Optimal. Leaf size=13 \[ \frac {4}{1-a x}+\log (x) \]
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Rubi [A]
time = 0.03, antiderivative size = 13, 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}{1-a x}+\log (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} \, dx &=\int \frac {e^{4 \tanh ^{-1}(a x)}}{x} \, dx\\ &=\int \frac {(1+a x)^2}{x (1-a x)^2} \, dx\\ &=\int \left (\frac {1}{x}+\frac {4 a}{(-1+a x)^2}\right ) \, dx\\ &=\frac {4}{1-a x}+\log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {4}{1-a x}+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 13, normalized size = 1.00
method | result | size |
default | \(-\frac {4}{a x -1}+\ln \left (x \right )\) | \(13\) |
norman | \(-\frac {4 a x}{a x -1}+\ln \left (x \right )\) | \(15\) |
risch | \(-\frac {4}{a x -1}+\ln \left (-x \right )\) | \(15\) |
meijerg | \(\frac {3 a x}{-a x +1}+\frac {2 a x}{-2 a x +2}+1+\ln \left (x \right )+\ln \left (-a \right )\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 12, normalized size = 0.92 \begin {gather*} -\frac {4}{a x - 1} + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 18, normalized size = 1.38 \begin {gather*} \frac {{\left (a x - 1\right )} \log \left (x\right ) - 4}{a x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 8, normalized size = 0.62 \begin {gather*} \log {\left (x \right )} - \frac {4}{a x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 57 vs.
\(2 (12) = 24\).
time = 0.40, size = 57, normalized size = 4.38 \begin {gather*} -a {\left (\frac {\log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a} - \frac {\log \left ({\left | -\frac {1}{a x - 1} - 1 \right |}\right )}{a} + \frac {4}{{\left (a x - 1\right )} a}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 12, normalized size = 0.92 \begin {gather*} \ln \left (x\right )-\frac {4}{a\,x-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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