Optimal. Leaf size=94 \[ -\frac {\tanh ^{-1}(a x)^3}{6 a^3}+\frac {\tanh ^{-1}(a x)}{4 a^3}+\frac {x}{4 a^2 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{2 a^3 \left (1-a^2 x^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6000, 5994, 199, 206} \[ \frac {x}{4 a^2 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{2 a^3 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{6 a^3}+\frac {\tanh ^{-1}(a x)}{4 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 199
Rule 206
Rule 5994
Rule 6000
Rubi steps
\begin {align*} \int \frac {x^2 \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx &=\frac {x \tanh ^{-1}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{6 a^3}-\frac {\int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{a}\\ &=-\frac {\tanh ^{-1}(a x)}{2 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{6 a^3}+\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx}{2 a^2}\\ &=\frac {x}{4 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{2 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{6 a^3}+\frac {\int \frac {1}{1-a^2 x^2} \, dx}{4 a^2}\\ &=\frac {x}{4 a^2 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)}{4 a^3}-\frac {\tanh ^{-1}(a x)}{2 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{6 a^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 93, normalized size = 0.99 \[ \frac {-3 \left (\left (a^2 x^2-1\right ) \log (1-a x)+\left (1-a^2 x^2\right ) \log (a x+1)+2 a x\right )+\left (4-4 a^2 x^2\right ) \tanh ^{-1}(a x)^3-12 a x \tanh ^{-1}(a x)^2+12 \tanh ^{-1}(a x)}{24 a^3 \left (a^2 x^2-1\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.57, size = 96, normalized size = 1.02 \[ -\frac {6 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + {\left (a^{2} x^{2} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 12 \, a x - 6 \, {\left (a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{48 \, {\left (a^{5} x^{2} - a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \operatorname {artanh}\left (a x\right )^{2}}{{\left (a^{2} x^{2} - 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.81, size = 1722, normalized size = 18.32 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.34, size = 273, normalized size = 2.90 \[ -\frac {1}{4} \, {\left (\frac {2 \, x}{a^{4} x^{2} - a^{2}} + \frac {\log \left (a x + 1\right )}{a^{3}} - \frac {\log \left (a x - 1\right )}{a^{3}}\right )} \operatorname {artanh}\left (a x\right )^{2} - \frac {{\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} - 3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) - {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} + 12 \, a x - 3 \, {\left (2 \, a^{2} x^{2} - {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 2\right )} \log \left (a x + 1\right ) + 6 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} a^{2}}{48 \, {\left (a^{7} x^{2} - a^{5}\right )}} + \frac {{\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} - 2 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) + {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} + 4\right )} a \operatorname {artanh}\left (a x\right )}{8 \, {\left (a^{6} x^{2} - a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.76, size = 231, normalized size = 2.46 \[ \frac {\ln \left (1-a\,x\right )}{4\,a^3-4\,a^5\,x^2}-\frac {{\ln \left (a\,x+1\right )}^3}{48\,a^3}+\frac {{\ln \left (1-a\,x\right )}^3}{48\,a^3}+\frac {x}{4\,a^2-4\,a^4\,x^2}-\frac {\ln \left (a\,x+1\right )}{4\,\left (a^3-a^5\,x^2\right )}+\frac {x\,{\ln \left (1-a\,x\right )}^2}{8\,a^2-8\,a^4\,x^2}-\frac {\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2}{16\,a^3}+\frac {{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )}{16\,a^3}+\frac {x\,{\ln \left (a\,x+1\right )}^2}{8\,\left (a^2-a^4\,x^2\right )}-\frac {x\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{4\,a^2-4\,a^4\,x^2}-\frac {\mathrm {atan}\left (a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4\,a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________