Optimal. Leaf size=54 \[ -\frac {1}{3 x^3}-\frac {2 a}{x^2}-\frac {8 a^2}{x}+\frac {4 a^3}{1-a x}+12 a^3 \log (x)-12 a^3 \log (1-a x) \]
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Rubi [A]
time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6261, 90}
\begin {gather*} \frac {4 a^3}{1-a x}+12 a^3 \log (x)-12 a^3 \log (1-a x)-\frac {8 a^2}{x}-\frac {2 a}{x^2}-\frac {1}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 6261
Rubi steps
\begin {align*} \int \frac {e^{4 \tanh ^{-1}(a x)}}{x^4} \, dx &=\int \frac {(1+a x)^2}{x^4 (1-a x)^2} \, dx\\ &=\int \left (\frac {1}{x^4}+\frac {4 a}{x^3}+\frac {8 a^2}{x^2}+\frac {12 a^3}{x}+\frac {4 a^4}{(-1+a x)^2}-\frac {12 a^4}{-1+a x}\right ) \, dx\\ &=-\frac {1}{3 x^3}-\frac {2 a}{x^2}-\frac {8 a^2}{x}+\frac {4 a^3}{1-a x}+12 a^3 \log (x)-12 a^3 \log (1-a x)\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 54, normalized size = 1.00 \begin {gather*} -\frac {1}{3 x^3}-\frac {2 a}{x^2}-\frac {8 a^2}{x}+\frac {4 a^3}{1-a x}+12 a^3 \log (x)-12 a^3 \log (1-a x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.08, size = 51, normalized size = 0.94
method | result | size |
default | \(-\frac {1}{3 x^{3}}-\frac {2 a}{x^{2}}-\frac {8 a^{2}}{x}+12 a^{3} \ln \left (x \right )-\frac {4 a^{3}}{a x -1}-12 a^{3} \ln \left (a x -1\right )\) | \(51\) |
risch | \(\frac {-12 a^{3} x^{3}+6 a^{2} x^{2}+\frac {5}{3} a x +\frac {1}{3}}{x^{3} \left (a x -1\right )}-12 a^{3} \ln \left (a x -1\right )+12 a^{3} \ln \left (-x \right )\) | \(55\) |
norman | \(\frac {\frac {1}{3}-6 a^{5} x^{5}+2 a x +\frac {23}{3} a^{2} x^{2}-12 a^{4} x^{4}}{\left (a^{2} x^{2}-1\right ) x^{3}}+12 a^{3} \ln \left (x \right )-12 a^{3} \ln \left (a x -1\right )\) | \(65\) |
meijerg | \(\frac {a^{4} \left (\frac {2 x \sqrt {-a^{2}}}{-2 a^{2} x^{2}+2}+\frac {\sqrt {-a^{2}}\, \arctanh \left (a x \right )}{a}\right )}{2 \sqrt {-a^{2}}}+2 a^{3} \left (\frac {2 a^{2} x^{2}}{-2 a^{2} x^{2}+2}-\ln \left (-a^{2} x^{2}+1\right )+1+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right )-\frac {3 a^{4} \left (-\frac {2 \left (-3 a^{2} x^{2}+2\right )}{x \sqrt {-a^{2}}\, \left (-2 a^{2} x^{2}+2\right )}+\frac {3 a \arctanh \left (a x \right )}{\sqrt {-a^{2}}}\right )}{\sqrt {-a^{2}}}-2 a^{3} \left (-\frac {3 a^{2} x^{2}}{-3 a^{2} x^{2}+3}+2 \ln \left (-a^{2} x^{2}+1\right )-1-4 \ln \left (x \right )-2 \ln \left (-a^{2}\right )+\frac {1}{a^{2} x^{2}}\right )+\frac {a^{4} \left (-\frac {2 \left (-15 a^{4} x^{4}+10 a^{2} x^{2}+2\right )}{3 x^{3} \left (-a^{2}\right )^{\frac {3}{2}} \left (-2 a^{2} x^{2}+2\right )}+\frac {5 a^{3} \arctanh \left (a x \right )}{\left (-a^{2}\right )^{\frac {3}{2}}}\right )}{2 \sqrt {-a^{2}}}\) | \(293\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 56, normalized size = 1.04 \begin {gather*} -12 \, a^{3} \log \left (a x - 1\right ) + 12 \, a^{3} \log \left (x\right ) - \frac {36 \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 5 \, a x - 1}{3 \, {\left (a x^{4} - x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 81, normalized size = 1.50 \begin {gather*} -\frac {36 \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 5 \, a x + 36 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (a x - 1\right ) - 36 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (x\right ) - 1}{3 \, {\left (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 = 0.16, size = 49, normalized size = 0.91 \begin {gather*} 12 a^{3} \left (\log {\left (x \right )} - \log {\left (x - \frac {1}{a} \right )}\right ) + \frac {- 36 a^{3} x^{3} + 18 a^{2} x^{2} + 5 a x + 1}{3 a x^{4} - 3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 55, normalized size = 1.02 \begin {gather*} -12 \, a^{3} \log \left ({\left | a x - 1 \right |}\right ) + 12 \, a^{3} \log \left ({\left | x \right |}\right ) - \frac {36 \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 5 \, a x - 1}{3 \, {\left (a x - 1\right )} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.82, size = 49, normalized size = 0.91 \begin {gather*} 24\,a^3\,\mathrm {atanh}\left (2\,a\,x-1\right )+\frac {-12\,a^3\,x^3+6\,a^2\,x^2+\frac {5\,a\,x}{3}+\frac {1}{3}}{a\,x^4-x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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