Optimal. Leaf size=42 \[ -\frac {x \tanh ^{-1}(a x)}{a^2}+\frac {\tanh ^{-1}(a x)^2}{2 a^3}-\frac {\log \left (1-a^2 x^2\right )}{2 a^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6127, 6021,
266, 6095} \begin {gather*} \frac {\tanh ^{-1}(a x)^2}{2 a^3}-\frac {x \tanh ^{-1}(a x)}{a^2}-\frac {\log \left (1-a^2 x^2\right )}{2 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 6021
Rule 6095
Rule 6127
Rubi steps
\begin {align*} \int \frac {x^2 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx &=-\frac {\int \tanh ^{-1}(a x) \, dx}{a^2}+\frac {\int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{a^2}\\ &=-\frac {x \tanh ^{-1}(a x)}{a^2}+\frac {\tanh ^{-1}(a x)^2}{2 a^3}+\frac {\int \frac {x}{1-a^2 x^2} \, dx}{a}\\ &=-\frac {x \tanh ^{-1}(a x)}{a^2}+\frac {\tanh ^{-1}(a x)^2}{2 a^3}-\frac {\log \left (1-a^2 x^2\right )}{2 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 42, normalized size = 1.00 \begin {gather*} -\frac {x \tanh ^{-1}(a x)}{a^2}+\frac {\tanh ^{-1}(a x)^2}{2 a^3}-\frac {\log \left (1-a^2 x^2\right )}{2 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(113\) vs.
\(2(38)=76\).
time = 0.30, size = 114, normalized size = 2.71
method | result | size |
risch | \(\frac {\ln \left (a x +1\right )^{2}}{8 a^{3}}-\frac {\left (2 a x +\ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{4 a^{3}}+\frac {\ln \left (-a x +1\right ) x}{2 a^{2}}+\frac {\ln \left (-a x +1\right )^{2}}{8 a^{3}}-\frac {\ln \left (a^{2} x^{2}-1\right )}{2 a^{3}}\) | \(80\) |
derivativedivides | \(\frac {-a x \arctanh \left (a x \right )-\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{2}+\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{8}-\frac {\ln \left (a x -1\right )}{2}-\frac {\ln \left (a x +1\right )}{2}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (a x +1\right )^{2}}{8}}{a^{3}}\) | \(114\) |
default | \(\frac {-a x \arctanh \left (a x \right )-\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{2}+\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{8}-\frac {\ln \left (a x -1\right )}{2}-\frac {\ln \left (a x +1\right )}{2}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (a x +1\right )^{2}}{8}}{a^{3}}\) | \(114\) |
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. 85 vs.
\(2 (38) = 76\).
time = 0.26, size = 85, normalized size = 2.02 \begin {gather*} -\frac {1}{2} \, {\left (\frac {2 \, x}{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 ) + \frac {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 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 56, normalized size = 1.33 \begin {gather*} -\frac {4 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) - \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, \log \left (a^{2} x^{2} - 1\right )}{8 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.81, size = 41, normalized size = 0.98 \begin {gather*} \begin {cases} - \frac {x \operatorname {atanh}{\left (a x \right )}}{a^{2}} - \frac {\log {\left (x - \frac {1}{a} \right )}}{a^{3}} + \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{2 a^{3}} - \frac {\operatorname {atanh}{\left (a x \right )}}{a^{3}} & \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 [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.
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Mupad [B]
time = 0.96, size = 82, normalized size = 1.95 \begin {gather*} \frac {{\ln \left (a\,x+1\right )}^2}{8\,a^3}-\ln \left (1-a\,x\right )\,\left (\frac {\ln \left (a\,x+1\right )}{4\,a^3}-\frac {x}{2\,a^2}\right )+\frac {{\ln \left (1-a\,x\right )}^2}{8\,a^3}-\frac {\ln \left (a^2\,x^2-1\right )}{2\,a^3}-\frac {x\,\ln \left (a\,x+1\right )}{2\,a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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