Optimal. Leaf size=94 \[ -\frac {1}{16 a \left (1-a^2 x^2\right )^2}-\frac {3}{16 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)^2}{16 a} \]
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Rubi [A]
time = 0.03, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6107, 6103,
267} \begin {gather*} -\frac {3}{16 a \left (1-a^2 x^2\right )}-\frac {1}{16 a \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3 \tanh ^{-1}(a x)^2}{16 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 6103
Rule 6107
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx &=-\frac {1}{16 a \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {1}{16 a \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)^2}{16 a}-\frac {1}{8} (3 a) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {1}{16 a \left (1-a^2 x^2\right )^2}-\frac {3}{16 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)^2}{16 a}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 65, normalized size = 0.69 \begin {gather*} \frac {-4+3 a^2 x^2+\left (10 a x-6 a^3 x^3\right ) \tanh ^{-1}(a x)+3 \left (-1+a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{16 a \left (-1+a^2 x^2\right )^2} \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. \(177\) vs.
\(2(84)=168\).
time = 0.77, size = 178, normalized size = 1.89
method | result | size |
derivativedivides | \(\frac {\frac {\arctanh \left (a x \right )}{16 \left (a x -1\right )^{2}}-\frac {3 \arctanh \left (a x \right )}{16 \left (a x -1\right )}-\frac {3 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{16}-\frac {\arctanh \left (a x \right )}{16 \left (a x +1\right )^{2}}-\frac {3 \arctanh \left (a x \right )}{16 \left (a x +1\right )}+\frac {3 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{16}+\frac {3 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{32}-\frac {3 \ln \left (a x -1\right )^{2}}{64}+\frac {3 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{32}-\frac {3 \ln \left (a x +1\right )^{2}}{64}-\frac {1}{64 \left (a x -1\right )^{2}}+\frac {7}{64 \left (a x -1\right )}-\frac {1}{64 \left (a x +1\right )^{2}}-\frac {7}{64 \left (a x +1\right )}}{a}\) | \(178\) |
default | \(\frac {\frac {\arctanh \left (a x \right )}{16 \left (a x -1\right )^{2}}-\frac {3 \arctanh \left (a x \right )}{16 \left (a x -1\right )}-\frac {3 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{16}-\frac {\arctanh \left (a x \right )}{16 \left (a x +1\right )^{2}}-\frac {3 \arctanh \left (a x \right )}{16 \left (a x +1\right )}+\frac {3 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{16}+\frac {3 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{32}-\frac {3 \ln \left (a x -1\right )^{2}}{64}+\frac {3 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{32}-\frac {3 \ln \left (a x +1\right )^{2}}{64}-\frac {1}{64 \left (a x -1\right )^{2}}+\frac {7}{64 \left (a x -1\right )}-\frac {1}{64 \left (a x +1\right )^{2}}-\frac {7}{64 \left (a x +1\right )}}{a}\) | \(178\) |
risch | \(\frac {3 \ln \left (a x +1\right )^{2}}{64 a}-\frac {\left (3 x^{4} \ln \left (-a x +1\right ) a^{4}+6 a^{3} x^{3}-6 x^{2} \ln \left (-a x +1\right ) a^{2}-10 a x +3 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{32 \left (a^{2} x^{2}-1\right )^{2} a}+\frac {3 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+12 a^{3} x^{3} \ln \left (-a x +1\right )-6 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+12 a^{2} x^{2}-20 a x \ln \left (-a x +1\right )+3 \ln \left (-a x +1\right )^{2}-16}{64 a \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}\) | \(200\) |
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. 182 vs.
\(2 (80) = 160\).
time = 0.26, size = 182, normalized size = 1.94 \begin {gather*} -\frac {1}{16} \, {\left (\frac {2 \, {\left (3 \, a^{2} x^{3} - 5 \, x\right )}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1} - \frac {3 \, \log \left (a x + 1\right )}{a} + \frac {3 \, \log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right ) + \frac {{\left (12 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 6 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 16\right )} a}{64 \, {\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 97, normalized size = 1.03 \begin {gather*} \frac {12 \, 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 (3 \, a^{3} x^{3} - 5 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 16}{64 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )}} \end {gather*}
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 \begin {gather*} - \int \frac {\operatorname {atanh}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx \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 = 1.37, size = 154, normalized size = 1.64 \begin {gather*} \frac {\frac {3\,a\,x^2}{2}-\frac {2}{a}}{8\,a^4\,x^4-16\,a^2\,x^2+8}-\ln \left (1-a\,x\right )\,\left (\frac {3\,\ln \left (a\,x+1\right )}{32\,a}+\frac {\frac {5\,x}{8}-\frac {3\,a^2\,x^3}{8}}{2\,a^4\,x^4-4\,a^2\,x^2+2}\right )+\frac {3\,{\ln \left (a\,x+1\right )}^2}{64\,a}+\frac {3\,{\ln \left (1-a\,x\right )}^2}{64\,a}+\frac {\ln \left (a\,x+1\right )\,\left (\frac {5\,x}{16\,a}-\frac {3\,a\,x^3}{16}\right )}{\frac {1}{a}-2\,a\,x^2+a^3\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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