Optimal. Leaf size=56 \[ -\frac {1}{4 a^3 (1-a x)}-\frac {1}{8 a^3 (a x+1)}+\frac {1}{8 a^3 (1-a x)^2}-\frac {\tanh ^{-1}(a x)}{8 a^3} \]
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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} \[ -\frac {1}{4 a^3 (1-a x)}-\frac {1}{8 a^3 (a x+1)}+\frac {1}{8 a^3 (1-a x)^2}-\frac {\tanh ^{-1}(a x)}{8 a^3} \]
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^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=\int \frac {x^2}{(1-a x)^3 (1+a x)^2} \, dx\\ &=\int \left (-\frac {1}{4 a^2 (-1+a x)^3}-\frac {1}{4 a^2 (-1+a x)^2}+\frac {1}{8 a^2 (1+a x)^2}+\frac {1}{8 a^2 \left (-1+a^2 x^2\right )}\right ) \, dx\\ &=\frac {1}{8 a^3 (1-a x)^2}-\frac {1}{4 a^3 (1-a x)}-\frac {1}{8 a^3 (1+a x)}+\frac {\int \frac {1}{-1+a^2 x^2} \, dx}{8 a^2}\\ &=\frac {1}{8 a^3 (1-a x)^2}-\frac {1}{4 a^3 (1-a x)}-\frac {1}{8 a^3 (1+a x)}-\frac {\tanh ^{-1}(a x)}{8 a^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 52, normalized size = 0.93 \[ \frac {a^2 x^2+3 a x-(a x-1)^2 (a x+1) \tanh ^{-1}(a x)-2}{8 a^3 (a x-1)^2 (a x+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.72, size = 100, normalized size = 1.79 \[ \frac {2 \, a^{2} x^{2} + 6 \, a x - {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right ) + {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x - 1\right ) - 4}{16 \, {\left (a^{6} x^{3} - a^{5} x^{2} - a^{4} x + a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 57, normalized size = 1.02 \[ -\frac {\log \left ({\left | a x + 1 \right |}\right )}{16 \, a^{3}} + \frac {\log \left ({\left | a x - 1 \right |}\right )}{16 \, a^{3}} + \frac {a^{2} x^{2} + 3 \, a x - 2}{8 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{2} a^{3}} \]
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 \[ \frac {1}{8 a^{3} \left (a x -1\right )^{2}}+\frac {1}{4 a^{3} \left (a x -1\right )}+\frac {\ln \left (a x -1\right )}{16 a^{3}}-\frac {1}{8 a^{3} \left (a x +1\right )}-\frac {\ln \left (a x +1\right )}{16 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 65, normalized size = 1.16 \[ \frac {a^{2} x^{2} + 3 \, a x - 2}{8 \, {\left (a^{6} x^{3} - a^{5} x^{2} - a^{4} x + a^{3}\right )}} - \frac {\log \left (a x + 1\right )}{16 \, a^{3}} + \frac {\log \left (a x - 1\right )}{16 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 54, normalized size = 0.96 \[ -\frac {\frac {3\,x}{8\,a^2}-\frac {1}{4\,a^3}+\frac {x^2}{8\,a}}{-a^3\,x^3+a^2\,x^2+a\,x-1}-\frac {\mathrm {atanh}\left (a\,x\right )}{8\,a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 63, normalized size = 1.12 \[ - \frac {- a^{2} x^{2} - 3 a x + 2}{8 a^{6} x^{3} - 8 a^{5} x^{2} - 8 a^{4} x + 8 a^{3}} - \frac {- \frac {\log {\left (x - \frac {1}{a} \right )}}{16} + \frac {\log {\left (x + \frac {1}{a} \right )}}{16}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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