Optimal. Leaf size=56 \[ \frac {1}{4 a (1-a x)}-\frac {1}{8 a (a x+1)}+\frac {1}{8 a (1-a x)^2}+\frac {3 \tanh ^{-1}(a x)}{8 a} \]
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Rubi [A] time = 0.06, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6140, 44, 207} \[ \frac {1}{4 a (1-a x)}-\frac {1}{8 a (a x+1)}+\frac {1}{8 a (1-a x)^2}+\frac {3 \tanh ^{-1}(a x)}{8 a} \]
Antiderivative was successfully verified.
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Rule 44
Rule 207
Rule 6140
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=\int \frac {1}{(1-a x)^3 (1+a x)^2} \, dx\\ &=\int \left (-\frac {1}{4 (-1+a x)^3}+\frac {1}{4 (-1+a x)^2}+\frac {1}{8 (1+a x)^2}-\frac {3}{8 \left (-1+a^2 x^2\right )}\right ) \, dx\\ &=\frac {1}{8 a (1-a x)^2}+\frac {1}{4 a (1-a x)}-\frac {1}{8 a (1+a x)}-\frac {3}{8} \int \frac {1}{-1+a^2 x^2} \, dx\\ &=\frac {1}{8 a (1-a x)^2}+\frac {1}{4 a (1-a x)}-\frac {1}{8 a (1+a x)}+\frac {3 \tanh ^{-1}(a x)}{8 a}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 0.95 \[ \frac {-3 a^2 x^2+3 a x+3 (a x-1)^2 (a x+1) \tanh ^{-1}(a x)+2}{8 a (a x-1)^2 (a x+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 99, normalized size = 1.77 \[ -\frac {6 \, a^{2} x^{2} - 6 \, a x - 3 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right ) + 3 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x - 1\right ) - 4}{16 \, {\left (a^{4} x^{3} - a^{3} x^{2} - a^{2} x + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 58, normalized size = 1.04 \[ \frac {3 \, \log \left ({\left | a x + 1 \right |}\right )}{16 \, a} - \frac {3 \, \log \left ({\left | a x - 1 \right |}\right )}{16 \, a} - \frac {3 \, a^{2} x^{2} - 3 \, a x - 2}{8 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{2} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 60, normalized size = 1.07 \[ \frac {1}{8 a \left (a x -1\right )^{2}}-\frac {1}{4 a \left (a x -1\right )}-\frac {3 \ln \left (a x -1\right )}{16 a}-\frac {1}{8 a \left (a x +1\right )}+\frac {3 \ln \left (a x +1\right )}{16 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 64, normalized size = 1.14 \[ -\frac {3 \, a^{2} x^{2} - 3 \, a x - 2}{8 \, {\left (a^{4} x^{3} - a^{3} x^{2} - a^{2} x + a\right )}} + \frac {3 \, \log \left (a x + 1\right )}{16 \, a} - \frac {3 \, \log \left (a x - 1\right )}{16 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 49, normalized size = 0.88 \[ \frac {3\,\mathrm {atanh}\left (a\,x\right )}{8\,a}-\frac {\frac {3\,x}{8}-\frac {3\,a\,x^2}{8}+\frac {1}{4\,a}}{-a^3\,x^3+a^2\,x^2+a\,x-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 65, normalized size = 1.16 \[ - \frac {3 a^{2} x^{2} - 3 a x - 2}{8 a^{4} x^{3} - 8 a^{3} x^{2} - 8 a^{2} x + 8 a} - \frac {\frac {3 \log {\left (x - \frac {1}{a} \right )}}{16} - \frac {3 \log {\left (x + \frac {1}{a} \right )}}{16}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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