Optimal. Leaf size=56 \[ \frac {1}{8 a (1-a x)^2}+\frac {1}{4 a (1-a x)}-\frac {1}{8 a (1+a x)}+\frac {3 \tanh ^{-1}(a x)}{8 a} \]
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Rubi [A]
time = 0.04, 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 = {6275, 46, 213}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 213
Rule 6275
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 = 0.02, size = 53, normalized size = 0.95 \begin {gather*} \frac {2+3 a x-3 a^2 x^2+3 (-1+a x)^2 (1+a x) \tanh ^{-1}(a x)}{8 a (-1+a x)^2 (1+a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 60, normalized size = 1.07
method | result | size |
risch | \(\frac {-\frac {3 x^{2} a}{8}+\frac {3 x}{8}+\frac {1}{4 a}}{\left (a x -1\right ) \left (a^{2} x^{2}-1\right )}+\frac {3 \ln \left (-a x -1\right )}{16 a}-\frac {3 \ln \left (a x -1\right )}{16 a}\) | \(59\) |
default | \(-\frac {1}{8 a \left (a x +1\right )}+\frac {3 \ln \left (a x +1\right )}{16 a}+\frac {1}{8 a \left (a x -1\right )^{2}}-\frac {1}{4 a \left (a x -1\right )}-\frac {3 \ln \left (a x -1\right )}{16 a}\) | \(60\) |
norman | \(\frac {\frac {1}{2} x^{2} a +\frac {5}{8} x -\frac {3}{8} a^{2} x^{3}-\frac {1}{4} a^{3} x^{4}}{\left (a^{2} x^{2}-1\right )^{2}}-\frac {3 \ln \left (a x -1\right )}{16 a}+\frac {3 \ln \left (a x +1\right )}{16 a}\) | \(62\) |
meijerg | \(\frac {\frac {x \sqrt {-a^{2}}\, \left (-3 a^{2} x^{2}+5\right )}{2 \left (-a^{2} x^{2}+1\right )^{2}}+\frac {3 \sqrt {-a^{2}}\, \arctanh \left (a x \right )}{2 a}}{4 \sqrt {-a^{2}}}+\frac {a \,x^{2} \left (-a^{2} x^{2}+2\right )}{4 \left (-a^{2} x^{2}+1\right )^{2}}\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 64, normalized size = 1.14 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs.
\(2 (46) = 92\).
time = 0.34, size = 99, normalized size = 1.77 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.16, size = 65, normalized size = 1.16 \begin {gather*} - \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 58, normalized size = 1.04 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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