Optimal. Leaf size=102 \[ -\frac {1}{3 x^3}-\frac {a}{2 x^2}-\frac {3 a^2}{x}+\frac {a^3}{8 (1-a x)^2}+\frac {5 a^3}{4 (1-a x)}-\frac {a^3}{8 (1+a x)}+3 a^3 \log (x)-\frac {59}{16} a^3 \log (1-a x)+\frac {11}{16} a^3 \log (1+a x) \]
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Rubi [A]
time = 0.10, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6285, 90}
\begin {gather*} \frac {5 a^3}{4 (1-a x)}-\frac {a^3}{8 (a x+1)}+\frac {a^3}{8 (1-a x)^2}+3 a^3 \log (x)-\frac {59}{16} a^3 \log (1-a x)+\frac {11}{16} a^3 \log (a x+1)-\frac {3 a^2}{x}-\frac {a}{2 x^2}-\frac {1}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 6285
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^4 \left (1-a^2 x^2\right )^{5/2}} \, dx &=\int \frac {1}{x^4 (1-a x)^3 (1+a x)^2} \, dx\\ &=\int \left (\frac {1}{x^4}+\frac {a}{x^3}+\frac {3 a^2}{x^2}+\frac {3 a^3}{x}-\frac {a^4}{4 (-1+a x)^3}+\frac {5 a^4}{4 (-1+a x)^2}-\frac {59 a^4}{16 (-1+a x)}+\frac {a^4}{8 (1+a x)^2}+\frac {11 a^4}{16 (1+a x)}\right ) \, dx\\ &=-\frac {1}{3 x^3}-\frac {a}{2 x^2}-\frac {3 a^2}{x}+\frac {a^3}{8 (1-a x)^2}+\frac {5 a^3}{4 (1-a x)}-\frac {a^3}{8 (1+a x)}+3 a^3 \log (x)-\frac {59}{16} a^3 \log (1-a x)+\frac {11}{16} a^3 \log (1+a x)\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 91, normalized size = 0.89 \begin {gather*} \frac {1}{48} \left (-\frac {16}{x^3}-\frac {24 a}{x^2}-\frac {144 a^2}{x}+\frac {60 a^3}{1-a x}+\frac {6 a^3}{(-1+a x)^2}-\frac {6 a^3}{1+a x}+144 a^3 \log (x)-177 a^3 \log (1-a x)+33 a^3 \log (1+a x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 86, normalized size = 0.84
method | result | size |
default | \(-\frac {a^{3}}{8 \left (a x +1\right )}+\frac {11 a^{3} \ln \left (a x +1\right )}{16}-\frac {1}{3 x^{3}}-\frac {a}{2 x^{2}}-\frac {3 a^{2}}{x}+3 a^{3} \ln \left (x \right )+\frac {a^{3}}{8 \left (a x -1\right )^{2}}-\frac {5 a^{3}}{4 \left (a x -1\right )}-\frac {59 a^{3} \ln \left (a x -1\right )}{16}\) | \(86\) |
norman | \(\frac {-\frac {1}{3}-\frac {9}{4} a^{7} x^{7}+3 x^{5} a^{5}-\frac {1}{2} a x -\frac {7}{3} a^{2} x^{2}+\frac {175}{24} a^{4} x^{4}-\frac {35}{8} x^{6} a^{6}}{\left (a^{2} x^{2}-1\right )^{2} x^{3}}+3 a^{3} \ln \left (x \right )-\frac {59 a^{3} \ln \left (a x -1\right )}{16}+\frac {11 a^{3} \ln \left (a x +1\right )}{16}\) | \(92\) |
risch | \(\frac {-\frac {35}{8} x^{5} a^{5}+\frac {23}{8} a^{4} x^{4}+\frac {53}{12} a^{3} x^{3}-\frac {13}{6} a^{2} x^{2}-\frac {1}{6} a x -\frac {1}{3}}{x^{3} \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}-\frac {59 a^{3} \ln \left (-a x +1\right )}{16}+\frac {11 a^{3} \ln \left (a x +1\right )}{16}+3 a^{3} \ln \left (-x \right )\) | \(94\) |
