Optimal. Leaf size=95 \[ \frac {1-a^2 x^2}{12 a^2}+\frac {x \tanh ^{-1}(a x)}{3 a}+\frac {x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{6 a}-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 a^2}+\frac {\log \left (1-a^2 x^2\right )}{6 a^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6141, 6089,
6021, 266} \begin {gather*} \frac {1-a^2 x^2}{12 a^2}+\frac {\log \left (1-a^2 x^2\right )}{6 a^2}-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 a^2}+\frac {x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{6 a}+\frac {x \tanh ^{-1}(a x)}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 6021
Rule 6089
Rule 6141
Rubi steps
\begin {align*} \int x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2 \, dx &=-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 a^2}+\frac {\int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx}{2 a}\\ &=\frac {1-a^2 x^2}{12 a^2}+\frac {x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{6 a}-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 a^2}+\frac {\int \tanh ^{-1}(a x) \, dx}{3 a}\\ &=\frac {1-a^2 x^2}{12 a^2}+\frac {x \tanh ^{-1}(a x)}{3 a}+\frac {x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{6 a}-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 a^2}-\frac {1}{3} \int \frac {x}{1-a^2 x^2} \, dx\\ &=\frac {1-a^2 x^2}{12 a^2}+\frac {x \tanh ^{-1}(a x)}{3 a}+\frac {x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{6 a}-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 a^2}+\frac {\log \left (1-a^2 x^2\right )}{6 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 66, normalized size = 0.69 \begin {gather*} \frac {-a^2 x^2+\left (6 a x-2 a^3 x^3\right ) \tanh ^{-1}(a x)-3 \left (-1+a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+2 \log \left (1-a^2 x^2\right )}{12 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 86, normalized size = 0.91
method | result | size |
derivativedivides | \(\frac {-\frac {a^{4} x^{4} \arctanh \left (a x \right )^{2}}{4}+\frac {a^{2} x^{2} \arctanh \left (a x \right )^{2}}{2}-\frac {\arctanh \left (a x \right )^{2}}{4}-\frac {a^{3} x^{3} \arctanh \left (a x \right )}{6}+\frac {a x \arctanh \left (a x \right )}{2}-\frac {a^{2} x^{2}}{12}+\frac {\ln \left (a x -1\right )}{6}+\frac {\ln \left (a x +1\right )}{6}}{a^{2}}\) | \(86\) |
default | \(\frac {-\frac {a^{4} x^{4} \arctanh \left (a x \right )^{2}}{4}+\frac {a^{2} x^{2} \arctanh \left (a x \right )^{2}}{2}-\frac {\arctanh \left (a x \right )^{2}}{4}-\frac {a^{3} x^{3} \arctanh \left (a x \right )}{6}+\frac {a x \arctanh \left (a x \right )}{2}-\frac {a^{2} x^{2}}{12}+\frac {\ln \left (a x -1\right )}{6}+\frac {\ln \left (a x +1\right )}{6}}{a^{2}}\) | \(86\) |
risch | \(-\frac {\left (a^{2} x^{2}-1\right )^{2} \ln \left (a x +1\right )^{2}}{16 a^{2}}+\frac {\left (3 x^{4} \ln \left (-a x +1\right ) a^{4}-2 a^{3} x^{3}-6 x^{2} \ln \left (-a x +1\right ) a^{2}+6 a x +3 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{24 a^{2}}-\frac {\ln \left (-a x +1\right )^{2} a^{2} x^{4}}{16}+\frac {a \,x^{3} \ln \left (-a x +1\right )}{12}+\frac {x^{2} \ln \left (-a x +1\right )^{2}}{8}-\frac {x^{2}}{12}-\frac {x \ln \left (-a x +1\right )}{4 a}-\frac {\ln \left (-a x +1\right )^{2}}{16 a^{2}}+\frac {\ln \left (a^{2} x^{2}-1\right )}{6 a^{2}}\) | \(180\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 74, normalized size = 0.78 \begin {gather*} -\frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{2}}{4 \, a^{2}} - \frac {{\left (x^{2} - \frac {2 \, \log \left (a x + 1\right )}{a^{2}} - \frac {2 \, \log \left (a x - 1\right )}{a^{2}}\right )} a + 2 \, {\left (a^{2} x^{3} - 3 \, x\right )} \operatorname {artanh}\left (a x\right )}{12 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 91, normalized size = 0.96 \begin {gather*} -\frac {4 \, a^{2} x^{2} + 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, {\left (a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 8 \, \log \left (a^{2} x^{2} - 1\right )}{48 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.29, size = 88, normalized size = 0.93 \begin {gather*} \begin {cases} - \frac {a^{2} x^{4} \operatorname {atanh}^{2}{\left (a x \right )}}{4} - \frac {a x^{3} \operatorname {atanh}{\left (a x \right )}}{6} + \frac {x^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{2} - \frac {x^{2}}{12} + \frac {x \operatorname {atanh}{\left (a x \right )}}{2 a} + \frac {\log {\left (x - \frac {1}{a} \right )}}{3 a^{2}} - \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{4 a^{2}} + \frac {\operatorname {atanh}{\left (a x \right )}}{3 a^{2}} & \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 305 vs.
\(2 (82) = 164\).
time = 0.40, size = 305, normalized size = 3.21 \begin {gather*} -\frac {1}{3} \, a {\left (\frac {{\left (\frac {3 \, {\left (a x + 1\right )}}{a x - 1} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{\frac {{\left (a x + 1\right )}^{3} a^{3}}{{\left (a x - 1\right )}^{3}} - \frac {3 \, {\left (a x + 1\right )}^{2} a^{3}}{{\left (a x - 1\right )}^{2}} + \frac {3 \, {\left (a x + 1\right )} a^{3}}{a x - 1} - a^{3}} + \frac {3 \, {\left (a x + 1\right )}^{2} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}}{{\left (\frac {{\left (a x + 1\right )}^{4} a^{3}}{{\left (a x - 1\right )}^{4}} - \frac {4 \, {\left (a x + 1\right )}^{3} a^{3}}{{\left (a x - 1\right )}^{3}} + \frac {6 \, {\left (a x + 1\right )}^{2} a^{3}}{{\left (a x - 1\right )}^{2}} - \frac {4 \, {\left (a x + 1\right )} a^{3}}{a x - 1} + a^{3}\right )} {\left (a x - 1\right )}^{2}} + \frac {a x + 1}{{\left (\frac {{\left (a x + 1\right )}^{2} a^{3}}{{\left (a x - 1\right )}^{2}} - \frac {2 \, {\left (a x + 1\right )} a^{3}}{a x - 1} + a^{3}\right )} {\left (a x - 1\right )}} + \frac {\log \left (-\frac {a x + 1}{a x - 1} + 1\right )}{a^{3}} - \frac {\log \left (-\frac {a x + 1}{a x - 1}\right )}{a^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.90, size = 77, normalized size = 0.81 \begin {gather*} \frac {x^2\,{\mathrm {atanh}\left (a\,x\right )}^2}{2}-\frac {{\mathrm {atanh}\left (a\,x\right )}^2}{4\,a^2}-\frac {x^2}{12}+\frac {\ln \left (a^2\,x^2-1\right )}{6\,a^2}+\frac {x\,\mathrm {atanh}\left (a\,x\right )}{2\,a}-\frac {a\,x^3\,\mathrm {atanh}\left (a\,x\right )}{6}-\frac {a^2\,x^4\,{\mathrm {atanh}\left (a\,x\right )}^2}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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