Optimal. Leaf size=192 \[ -\frac {3 \tanh ^{-1}(a x)^3}{32 a^4}+\frac {27 \tanh ^{-1}(a x)}{256 a^4}+\frac {x^4 \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {3 x^4 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}-\frac {3 x^3}{128 a \left (1-a^2 x^2\right )^2}-\frac {3 x^3 \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}-\frac {9 \tanh ^{-1}(a x)}{32 a^4 \left (1-a^2 x^2\right )}+\frac {45 x}{256 a^3 \left (1-a^2 x^2\right )}+\frac {9 x \tanh ^{-1}(a x)^2}{32 a^3 \left (1-a^2 x^2\right )} \]
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Rubi [A] time = 0.27, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6008, 6004, 6000, 5994, 199, 206, 288} \[ -\frac {3 x^3}{128 a \left (1-a^2 x^2\right )^2}+\frac {45 x}{256 a^3 \left (1-a^2 x^2\right )}+\frac {x^4 \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {3 x^4 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}-\frac {3 x^3 \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {9 x \tanh ^{-1}(a x)^2}{32 a^3 \left (1-a^2 x^2\right )}-\frac {9 \tanh ^{-1}(a x)}{32 a^4 \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^3}{32 a^4}+\frac {27 \tanh ^{-1}(a x)}{256 a^4} \]
Antiderivative was successfully verified.
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Rule 199
Rule 206
Rule 288
Rule 5994
Rule 6000
Rule 6004
Rule 6008
Rubi steps
\begin {align*} \int \frac {x^3 \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx &=\frac {x^4 \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{4} (3 a) \int \frac {x^4 \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx\\ &=\frac {3 x^4 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}-\frac {3 x^3 \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {x^4 \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {9 \int \frac {x^2 \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{16 a}-\frac {1}{32} (3 a) \int \frac {x^4}{\left (1-a^2 x^2\right )^3} \, dx\\ &=-\frac {3 x^3}{128 a \left (1-a^2 x^2\right )^2}+\frac {3 x^4 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}-\frac {3 x^3 \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {9 x \tanh ^{-1}(a x)^2}{32 a^3 \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^3}{32 a^4}+\frac {x^4 \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {9 \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{16 a^2}+\frac {9 \int \frac {x^2}{\left (1-a^2 x^2\right )^2} \, dx}{128 a}\\ &=-\frac {3 x^3}{128 a \left (1-a^2 x^2\right )^2}+\frac {9 x}{256 a^3 \left (1-a^2 x^2\right )}+\frac {3 x^4 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}-\frac {9 \tanh ^{-1}(a x)}{32 a^4 \left (1-a^2 x^2\right )}-\frac {3 x^3 \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {9 x \tanh ^{-1}(a x)^2}{32 a^3 \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^3}{32 a^4}+\frac {x^4 \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {9 \int \frac {1}{1-a^2 x^2} \, dx}{256 a^3}+\frac {9 \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx}{32 a^3}\\ &=-\frac {3 x^3}{128 a \left (1-a^2 x^2\right )^2}+\frac {45 x}{256 a^3 \left (1-a^2 x^2\right )}-\frac {9 \tanh ^{-1}(a x)}{256 a^4}+\frac {3 x^4 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}-\frac {9 \tanh ^{-1}(a x)}{32 a^4 \left (1-a^2 x^2\right )}-\frac {3 x^3 \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {9 x \tanh ^{-1}(a x)^2}{32 a^3 \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^3}{32 a^4}+\frac {x^4 \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {9 \int \frac {1}{1-a^2 x^2} \, dx}{64 a^3}\\ &=-\frac {3 x^3}{128 a \left (1-a^2 x^2\right )^2}+\frac {45 x}{256 a^3 \left (1-a^2 x^2\right )}+\frac {27 \tanh ^{-1}(a x)}{256 a^4}+\frac {3 x^4 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}-\frac {9 \tanh ^{-1}(a x)}{32 a^4 \left (1-a^2 x^2\right )}-\frac {3 x^3 \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {9 x \tanh ^{-1}(a x)^2}{32 a^3 \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^3}{32 a^4}+\frac {x^4 \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 135, normalized size = 0.70 \[ \frac {-48 a x \left (5 a^2 x^2-3\right ) \tanh ^{-1}(a x)^2+48 \left (5 a^2 x^2-4\right ) \tanh ^{-1}(a x)+16 \left (5 a^4 x^4+6 a^2 x^2-3\right ) \tanh ^{-1}(a x)^3+3 \left (-34 a^3 x^3-17 \left (a^2 x^2-1\right )^2 \log (1-a x)+17 \left (a^2 x^2-1\right )^2 \log (a x+1)+30 a x\right )}{512 a^4 \left (a^2 x^2-1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.52, size = 140, normalized size = 0.73 \[ -\frac {102 \, a^{3} x^{3} - 2 \, {\left (5 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 3\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 12 \, {\left (5 \, a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 90 \, a x - 3 \, {\left (17 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 15\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{512 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 341, normalized size = 1.78 \[ \frac {1}{2048} \, {\left (4 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {4 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}}{{\left (a x + 1\right )}^{2} a^{5}} + \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{5}} + \frac {4 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{5}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 6 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {8 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}}{{\left (a x + 1\right )}^{2} a^{5}} - \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{5}} - \frac {8 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{5}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 6 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {16 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}}{{\left (a x + 1\right )}^{2} a^{5}} + \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{5}} + \frac {16 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{5}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + \frac {3 \, {\left (a x - 1\right )}^{2} {\left (\frac {32 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}}{{\left (a x + 1\right )}^{2} a^{5}} - \frac {3 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{5}} - \frac {96 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{5}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.90, size = 2634, normalized size = 13.72 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 437, normalized size = 2.28 \[ -\frac {3}{64} \, a {\left (\frac {2 \, {\left (5 \, a^{2} x^{3} - 3 \, x\right )}}{a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}} - \frac {5 \, \log \left (a x + 1\right )}{a^{5}} + \frac {5 \, \log \left (a x - 1\right )}{a^{5}}\right )} \operatorname {artanh}\left (a x\right )^{2} + \frac {{\left (2 \, a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{3}}{4 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} - \frac {1}{512} \, {\left (\frac {{\left (102 \, a^{3} x^{3} - 10 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{3} + 30 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) + 10 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{3} - 90 \, a x - 3 \, {\left (17 \, a^{4} x^{4} - 34 \, a^{2} x^{2} + 10 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} + 17\right )} \log \left (a x + 1\right ) + 51 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} a^{2}}{a^{11} x^{4} - 2 \, a^{9} x^{2} + a^{7}} - \frac {12 \, {\left (20 \, a^{2} x^{2} - 5 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 10 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 5 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 16\right )} a \operatorname {artanh}\left (a x\right )}{a^{10} x^{4} - 2 \, a^{8} x^{2} + a^{6}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.09, size = 414, normalized size = 2.16 \[ \frac {48\,\ln \left (1-a\,x\right )-48\,\ln \left (a\,x+1\right )+51\,\mathrm {atanh}\left (a\,x\right )+45\,a\,x-3\,{\ln \left (a\,x+1\right )}^3+3\,{\ln \left (1-a\,x\right )}^3-9\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2+9\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )-51\,a^3\,x^3+6\,a^2\,x^2\,{\ln \left (a\,x+1\right )}^3-6\,a^2\,x^2\,{\ln \left (1-a\,x\right )}^3-30\,a^3\,x^3\,{\ln \left (a\,x+1\right )}^2-30\,a^3\,x^3\,{\ln \left (1-a\,x\right )}^2+5\,a^4\,x^4\,{\ln \left (a\,x+1\right )}^3-5\,a^4\,x^4\,{\ln \left (1-a\,x\right )}^3-102\,a^2\,x^2\,\mathrm {atanh}\left (a\,x\right )+51\,a^4\,x^4\,\mathrm {atanh}\left (a\,x\right )+18\,a\,x\,{\ln \left (a\,x+1\right )}^2+18\,a\,x\,{\ln \left (1-a\,x\right )}^2+60\,a^2\,x^2\,\ln \left (a\,x+1\right )-60\,a^2\,x^2\,\ln \left (1-a\,x\right )-36\,a\,x\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )+18\,a^2\,x^2\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2-18\,a^2\,x^2\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )+15\,a^4\,x^4\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2-15\,a^4\,x^4\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )+60\,a^3\,x^3\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{256\,a^4\,{\left (a^2\,x^2-1\right )}^2} \]
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 \[ - \int \frac {x^{3} \operatorname {atanh}^{3}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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