Optimal. Leaf size=56 \[ -\frac {1}{2 a^4 (1-a x)}+\frac {1}{8 a^4 (a x+1)}+\frac {1}{8 a^4 (1-a x)^2}+\frac {3 \tanh ^{-1}(a x)}{8 a^4} \]
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Rubi [A] time = 0.12, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6150, 88, 207} \[ -\frac {1}{2 a^4 (1-a x)}+\frac {1}{8 a^4 (a x+1)}+\frac {1}{8 a^4 (1-a x)^2}+\frac {3 \tanh ^{-1}(a x)}{8 a^4} \]
Antiderivative was successfully verified.
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Rule 88
Rule 207
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=\int \frac {x^3}{(1-a x)^3 (1+a x)^2} \, dx\\ &=\int \left (-\frac {1}{4 a^3 (-1+a x)^3}-\frac {1}{2 a^3 (-1+a x)^2}-\frac {1}{8 a^3 (1+a x)^2}-\frac {3}{8 a^3 \left (-1+a^2 x^2\right )}\right ) \, dx\\ &=\frac {1}{8 a^4 (1-a x)^2}-\frac {1}{2 a^4 (1-a x)}+\frac {1}{8 a^4 (1+a x)}-\frac {3 \int \frac {1}{-1+a^2 x^2} \, dx}{8 a^3}\\ &=\frac {1}{8 a^4 (1-a x)^2}-\frac {1}{2 a^4 (1-a x)}+\frac {1}{8 a^4 (1+a x)}+\frac {3 \tanh ^{-1}(a x)}{8 a^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 0.95 \[ \frac {5 a^2 x^2-a x+3 (a x-1)^2 (a x+1) \tanh ^{-1}(a x)-2}{8 a^4 (a x-1)^2 (a x+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 101, normalized size = 1.80 \[ \frac {10 \, a^{2} x^{2} - 2 \, a x + 3 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right ) - 3 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x - 1\right ) - 4}{16 \, {\left (a^{7} x^{3} - a^{6} x^{2} - a^{5} x + a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 58, normalized size = 1.04 \[ \frac {3 \, \log \left ({\left | a x + 1 \right |}\right )}{16 \, a^{4}} - \frac {3 \, \log \left ({\left | a x - 1 \right |}\right )}{16 \, a^{4}} + \frac {5 \, a^{2} x^{2} - a x - 2}{8 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{2} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 60, normalized size = 1.07 \[ \frac {1}{8 a^{4} \left (a x -1\right )^{2}}+\frac {1}{2 a^{4} \left (a x -1\right )}-\frac {3 \ln \left (a x -1\right )}{16 a^{4}}+\frac {1}{8 a^{4} \left (a x +1\right )}+\frac {3 \ln \left (a x +1\right )}{16 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 66, normalized size = 1.18 \[ \frac {5 \, a^{2} x^{2} - a x - 2}{8 \, {\left (a^{7} x^{3} - a^{6} x^{2} - a^{5} x + a^{4}\right )}} + \frac {3 \, \log \left (a x + 1\right )}{16 \, a^{4}} - \frac {3 \, \log \left (a x - 1\right )}{16 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 53, normalized size = 0.95 \[ \frac {\frac {x}{8\,a^3}+\frac {1}{4\,a^4}-\frac {5\,x^2}{8\,a^2}}{-a^3\,x^3+a^2\,x^2+a\,x-1}+\frac {3\,\mathrm {atanh}\left (a\,x\right )}{8\,a^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 66, normalized size = 1.18 \[ - \frac {- 5 a^{2} x^{2} + a x + 2}{8 a^{7} x^{3} - 8 a^{6} x^{2} - 8 a^{5} x + 8 a^{4}} - \frac {\frac {3 \log {\left (x - \frac {1}{a} \right )}}{16} - \frac {3 \log {\left (x + \frac {1}{a} \right )}}{16}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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