Optimal. Leaf size=27 \[ -\frac {1}{2} \log (\coth (x)) \sqrt {\log ^2(\coth (x))+1}-\frac {1}{2} \sinh ^{-1}(\log (\coth (x))) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6696, 195, 215} \[ -\frac {1}{2} \log (\coth (x)) \sqrt {\log ^2(\coth (x))+1}-\frac {1}{2} \sinh ^{-1}(\log (\coth (x))) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 215
Rule 6696
Rubi steps
\begin {align*} \int \text {csch}(x) \sqrt {1+\log ^2(\coth (x))} \text {sech}(x) \, dx &=-\operatorname {Subst}\left (\int \sqrt {1+x^2} \, dx,x,\log (\coth (x))\right )\\ &=-\frac {1}{2} \log (\coth (x)) \sqrt {1+\log ^2(\coth (x))}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\log (\coth (x))\right )\\ &=-\frac {1}{2} \sinh ^{-1}(\log (\coth (x)))-\frac {1}{2} \log (\coth (x)) \sqrt {1+\log ^2(\coth (x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 27, normalized size = 1.00 \[ -\frac {1}{2} \log (\coth (x)) \sqrt {\log ^2(\coth (x))+1}-\frac {1}{2} \sinh ^{-1}(\log (\coth (x))) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.46, size = 53, normalized size = 1.96 \[ -\frac {1}{2} \, \sqrt {\log \left (\frac {\cosh \relax (x)}{\sinh \relax (x)}\right )^{2} + 1} \log \left (\frac {\cosh \relax (x)}{\sinh \relax (x)}\right ) + \frac {1}{2} \, \log \left (\sqrt {\log \left (\frac {\cosh \relax (x)}{\sinh \relax (x)}\right )^{2} + 1} - \log \left (\frac {\cosh \relax (x)}{\sinh \relax (x)}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\log \left (\coth \relax (x)\right )^{2} + 1} \operatorname {csch}\relax (x) \operatorname {sech}\relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.24, size = 22, normalized size = 0.81 \[ -\frac {\arcsinh \left (\ln \left (\coth \relax (x )\right )\right )}{2}-\frac {\ln \left (\coth \relax (x )\right ) \sqrt {1+\ln \left (\coth \relax (x )\right )^{2}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\log \left (\coth \relax (x)\right )^{2} + 1} \operatorname {csch}\relax (x) \operatorname {sech}\relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.88, size = 21, normalized size = 0.78 \[ -\frac {\mathrm {asinh}\left (\ln \left (\mathrm {coth}\relax (x)\right )\right )}{2}-\frac {\ln \left (\mathrm {coth}\relax (x)\right )\,\sqrt {{\ln \left (\mathrm {coth}\relax (x)\right )}^2+1}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\log {\left (\coth {\relax (x )} \right )}^{2} + 1} \operatorname {csch}{\relax (x )} \operatorname {sech}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________