Optimal. Leaf size=119 \[ -\frac {3 x}{8 a \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 x \tanh ^{-1}(a x)^2}{4 a \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{4 a^2}-\frac {3 \tanh ^{-1}(a x)}{8 a^2} \]
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Rubi [A] time = 0.11, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5994, 5956, 199, 206} \[ -\frac {3 x}{8 a \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 x \tanh ^{-1}(a x)^2}{4 a \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{4 a^2}-\frac {3 \tanh ^{-1}(a x)}{8 a^2} \]
Antiderivative was successfully verified.
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Rule 199
Rule 206
Rule 5956
Rule 5994
Rubi steps
\begin {align*} \int \frac {x \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx &=\frac {\tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{2 a}\\ &=-\frac {3 x \tanh ^{-1}(a x)^2}{4 a \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{4 a^2}+\frac {\tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}+\frac {3}{2} \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac {3 \tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )}-\frac {3 x \tanh ^{-1}(a x)^2}{4 a \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{4 a^2}+\frac {\tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx}{4 a}\\ &=-\frac {3 x}{8 a \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )}-\frac {3 x \tanh ^{-1}(a x)^2}{4 a \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{4 a^2}+\frac {\tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \int \frac {1}{1-a^2 x^2} \, dx}{8 a}\\ &=-\frac {3 x}{8 a \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)}{8 a^2}+\frac {3 \tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )}-\frac {3 x \tanh ^{-1}(a x)^2}{4 a \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{4 a^2}+\frac {\tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 91, normalized size = 0.76 \[ \frac {3 \left (a^2 x^2-1\right ) \log (1-a x)-3 \left (a^2 x^2-1\right ) \log (a x+1)-4 \left (a^2 x^2+1\right ) \tanh ^{-1}(a x)^3+6 a x+12 a x \tanh ^{-1}(a x)^2-12 \tanh ^{-1}(a x)}{16 a^2 \left (a^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 97, normalized size = 0.82 \[ \frac {6 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - {\left (a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 12 \, a x - 6 \, {\left (a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{32 \, {\left (a^{4} x^{2} - a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 192, normalized size = 1.61 \[ -\frac {1}{64} \, {\left ({\left (\frac {a x + 1}{{\left (a x - 1\right )} a^{3}} + \frac {a x - 1}{{\left (a x + 1\right )} a^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} - 3 \, {\left (\frac {a x + 1}{{\left (a x - 1\right )} a^{3}} - \frac {a x - 1}{{\left (a x + 1\right )} a^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 6 \, {\left (\frac {a x + 1}{{\left (a x - 1\right )} a^{3}} + \frac {a x - 1}{{\left (a x + 1\right )} a^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - \frac {6 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{3}} + \frac {6 \, {\left (a x - 1\right )}}{{\left (a x + 1\right )} a^{3}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.81, size = 1708, normalized size = 14.35 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 298, normalized size = 2.50 \[ \frac {3 \, {\left (\frac {2 \, x}{a^{2} x^{2} - 1} - \frac {\log \left (a x + 1\right )}{a} + \frac {\log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right )^{2}}{8 \, a} - \frac {\frac {{\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} - 3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) - {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} - 12 \, a x + 3 \, {\left (2 \, a^{2} x^{2} + {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 2\right )} \log \left (a x + 1\right ) - 6 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} a^{2}}{a^{5} x^{2} - a^{3}} - \frac {6 \, {\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} - 2 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) + {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 4\right )} a \operatorname {artanh}\left (a x\right )}{a^{4} x^{2} - a^{2}}}{32 \, a} - \frac {\operatorname {artanh}\left (a x\right )^{3}}{2 \, {\left (a^{2} x^{2} - 1\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.76, size = 239, normalized size = 2.01 \[ -\frac {6\,\ln \left (1-a\,x\right )-6\,\ln \left (a\,x+1\right )+12\,a\,x-{\ln \left (a\,x+1\right )}^3+{\ln \left (1-a\,x\right )}^3-3\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2+3\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )-a^2\,x^2\,\left (6\,\ln \left (a\,x+1\right )-6\,\ln \left (1-a\,x\right )\right )-a^2\,x^2\,{\ln \left (a\,x+1\right )}^3+a^2\,x^2\,{\ln \left (1-a\,x\right )}^3+6\,a\,x\,{\ln \left (a\,x+1\right )}^2+6\,a\,x\,{\ln \left (1-a\,x\right )}^2-12\,a\,x\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )-3\,a^2\,x^2\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2+3\,a^2\,x^2\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )}{32\,a^2-32\,a^4\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \operatorname {atanh}^{3}{\left (a x \right )}}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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