Optimal. Leaf size=55 \[ a \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{x}+2 a \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-a \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6038, 6136,
6080, 2497} \begin {gather*} -a \text {Li}_2\left (\frac {2}{a x+1}-1\right )+a \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{x}+2 a \log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2497
Rule 6038
Rule 6080
Rule 6136
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a x)^2}{x^2} \, dx &=-\frac {\coth ^{-1}(a x)^2}{x}+(2 a) \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx\\ &=a \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{x}+(2 a) \int \frac {\coth ^{-1}(a x)}{x (1+a x)} \, dx\\ &=a \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{x}+2 a \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\left (2 a^2\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=a \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{x}+2 a \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-a \text {Li}_2\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 49, normalized size = 0.89 \begin {gather*} \frac {(-1+a x) \coth ^{-1}(a x)^2}{x}+2 a \coth ^{-1}(a x) \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )-a \text {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(144\) vs.
\(2(55)=110\).
time = 0.07, size = 145, normalized size = 2.64
method | result | size |
derivativedivides | \(a \left (-\frac {\mathrm {arccoth}\left (a x \right )^{2}}{a x}-\mathrm {arccoth}\left (a x \right ) \ln \left (a x +1\right )-\mathrm {arccoth}\left (a x \right ) \ln \left (a x -1\right )+2 \ln \left (a x \right ) \mathrm {arccoth}\left (a x \right )+\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (a x -1\right )^{2}}{4}+\frac {\ln \left (a x +1\right )^{2}}{4}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}-\dilog \left (a x +1\right )-\ln \left (a x \right ) \ln \left (a x +1\right )-\dilog \left (a x \right )\right )\) | \(145\) |
default | \(a \left (-\frac {\mathrm {arccoth}\left (a x \right )^{2}}{a x}-\mathrm {arccoth}\left (a x \right ) \ln \left (a x +1\right )-\mathrm {arccoth}\left (a x \right ) \ln \left (a x -1\right )+2 \ln \left (a x \right ) \mathrm {arccoth}\left (a x \right )+\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (a x -1\right )^{2}}{4}+\frac {\ln \left (a x +1\right )^{2}}{4}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}-\dilog \left (a x +1\right )-\ln \left (a x \right ) \ln \left (a x +1\right )-\dilog \left (a x \right )\right )\) | \(145\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 146 vs.
\(2 (54) = 108\).
time = 0.26, size = 146, normalized size = 2.65 \begin {gather*} \frac {1}{4} \, a^{2} {\left (\frac {\log \left (a x + 1\right )^{2} - 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - \log \left (a x - 1\right )^{2}}{a} + \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} - \frac {4 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a} + \frac {4 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )}}{a}\right )} - a {\left (\log \left (a^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} \operatorname {arcoth}\left (a x\right ) - \frac {\operatorname {arcoth}\left (a x\right )^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acoth}^{2}{\left (a x \right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {acoth}\left (a\,x\right )}^2}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________