Optimal. Leaf size=25 \[ c x-\frac {c \log (x)}{a}+\frac {4 c \log (1-a x)}{a} \]
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Rubi [A]
time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6266, 6264, 84}
\begin {gather*} -\frac {c \log (x)}{a}+\frac {4 c \log (1-a x)}{a}+c x \end {gather*}
Antiderivative was successfully verified.
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Rule 84
Rule 6264
Rule 6266
Rubi steps
\begin {align*} \int e^{4 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx &=-\frac {c \int \frac {e^{4 \tanh ^{-1}(a x)} (1-a x)}{x} \, dx}{a}\\ &=-\frac {c \int \frac {(1+a x)^2}{x (1-a x)} \, dx}{a}\\ &=-\frac {c \int \left (-a+\frac {1}{x}-\frac {4 a}{-1+a x}\right ) \, dx}{a}\\ &=c x-\frac {c \log (x)}{a}+\frac {4 c \log (1-a x)}{a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 25, normalized size = 1.00 \begin {gather*} c x-\frac {c \log (x)}{a}+\frac {4 c \log (1-a x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.85, size = 22, normalized size = 0.88
method | result | size |
default | \(\frac {c \left (a x -\ln \left (x \right )+4 \ln \left (a x -1\right )\right )}{a}\) | \(22\) |
risch | \(c x -\frac {c \ln \left (x \right )}{a}+\frac {4 c \ln \left (-a x +1\right )}{a}\) | \(26\) |
norman | \(\frac {c \,x^{3} a^{2}-c x}{a^{2} x^{2}-1}-\frac {c \ln \left (x \right )}{a}+\frac {4 c \ln \left (a x -1\right )}{a}\) | \(47\) |
meijerg | \(\frac {c \left (\frac {x \left (-a^{2}\right )^{\frac {5}{2}} \left (-10 a^{2} x^{2}+15\right )}{5 a^{4} \left (-a^{2} x^{2}+1\right )}-\frac {3 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{2 \sqrt {-a^{2}}}+\frac {3 c \left (\frac {a^{2} x^{2}}{-a^{2} x^{2}+1}+\ln \left (-a^{2} x^{2}+1\right )\right )}{2 a}-\frac {c \left (\frac {x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \left (-a^{2} x^{2}+1\right )}-\frac {\left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{\sqrt {-a^{2}}}-\frac {a c \,x^{2}}{-a^{2} x^{2}+1}-\frac {3 c \left (\frac {2 x \sqrt {-a^{2}}}{-2 a^{2} x^{2}+2}+\frac {\sqrt {-a^{2}}\, \arctanh \left (a x \right )}{a}\right )}{2 \sqrt {-a^{2}}}-\frac {c \left (\frac {2 a^{2} x^{2}}{-2 a^{2} x^{2}+2}-\ln \left (-a^{2} x^{2}+1\right )+1+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right )}{2 a}\) | \(270\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 24, normalized size = 0.96 \begin {gather*} c x + \frac {4 \, c \log \left (a x - 1\right )}{a} - \frac {c \log \left (x\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 23, normalized size = 0.92 \begin {gather*} \frac {a c x + 4 \, c \log \left (a x - 1\right ) - c \log \left (x\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 17, normalized size = 0.68 \begin {gather*} c x + \frac {c \left (- \log {\left (x \right )} + 4 \log {\left (x - \frac {1}{a} \right )}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 26, normalized size = 1.04 \begin {gather*} c x + \frac {4 \, c \log \left ({\left | a x - 1 \right |}\right )}{a} - \frac {c \log \left ({\left | x \right |}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 24, normalized size = 0.96 \begin {gather*} c\,x-\frac {c\,\ln \left (x\right )}{a}+\frac {4\,c\,\ln \left (a\,x-1\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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