Optimal. Leaf size=13 \[ \frac {\tanh ^{-1}(a x)^3}{3 a} \]
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Rubi [A]
time = 0.02, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {6095}
\begin {gather*} \frac {\tanh ^{-1}(a x)^3}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 6095
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx &=\frac {\tanh ^{-1}(a x)^3}{3 a}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}(a x)^3}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 12, normalized size = 0.92
method | result | size |
derivativedivides | \(\frac {\arctanh \left (a x \right )^{3}}{3 a}\) | \(12\) |
default | \(\frac {\arctanh \left (a x \right )^{3}}{3 a}\) | \(12\) |
risch | \(\frac {\ln \left (a x +1\right )^{3}}{24 a}-\frac {\ln \left (-a x +1\right ) \ln \left (a x +1\right )^{2}}{8 a}+\frac {\ln \left (-a x +1\right )^{2} \ln \left (a x +1\right )}{8 a}-\frac {\ln \left (-a x +1\right )^{3}}{24 a}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 127 vs.
\(2 (11) = 22\).
time = 0.26, size = 127, normalized size = 9.77 \begin {gather*} \frac {1}{2} \, {\left (\frac {\log \left (a x + 1\right )}{a} - \frac {\log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right )^{2} - \frac {{\left (\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}\right )} \operatorname {artanh}\left (a x\right )}{4 \, a} + \frac {\log \left (a x + 1\right )^{3} - 3 \, \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) + 3 \, \log \left (a x + 1\right ) \log \left (a x - 1\right )^{2} - \log \left (a x - 1\right )^{3}}{24 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 22, normalized size = 1.69 \begin {gather*} \frac {\log \left (-\frac {a x + 1}{a x - 1}\right )^{3}}{24 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.00, size = 10, normalized size = 0.77 \begin {gather*} \begin {cases} \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{3 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 22, normalized size = 1.69 \begin {gather*} \frac {\log \left (-\frac {a x + 1}{a x - 1}\right )^{3}}{24 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.95, size = 68, normalized size = 5.23 \begin {gather*} \frac {{\ln \left (a\,x+1\right )}^3}{24\,a}-\frac {{\ln \left (1-a\,x\right )}^3}{24\,a}+\frac {\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2}{8\,a}-\frac {{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )}{8\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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