Optimal. Leaf size=127 \[ \frac {x^4}{32 \left (1-a^2 x^2\right )^2}-\frac {3}{32 a^4 \left (1-a^2 x^2\right )}-\frac {x^3 \tanh ^{-1}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{16 a^3 \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^2}{32 a^4}+\frac {x^4 \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2} \]
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Rubi [A]
time = 0.13, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6155, 6149,
6145, 6095} \begin {gather*} -\frac {3 \tanh ^{-1}(a x)^2}{32 a^4}+\frac {x^4}{32 \left (1-a^2 x^2\right )^2}+\frac {x^4 \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {x^3 \tanh ^{-1}(a x)}{8 a \left (1-a^2 x^2\right )^2}-\frac {3}{32 a^4 \left (1-a^2 x^2\right )}+\frac {3 x \tanh ^{-1}(a x)}{16 a^3 \left (1-a^2 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 6095
Rule 6145
Rule 6149
Rule 6155
Rubi steps
\begin {align*} \int \frac {x^3 \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx &=\frac {x^4 \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{2} a \int \frac {x^4 \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx\\ &=\frac {x^4}{32 \left (1-a^2 x^2\right )^2}-\frac {x^3 \tanh ^{-1}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {x^4 \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}+\frac {3 \int \frac {x^2 \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{8 a}\\ &=\frac {x^4}{32 \left (1-a^2 x^2\right )^2}-\frac {3}{32 a^4 \left (1-a^2 x^2\right )}-\frac {x^3 \tanh ^{-1}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{16 a^3 \left (1-a^2 x^2\right )}+\frac {x^4 \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{16 a^3}\\ &=\frac {x^4}{32 \left (1-a^2 x^2\right )^2}-\frac {3}{32 a^4 \left (1-a^2 x^2\right )}-\frac {x^3 \tanh ^{-1}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{16 a^3 \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^2}{32 a^4}+\frac {x^4 \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 71, normalized size = 0.56 \begin {gather*} \frac {-4+5 a^2 x^2+\left (6 a x-10 a^3 x^3\right ) \tanh ^{-1}(a x)+\left (-3+6 a^2 x^2+5 a^4 x^4\right ) \tanh ^{-1}(a x)^2}{32 a^4 \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. \(237\) vs.
\(2(115)=230\).
time = 1.58, size = 238, normalized size = 1.87
method | result | size |
derivativedivides | \(\frac {\frac {\arctanh \left (a x \right )^{2}}{16 \left (a x +1\right )^{2}}-\frac {3 \arctanh \left (a x \right )^{2}}{16 \left (a x +1\right )}+\frac {\arctanh \left (a x \right )^{2}}{16 \left (a x -1\right )^{2}}+\frac {3 \arctanh \left (a x \right )^{2}}{16 \left (a x -1\right )}+\frac {\arctanh \left (a x \right )}{32 \left (a x +1\right )^{2}}-\frac {5 \arctanh \left (a x \right )}{32 \left (a x +1\right )}+\frac {5 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{32}-\frac {\arctanh \left (a x \right )}{32 \left (a x -1\right )^{2}}-\frac {5 \arctanh \left (a x \right )}{32 \left (a x -1\right )}-\frac {5 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{32}-\frac {5 \ln \left (a x -1\right )^{2}}{128}+\frac {5 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{64}+\frac {5 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{64}-\frac {5 \ln \left (a x +1\right )^{2}}{128}+\frac {1}{128 \left (a x +1\right )^{2}}-\frac {9}{128 \left (a x +1\right )}+\frac {1}{128 \left (a x -1\right )^{2}}+\frac {9}{128 \left (a x -1\right )}}{a^{4}}\) | \(238\) |
default | \(\frac {\frac {\arctanh \left (a x \right )^{2}}{16 \left (a x +1\right )^{2}}-\frac {3 \arctanh \left (a x \right )^{2}}{16 \left (a x +1\right )}+\frac {\arctanh \left (a x \right )^{2}}{16 \left (a x -1\right )^{2}}+\frac {3 \arctanh \left (a x \right )^{2}}{16 \left (a x -1\right )}+\frac {\arctanh \left (a x \right )}{32 \left (a x +1\right )^{2}}-\frac {5 \arctanh \left (a x \right )}{32 \left (a x +1\right )}+\frac {5 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{32}-\frac {\arctanh \left (a x \right )}{32 \left (a x -1\right )^{2}}-\frac {5 \arctanh \left (a x \right )}{32 \left (a x -1\right )}-\frac {5 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{32}-\frac {5 \ln \left (a x -1\right )^{2}}{128}+\frac {5 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{64}+\frac {5 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{64}-\frac {5 \ln \left (a x +1\right )^{2}}{128}+\frac {1}{128 \left (a x +1\right )^{2}}-\frac {9}{128 \left (a x +1\right )}+\frac {1}{128 \left (a x -1\right )^{2}}+\frac {9}{128 \left (a x -1\right )}}{a^{4}}\) | \(238\) |
risch | \(\frac {\left (5 a^{4} x^{4}+6 a^{2} x^{2}-3\right ) \ln \left (a x +1\right )^{2}}{128 a^{4} \left (a^{2} x^{2}-1\right )^{2}}-\frac {\left (5 x^{4} \ln \left (-a x +1\right ) a^{4}+10 a^{3} x^{3}+6 x^{2} \ln \left (-a x +1\right ) a^{2}-6 a x -3 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{64 a^{4} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}+\frac {5 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+20 a^{3} x^{3} \ln \left (-a x +1\right )+6 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+20 a^{2} x^{2}-12 a x \ln \left (-a x +1\right )-3 \ln \left (-a x +1\right )^{2}-16}{128 a^{4} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}\) | \(243\) |
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. 226 vs.
