Optimal. Leaf size=61 \[ -\frac{1}{2} a^2 \log \left (1-a^2 x^2\right )+a^2 \log (x)+\frac{1}{2} a^2 \coth ^{-1}(a x)^2-\frac{\coth ^{-1}(a x)^2}{2 x^2}-\frac{a \coth ^{-1}(a x)}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0985716, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5917, 5983, 266, 36, 29, 31, 5949} \[ -\frac{1}{2} a^2 \log \left (1-a^2 x^2\right )+a^2 \log (x)+\frac{1}{2} a^2 \coth ^{-1}(a x)^2-\frac{\coth ^{-1}(a x)^2}{2 x^2}-\frac{a \coth ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5917
Rule 5983
Rule 266
Rule 36
Rule 29
Rule 31
Rule 5949
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(a x)^2}{x^3} \, dx &=-\frac{\coth ^{-1}(a x)^2}{2 x^2}+a \int \frac{\coth ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac{\coth ^{-1}(a x)^2}{2 x^2}+a \int \frac{\coth ^{-1}(a x)}{x^2} \, dx+a^3 \int \frac{\coth ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-\frac{a \coth ^{-1}(a x)}{x}+\frac{1}{2} a^2 \coth ^{-1}(a x)^2-\frac{\coth ^{-1}(a x)^2}{2 x^2}+a^2 \int \frac{1}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac{a \coth ^{-1}(a x)}{x}+\frac{1}{2} a^2 \coth ^{-1}(a x)^2-\frac{\coth ^{-1}(a x)^2}{2 x^2}+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{a \coth ^{-1}(a x)}{x}+\frac{1}{2} a^2 \coth ^{-1}(a x)^2-\frac{\coth ^{-1}(a x)^2}{2 x^2}+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{2} a^4 \operatorname{Subst}\left (\int \frac{1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac{a \coth ^{-1}(a x)}{x}+\frac{1}{2} a^2 \coth ^{-1}(a x)^2-\frac{\coth ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)-\frac{1}{2} a^2 \log \left (1-a^2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0170421, size = 57, normalized size = 0.93 \[ -\frac{1}{2} a^2 \log \left (1-a^2 x^2\right )+\frac{\left (a^2 x^2-1\right ) \coth ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)-\frac{a \coth ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.055, size = 164, normalized size = 2.7 \begin{align*} -{\frac{ \left ({\rm arccoth} \left (ax\right ) \right ) ^{2}}{2\,{x}^{2}}}-{\frac{a{\rm arccoth} \left (ax\right )}{x}}-{\frac{{a}^{2}{\rm arccoth} \left (ax\right )\ln \left ( ax-1 \right ) }{2}}+{\frac{{a}^{2}{\rm arccoth} \left (ax\right )\ln \left ( ax+1 \right ) }{2}}-{\frac{{a}^{2} \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{8}}+{\frac{{a}^{2}\ln \left ( ax-1 \right ) }{4}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{{a}^{2}\ln \left ( ax-1 \right ) }{2}}+{a}^{2}\ln \left ( ax \right ) -{\frac{{a}^{2}\ln \left ( ax+1 \right ) }{2}}+{\frac{{a}^{2}\ln \left ( ax+1 \right ) }{4}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) }-{\frac{{a}^{2}}{4}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{{a}^{2} \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.971504, size = 130, normalized size = 2.13 \begin{align*} \frac{1}{8} \,{\left (2 \,{\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2} - \log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right ) + 8 \, \log \left (x\right )\right )} a^{2} + \frac{1}{2} \,{\left (a \log \left (a x + 1\right ) - a \log \left (a x - 1\right ) - \frac{2}{x}\right )} a \operatorname{arcoth}\left (a x\right ) - \frac{\operatorname{arcoth}\left (a x\right )^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.89944, size = 181, normalized size = 2.97 \begin{align*} -\frac{4 \, a^{2} x^{2} \log \left (a^{2} x^{2} - 1\right ) - 8 \, a^{2} x^{2} \log \left (x\right ) + 4 \, a x \log \left (\frac{a x + 1}{a x - 1}\right ) -{\left (a^{2} x^{2} - 1\right )} \log \left (\frac{a x + 1}{a x - 1}\right )^{2}}{8 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.0102, size = 56, normalized size = 0.92 \begin{align*} a^{2} \log{\left (x \right )} - a^{2} \log{\left (a x + 1 \right )} + \frac{a^{2} \operatorname{acoth}^{2}{\left (a x \right )}}{2} + a^{2} \operatorname{acoth}{\left (a x \right )} - \frac{a \operatorname{acoth}{\left (a x \right )}}{x} - \frac{\operatorname{acoth}^{2}{\left (a x \right )}}{2 x^{2}} \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 \frac{\operatorname{arcoth}\left (a x\right )^{2}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]