Optimal. Leaf size=32 \[ -\frac {1}{x}+\frac {4 a}{1-a x}+4 a \log (x)-4 a \log (1-a x) \]
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Rubi [A]
time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6302, 6261, 90}
\begin {gather*} \frac {4 a}{1-a x}+4 a \log (x)-4 a \log (1-a x)-\frac {1}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 6261
Rule 6302
Rubi steps
\begin {align*} \int \frac {e^{4 \coth ^{-1}(a x)}}{x^2} \, dx &=\int \frac {e^{4 \tanh ^{-1}(a x)}}{x^2} \, dx\\ &=\int \frac {(1+a x)^2}{x^2 (1-a x)^2} \, dx\\ &=\int \left (\frac {1}{x^2}+\frac {4 a}{x}+\frac {4 a^2}{(-1+a x)^2}-\frac {4 a^2}{-1+a x}\right ) \, dx\\ &=-\frac {1}{x}+\frac {4 a}{1-a x}+4 a \log (x)-4 a \log (1-a x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 32, normalized size = 1.00 \begin {gather*} -\frac {1}{x}+\frac {4 a}{1-a x}+4 a \log (x)-4 a \log (1-a x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 31, normalized size = 0.97
method | result | size |
default | \(-\frac {1}{x}+4 a \ln \left (x \right )-\frac {4 a}{a x -1}-4 a \ln \left (a x -1\right )\) | \(31\) |
risch | \(\frac {-5 a x +1}{x \left (a x -1\right )}+4 a \ln \left (-x \right )-4 a \ln \left (a x -1\right )\) | \(35\) |
norman | \(\frac {-5 a^{2} x^{2}+1}{x \left (a x -1\right )}+4 a \ln \left (x \right )-4 a \ln \left (a x -1\right )\) | \(37\) |
meijerg | \(\frac {a^{2} x}{-a x +1}+2 a \left (\frac {2 a x}{-2 a x +2}-\ln \left (-a x +1\right )+1+\ln \left (x \right )+\ln \left (-a \right )\right )-a \left (-\frac {3 a x}{-3 a x +3}+2 \ln \left (-a x +1\right )-1-2 \ln \left (x \right )-2 \ln \left (-a \right )+\frac {1}{a x}\right )\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 34, normalized size = 1.06 \begin {gather*} -4 \, a \log \left (a x - 1\right ) + 4 \, a \log \left (x\right ) - \frac {5 \, a x - 1}{a x^{2} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 55, normalized size = 1.72 \begin {gather*} -\frac {5 \, a x + 4 \, {\left (a^{2} x^{2} - a x\right )} \log \left (a x - 1\right ) - 4 \, {\left (a^{2} x^{2} - a x\right )} \log \left (x\right ) - 1}{a x^{2} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 26, normalized size = 0.81 \begin {gather*} 4 a \left (\log {\left (x \right )} - \log {\left (x - \frac {1}{a} \right )}\right ) + \frac {- 5 a x + 1}{a x^{2} - x} \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 = 1.25 \begin {gather*} 4 \, a \log \left ({\left | -\frac {1}{a x - 1} - 1 \right |}\right ) - \frac {4 \, a}{a x - 1} + \frac {a}{\frac {1}{a x - 1} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 28, normalized size = 0.88 \begin {gather*} 8\,a\,\mathrm {atanh}\left (2\,a\,x-1\right )+\frac {5\,a\,x-1}{x-a\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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