meijerg | \(\frac {a^{4} \left (-\frac {105 x^{6} a^{6}-175 a^{4} x^{4}+56 a^{2} x^{2}+8}{6 x^{3} \left (-a^{2}\right )^{\frac {3}{2}} \left (-a^{2} x^{2}+1\right )^{2}}+\frac {35 a^{3} \arctanh \left (a x \right )}{2 \left (-a^{2}\right )^{\frac {3}{2}}}\right )}{4 \sqrt {-a^{2}}}-\frac {a^{3} \left (-\frac {a^{2} x^{2} \left (-5 a^{2} x^{2}+6\right )}{\left (-a^{2} x^{2}+1\right )^{2}}+6 \ln \left (-a^{2} x^{2}+1\right )-5-12 \ln \left (x \right )-6 \ln \left (-a^{2}\right )+\frac {2}{a^{2} x^{2}}\right )}{4}\) | \(151\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 97, normalized size = 0.95 \begin {gather*} \frac {11}{16} \, a^{3} \log \left (a x + 1\right ) - \frac {59}{16} \, a^{3} \log \left (a x - 1\right ) + 3 \, a^{3} \log \left (x\right ) - \frac {105 \, a^{5} x^{5} - 69 \, a^{4} x^{4} - 106 \, a^{3} x^{3} + 52 \, a^{2} x^{2} + 4 \, a x + 8}{24 \, {\left (a^{3} x^{6} - a^{2} x^{5} - a x^{4} + x^{3}\right )}} \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. 180 vs.
\(2 (86) = 172\).
time = 0.34, size = 180, normalized size = 1.76 \begin {gather*} -\frac {210 \, a^{5} x^{5} - 138 \, a^{4} x^{4} - 212 \, a^{3} x^{3} + 104 \, a^{2} x^{2} + 8 \, a x - 33 \, {\left (a^{6} x^{6} - a^{5} x^{5} - a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (a x + 1\right ) + 177 \, {\left (a^{6} x^{6} - a^{5} x^{5} - a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (a x - 1\right ) - 144 \, {\left (a^{6} x^{6} - a^{5} x^{5} - a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (x\right ) + 16}{48 \, {\left (a^{3} x^{6} - a^{2} x^{5} - a x^{4} + x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.48, size = 104, normalized size = 1.02 \begin {gather*} 3 a^{3} \log {\left (x \right )} - \frac {59 a^{3} \log {\left (x - \frac {1}{a} \right )}}{16} + \frac {11 a^{3} \log {\left (x + \frac {1}{a} \right )}}{16} - \frac {105 a^{5} x^{5} - 69 a^{4} x^{4} - 106 a^{3} x^{3} + 52 a^{2} x^{2} + 4 a x + 8}{24 a^{3} x^{6} - 24 a^{2} x^{5} - 24 a x^{4} + 24 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 90, normalized size = 0.88 \begin {gather*} \frac {11}{16} \, a^{3} \log \left ({\left | a x + 1 \right |}\right ) - \frac {59}{16} \, a^{3} \log \left ({\left | a x - 1 \right |}\right ) + 3 \, a^{3} \log \left ({\left | x \right |}\right ) - \frac {105 \, a^{5} x^{5} - 69 \, a^{4} x^{4} - 106 \, a^{3} x^{3} + 52 \, a^{2} x^{2} + 4 \, a x + 8}{24 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.94, size = 97, normalized size = 0.95 \begin {gather*} 3\,a^3\,\ln \left (x\right )-\frac {59\,a^3\,\ln \left (a\,x-1\right )}{16}+\frac {11\,a^3\,\ln \left (a\,x+1\right )}{16}+\frac {\frac {35\,a^5\,x^5}{8}-\frac {23\,a^4\,x^4}{8}-\frac {53\,a^3\,x^3}{12}+\frac {13\,a^2\,x^2}{6}+\frac {a\,x}{6}+\frac {1}{3}}{-a^3\,x^6+a^2\,x^5+a\,x^4-x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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