Optimal. Leaf size=21 \[ b x-\log (x) \left (b x-\coth ^{-1}(\coth (a+b x))\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2158, 29} \[ b x-\log (x) \left (b x-\coth ^{-1}(\coth (a+b x))\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 2158
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(\coth (a+b x))}{x} \, dx &=b x-\left (b x-\coth ^{-1}(\coth (a+b x))\right ) \int \frac {1}{x} \, dx\\ &=b x-\left (b x-\coth ^{-1}(\coth (a+b x))\right ) \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 19, normalized size = 0.90 \[ \log (x) \left (\coth ^{-1}(\coth (a+b x))-b x\right )+b x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 8, normalized size = 0.38 \[ b x + a \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.13, size = 9, normalized size = 0.43 \[ b x + a \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.31, size = 27, normalized size = 1.29 \[ b x +a \ln \relax (x )+\ln \relax (x ) \left (\mathrm {arccoth}\left (\coth \left (b x +a \right )\right )-b x -a \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 8, normalized size = 0.38 \[ b x + a \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.55, size = 58, normalized size = 2.76 \[ b\,x-\frac {\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \relax (x)}{2}+\frac {\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \relax (x)}{2}-b\,x\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acoth}{\left (\coth {\left (a + b x \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________