Optimal. Leaf size=41 \[ -\frac {1}{3 x^3}-\frac {a}{x^2}-\frac {2 a^2}{x}+2 a^3 \log (x)-2 a^3 \log (1-a x) \]
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Rubi [A]
time = 0.03, antiderivative size = 41, 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, 78}
\begin {gather*} 2 a^3 \log (x)-2 a^3 \log (1-a x)-\frac {2 a^2}{x}-\frac {a}{x^2}-\frac {1}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 6261
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{x^4} \, dx &=\int \frac {1+a x}{x^4 (1-a x)} \, dx\\ &=\int \left (\frac {1}{x^4}+\frac {2 a}{x^3}+\frac {2 a^2}{x^2}+\frac {2 a^3}{x}-\frac {2 a^4}{-1+a x}\right ) \, dx\\ &=-\frac {1}{3 x^3}-\frac {a}{x^2}-\frac {2 a^2}{x}+2 a^3 \log (x)-2 a^3 \log (1-a x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 41, normalized size = 1.00 \begin {gather*} -\frac {1}{3 x^3}-\frac {a}{x^2}-\frac {2 a^2}{x}+2 a^3 \log (x)-2 a^3 \log (1-a x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.99, size = 39, normalized size = 0.95
method | result | size |
norman | \(\frac {-\frac {1}{3}-2 a^{2} x^{2}-a x}{x^{3}}+2 a^{3} \ln \left (x \right )-2 a^{3} \ln \left (a x -1\right )\) | \(38\) |
default | \(-\frac {1}{3 x^{3}}-\frac {a}{x^{2}}-\frac {2 a^{2}}{x}+2 a^{3} \ln \left (x \right )-2 a^{3} \ln \left (a x -1\right )\) | \(39\) |
risch | \(\frac {-\frac {1}{3}-2 a^{2} x^{2}-a x}{x^{3}}+2 a^{3} \ln \left (-x \right )-2 a^{3} \ln \left (a x -1\right )\) | \(40\) |
meijerg | \(-\frac {a^{4} \left (-\frac {2}{x \sqrt {-a^{2}}}+\frac {2 a \arctanh \left (a x \right )}{\sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}}-a^{3} \left (\ln \left (-a^{2} x^{2}+1\right )-2 \ln \left (x \right )-\ln \left (-a^{2}\right )+\frac {1}{a^{2} x^{2}}\right )+\frac {a^{4} \left (-\frac {2 a^{2}}{x \left (-a^{2}\right )^{\frac {3}{2}}}-\frac {2}{3 x^{3} \left (-a^{2}\right )^{\frac {3}{2}}}+\frac {2 a^{3} \arctanh \left (a x \right )}{\left (-a^{2}\right )^{\frac {3}{2}}}\right )}{2 \sqrt {-a^{2}}}\) | \(133\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 38, normalized size = 0.93 \begin {gather*} -2 \, a^{3} \log \left (a x - 1\right ) + 2 \, a^{3} \log \left (x\right ) - \frac {6 \, a^{2} x^{2} + 3 \, a x + 1}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 43, normalized size = 1.05 \begin {gather*} -\frac {6 \, a^{3} x^{3} \log \left (a x - 1\right ) - 6 \, a^{3} x^{3} \log \left (x\right ) + 6 \, a^{2} x^{2} + 3 \, a x + 1}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 36, normalized size = 0.88 \begin {gather*} - 2 a^{3} \left (- \log {\left (x \right )} + \log {\left (x - \frac {1}{a} \right )}\right ) - \frac {6 a^{2} x^{2} + 3 a x + 1}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 40, normalized size = 0.98 \begin {gather*} -2 \, a^{3} \log \left ({\left | a x - 1 \right |}\right ) + 2 \, a^{3} \log \left ({\left | x \right |}\right ) - \frac {6 \, a^{2} x^{2} + 3 \, a x + 1}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 31, normalized size = 0.76 \begin {gather*} 4\,a^3\,\mathrm {atanh}\left (2\,a\,x-1\right )-\frac {2\,a^2\,x^2+a\,x+\frac {1}{3}}{x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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