Optimal. Leaf size=39 \[ -\frac {1}{3 x^3}+\frac {a}{x^2}-\frac {2 a^2}{x}-2 a^3 \log (x)+2 a^3 \log (1+a x) \]
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Rubi [A]
time = 0.02, antiderivative size = 39, 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 (a x+1)-\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 = 39, 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 = 0.75, size = 38, normalized size = 0.97
method | result | size |
default | \(-\frac {1}{3 x^{3}}+\frac {a}{x^{2}}-\frac {2 a^{2}}{x}-2 a^{3} \ln \left (x \right )+2 a^{3} \ln \left (a x +1\right )\) | \(38\) |
risch | \(\frac {-2 a^{2} x^{2}+a x -\frac {1}{3}}{x^{3}}-2 a^{3} \ln \left (x \right )+2 a^{3} \ln \left (-a x -1\right )\) | \(38\) |
norman | \(\frac {-\frac {1}{3}+2 a^{4} x^{4}+\frac {2}{3} a x -a^{2} x^{2}}{x^{3} \left (a x +1\right )}-2 a^{3} \ln \left (x \right )+2 a^{3} \ln \left (a x +1\right )\) | \(53\) |
meijerg | \(-a^{3} \left (\frac {3 a x}{3 a x +3}+2 \ln \left (a x +1\right )-1-2 \ln \left (x \right )-2 \ln \left (a \right )-\frac {1}{a x}\right )+a^{3} \left (\frac {5 a x}{5 a x +5}+4 \ln \left (a x +1\right )-1-4 \ln \left (x \right )-4 \ln \left (a \right )-\frac {1}{3 a^{3} x^{3}}+\frac {1}{a^{2} x^{2}}-\frac {3}{a x}\right )\) | \(102\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 38, normalized size = 0.97 \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 = 1.10 \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 = 0.09, size = 36, normalized size = 0.92 \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 = 0.41, size = 62, normalized size = 1.59 \begin {gather*} -2 \, a^{3} \log \left ({\left | -\frac {1}{a x + 1} + 1 \right |}\right ) + \frac {10 \, a^{3} - \frac {24 \, a^{3}}{a x + 1} + \frac {15 \, a^{3}}{{\left (a x + 1\right )}^{2}}}{3 \, {\left (\frac {1}{a x + 1} - 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 32, normalized size = 0.82 \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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