Optimal. Leaf size=33 \[ \frac {c}{a^2 x}-\frac {4 c \log (x)}{a}+\frac {8 c \log (1-a x)}{a}+c x \]
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Rubi [A] time = 0.08, antiderivative size = 33, 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 = {6157, 6150, 88} \[ \frac {c}{a^2 x}-\frac {4 c \log (x)}{a}+\frac {8 c \log (1-a x)}{a}+c x \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6157
Rubi steps
\begin {align*} \int e^{4 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx &=-\frac {c \int \frac {e^{4 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )}{x^2} \, dx}{a^2}\\ &=-\frac {c \int \frac {(1+a x)^3}{x^2 (1-a x)} \, dx}{a^2}\\ &=-\frac {c \int \left (-a^2+\frac {1}{x^2}+\frac {4 a}{x}-\frac {8 a^2}{-1+a x}\right ) \, dx}{a^2}\\ &=\frac {c}{a^2 x}+c x-\frac {4 c \log (x)}{a}+\frac {8 c \log (1-a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 1.00 \[ \frac {c}{a^2 x}-\frac {4 c \log (x)}{a}+\frac {8 c \log (1-a x)}{a}+c x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 35, normalized size = 1.06 \[ \frac {a^{2} c x^{2} + 8 \, a c x \log \left (a x - 1\right ) - 4 \, a c x \log \relax (x) + c}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 34, normalized size = 1.03 \[ c x + \frac {8 \, c \log \left ({\left | a x - 1 \right |}\right )}{a} - \frac {4 \, c \log \left ({\left | x \right |}\right )}{a} + \frac {c}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 33, normalized size = 1.00 \[ c x +\frac {c}{a^{2} x}-\frac {4 c \ln \relax (x )}{a}+\frac {8 c \ln \left (a x -1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 32, normalized size = 0.97 \[ c x + \frac {8 \, c \log \left (a x - 1\right )}{a} - \frac {4 \, c \log \relax (x)}{a} + \frac {c}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 32, normalized size = 0.97 \[ c\,x+\frac {c}{a^2\,x}-\frac {4\,c\,\ln \relax (x)}{a}+\frac {8\,c\,\ln \left (a\,x-1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 26, normalized size = 0.79 \[ c x + \frac {4 c \left (- \log {\relax (x )} + 2 \log {\left (x - \frac {1}{a} \right )}\right )}{a} + \frac {c}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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