Optimal. Leaf size=56 \[ \frac {1}{8 a^4 (1-a x)^2}-\frac {1}{2 a^4 (1-a x)}+\frac {1}{8 a^4 (1+a x)}+\frac {3 \tanh ^{-1}(a x)}{8 a^4} \]
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Rubi [A]
time = 0.08, 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 = {6285, 90, 213}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 213
Rule 6285
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 = 0.02, size = 53, normalized size = 0.95 \begin {gather*} \frac {-2-a x+5 a^2 x^2+3 (-1+a x)^2 (1+a x) \tanh ^{-1}(a x)}{8 a^4 (-1+a x)^2 (1+a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 60, normalized size = 1.07
method | result | size |
norman | \(\frac {\frac {x^{4}}{4}-\frac {3 x}{8 a^{3}}+\frac {5 x^{3}}{8 a}}{\left (a^{2} x^{2}-1\right )^{2}}-\frac {3 \ln \left (a x -1\right )}{16 a^{4}}+\frac {3 \ln \left (a x +1\right )}{16 a^{4}}\) | \(56\) |
default | \(\frac {1}{8 a^{4} \left (a x +1\right )}+\frac {3 \ln \left (a x +1\right )}{16 a^{4}}+\frac {1}{8 a^{4} \left (a x -1\right )^{2}}+\frac {1}{2 a^{4} \left (a x -1\right )}-\frac {3 \ln \left (a x -1\right )}{16 a^{4}}\) | \(60\) |
risch | \(\frac {\frac {5 x^{2}}{8 a^{2}}-\frac {x}{8 a^{3}}-\frac {1}{4 a^{4}}}{\left (a x -1\right ) \left (a^{2} x^{2}-1\right )}-\frac {3 \ln \left (a x -1\right )}{16 a^{4}}+\frac {3 \ln \left (-a x -1\right )}{16 a^{4}}\) | \(64\) |
meijerg | \(\frac {x^{4}}{4 \left (-a^{2} x^{2}+1\right )^{2}}+\frac {-\frac {x \left (-a^{2}\right )^{\frac {5}{2}} \left (-25 a^{2} x^{2}+15\right )}{10 a^{4} \left (-a^{2} x^{2}+1\right )^{2}}+\frac {3 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{2 a^{5}}}{4 a^{3} \sqrt {-a^{2}}}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 66, normalized size = 1.18 \begin {gather*} \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}} \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. 101 vs.
\(2 (46) = 92\).
time = 0.33, size = 101, normalized size = 1.80 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 66, normalized size = 1.18 \begin {gather*} - \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 58, normalized size = 1.04 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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