\it csgn} \left ( ia \right ) x-{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( i{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}-1 \right ) \right ) \right ) ^{3}x+\ln \left ( a \right ) x-\ln \left ( 2 \right ) x+{\frac{{x}^{2}}{2}}+{\frac{i}{2}}\pi \,{\it csgn} \left ( i{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}-1 \right ) \right ) \left ({\it csgn} \left ( ia \left ({{\rm e}^{2\,x}}-1 \right ){{\rm e}^{-x}} \right ) \right ) ^{2}x-{\frac{i}{2}}\pi \,{\it csgn} \left ( i{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}-1 \right ) \right ){\it csgn} \left ( ia \left ({{\rm e}^{2\,x}}-1 \right ){{\rm e}^{-x}} \right ){\it csgn} \left ( ia \right ) x+\ln \left ({{\rm e}^{x}} \right ) \ln \left ({{\rm e}^{2\,x}}-1 \right ) +{\it dilog} \left ({{\rm e}^{x}} \right ) -{\it dilog} \left ({{\rm e}^{x}}+1 \right ) -\ln \left ({{\rm e}^{x}} \right ) \ln \left ({{\rm e}^{x}}+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.2852, size = 58, normalized size = 1.49 \begin{align*} \frac{1}{2} \, x^{2} + x \log \left (a \sinh \left (x\right )\right ) - x \log \left (e^{x} + 1\right ) - x \log \left (-e^{x} + 1\right ) -{\rm Li}_2\left (-e^{x}\right ) -{\rm Li}_2\left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.02053, size = 197, normalized size = 5.05 \begin{align*} \frac{1}{2} \, x^{2} + x \log \left (a \sinh \left (x\right )\right ) - x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) -{\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) -{\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \log{\left (a \sinh{\left (x \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \log \left (a \sinh \left (x\right )\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]