Optimal. Leaf size=77 \[ -\frac {3 \tanh ^{-1}(a x)}{32 a^4}+\frac {x^4 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {x^3}{16 a \left (1-a^2 x^2\right )^2}+\frac {3 x}{32 a^3 \left (1-a^2 x^2\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6008, 288, 206} \[ -\frac {x^3}{16 a \left (1-a^2 x^2\right )^2}+\frac {3 x}{32 a^3 \left (1-a^2 x^2\right )}+\frac {x^4 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \tanh ^{-1}(a x)}{32 a^4} \]
Antiderivative was successfully verified.
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Rule 206
Rule 288
Rule 6008
Rubi steps
\begin {align*} \int \frac {x^3 \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx &=\frac {x^4 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{4} a \int \frac {x^4}{\left (1-a^2 x^2\right )^3} \, dx\\ &=-\frac {x^3}{16 a \left (1-a^2 x^2\right )^2}+\frac {x^4 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3 \int \frac {x^2}{\left (1-a^2 x^2\right )^2} \, dx}{16 a}\\ &=-\frac {x^3}{16 a \left (1-a^2 x^2\right )^2}+\frac {3 x}{32 a^3 \left (1-a^2 x^2\right )}+\frac {x^4 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \int \frac {1}{1-a^2 x^2} \, dx}{32 a^3}\\ &=-\frac {x^3}{16 a \left (1-a^2 x^2\right )^2}+\frac {3 x}{32 a^3 \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)}{32 a^4}+\frac {x^4 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 98, normalized size = 1.27 \[ -\frac {5 \log (1-a x)}{64 a^4}+\frac {5 \log (a x+1)}{64 a^4}+\frac {\left (2 a^2 x^2-1\right ) \tanh ^{-1}(a x)}{4 a^4 \left (a^2 x^2-1\right )^2}-\frac {5 x}{32 a^3 \left (a^2 x^2-1\right )}-\frac {x}{16 a^3 \left (a^2 x^2-1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 71, normalized size = 0.92 \[ -\frac {10 \, a^{3} x^{3} - 6 \, a x - {\left (5 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 3\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{64 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 239, normalized size = 3.10 \[ \frac {1}{256} \, {\left (2 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {4 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}}{{\left (a x + 1\right )}^{2} a^{5}} + \frac {\frac {{\left (a x + 1\right )}^{2} a^{5}}{{\left (a x - 1\right )}^{2}} + \frac {4 \, {\left (a x + 1\right )} a^{5}}{a x - 1}}{a^{10}}\right )} \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}\right ) + \frac {{\left (a x - 1\right )}^{2} {\left (\frac {8 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}}{{\left (a x + 1\right )}^{2} a^{5}} - \frac {\frac {{\left (a x + 1\right )}^{2} a^{5}}{{\left (a x - 1\right )}^{2}} + \frac {8 \, {\left (a x + 1\right )} a^{5}}{a x - 1}}{a^{10}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 136, normalized size = 1.77 \[ \frac {\arctanh \left (a x \right )}{16 a^{4} \left (a x -1\right )^{2}}+\frac {3 \arctanh \left (a x \right )}{16 a^{4} \left (a x -1\right )}+\frac {\arctanh \left (a x \right )}{16 a^{4} \left (a x +1\right )^{2}}-\frac {3 \arctanh \left (a x \right )}{16 a^{4} \left (a x +1\right )}-\frac {1}{64 a^{4} \left (a x -1\right )^{2}}-\frac {5}{64 a^{4} \left (a x -1\right )}-\frac {5 \ln \left (a x -1\right )}{64 a^{4}}+\frac {1}{64 a^{4} \left (a x +1\right )^{2}}-\frac {5}{64 a^{4} \left (a x +1\right )}+\frac {5 \ln \left (a x +1\right )}{64 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 99, normalized size = 1.29 \[ -\frac {1}{64} \, a {\left (\frac {2 \, {\left (5 \, a^{2} x^{3} - 3 \, x\right )}}{a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}} - \frac {5 \, \log \left (a x + 1\right )}{a^{5}} + \frac {5 \, \log \left (a x - 1\right )}{a^{5}}\right )} + \frac {{\left (2 \, a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )}{4 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.39, size = 83, normalized size = 1.08 \[ \frac {5\,\mathrm {atanh}\left (a\,x\right )}{32\,a^4}+\frac {\frac {\ln \left (1-a\,x\right )}{8}-\frac {\ln \left (a\,x+1\right )}{8}+\frac {3\,a\,x}{32}+x^2\,\left (\frac {a^2\,\ln \left (a\,x+1\right )}{4}-\frac {a^2\,\ln \left (1-a\,x\right )}{4}\right )-\frac {5\,a^3\,x^3}{32}}{a^4\,{\left (a^2\,x^2-1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.41, size = 158, normalized size = 2.05 \[ \begin {cases} \frac {5 a^{4} x^{4} \operatorname {atanh}{\left (a x \right )}}{32 a^{8} x^{4} - 64 a^{6} x^{2} + 32 a^{4}} - \frac {5 a^{3} x^{3}}{32 a^{8} x^{4} - 64 a^{6} x^{2} + 32 a^{4}} + \frac {6 a^{2} x^{2} \operatorname {atanh}{\left (a x \right )}}{32 a^{8} x^{4} - 64 a^{6} x^{2} + 32 a^{4}} + \frac {3 a x}{32 a^{8} x^{4} - 64 a^{6} x^{2} + 32 a^{4}} - \frac {3 \operatorname {atanh}{\left (a x \right )}}{32 a^{8} x^{4} - 64 a^{6} x^{2} + 32 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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