\(2 (110) = 220\).
time = 0.27, size = 226, normalized size = 1.78 \begin {gather*} -\frac {1}{32} \, a {\left (\frac {2 \, {\left (5 \, a^{2} x^{3} - 3 \, x\right )}}{a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}} - \frac {5 \, \log \left (a x + 1\right )}{a^{5}} + \frac {5 \, \log \left (a x - 1\right )}{a^{5}}\right )} \operatorname {artanh}\left (a x\right ) + \frac {{\left (20 \, a^{2} x^{2} - 5 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 10 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 5 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 16\right )} a^{2}}{128 \, {\left (a^{10} x^{4} - 2 \, a^{8} x^{2} + a^{6}\right )}} + \frac {{\left (2 \, a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{2}}{4 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 99, normalized size = 0.78 \begin {gather*} \frac {20 \, a^{2} x^{2} + {\left (5 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 3\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 4 \, {\left (5 \, a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 16}{128 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\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 {x^{3} \operatorname {atanh}^{2}{\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 250 vs.
\(2 (110) = 220\).
time = 0.40, size = 250, normalized size = 1.97 \begin {gather*} \frac {1}{512} \, {\left (2 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {4 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}}{{\left (a x + 1\right )}^{2} a^{5}} + \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{5}} + \frac {4 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{5}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 2 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {8 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}}{{\left (a x + 1\right )}^{2} a^{5}} - \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{5}} - \frac {8 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{5}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + \frac {{\left (a x - 1\right )}^{2} {\left (\frac {16 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}}{{\left (a x + 1\right )}^{2} a^{5}} + \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{5}} + \frac {16 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{5}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.53, size = 372, normalized size = 2.93 \begin {gather*} {\ln \left (a\,x+1\right )}^2\,\left (\frac {5}{128\,a^4}-\frac {\frac {1}{16\,a^5}-\frac {x^2}{8\,a^3}}{\frac {1}{a}-2\,a\,x^2+a^3\,x^4}\right )-\ln \left (1-a\,x\right )\,\left (\frac {\frac {3\,x}{8}+a\,x^2-\frac {3}{4\,a}-\frac {5\,a^2\,x^3}{8}}{8\,a^7\,x^4-16\,a^5\,x^2+8\,a^3}+\frac {\frac {3\,x}{8}-a\,x^2+\frac {3}{4\,a}-\frac {5\,a^2\,x^3}{8}}{8\,a^7\,x^4-16\,a^5\,x^2+8\,a^3}-\ln \left (a\,x+1\right )\,\left (\frac {\frac {1}{4\,a^4}-\frac {x^2}{2\,a^2}}{2\,a^4\,x^4-4\,a^2\,x^2+2}-\frac {5\,\left (a^4\,x^4-2\,a^2\,x^2+1\right )}{32\,a^4\,\left (2\,a^4\,x^4-4\,a^2\,x^2+2\right )}\right )\right )-{\ln \left (1-a\,x\right )}^2\,\left (\frac {\frac {1}{4\,a^4}-\frac {x^2}{2\,a^2}}{4\,a^4\,x^4-8\,a^2\,x^2+4}-\frac {5}{128\,a^4}\right )-\frac {\frac {2}{a^2}-\frac {5\,x^2}{2}}{16\,a^6\,x^4-32\,a^4\,x^2+16\,a^2}+\frac {\ln \left (a\,x+1\right )\,\left (\frac {3\,x}{32\,a^4}-\frac {5\,x^3}{32\,a^2}\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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