Optimal. Leaf size=71 \[ \frac {a^3}{c (1-a x)}+\frac {4 a^3 \log (x)}{c}-\frac {4 a^3 \log (1-a x)}{c}-\frac {3 a^2}{c x}-\frac {a}{c x^2}-\frac {1}{3 c x^3} \]
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Rubi [A] time = 0.11, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6150, 44} \[ \frac {a^3}{c (1-a x)}-\frac {3 a^2}{c x}+\frac {4 a^3 \log (x)}{c}-\frac {4 a^3 \log (1-a x)}{c}-\frac {a}{c x^2}-\frac {1}{3 c x^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{x^4 \left (c-a^2 c x^2\right )} \, dx &=\frac {\int \frac {1}{x^4 (1-a x)^2} \, dx}{c}\\ &=\frac {\int \left (\frac {1}{x^4}+\frac {2 a}{x^3}+\frac {3 a^2}{x^2}+\frac {4 a^3}{x}+\frac {a^4}{(-1+a x)^2}-\frac {4 a^4}{-1+a x}\right ) \, dx}{c}\\ &=-\frac {1}{3 c x^3}-\frac {a}{c x^2}-\frac {3 a^2}{c x}+\frac {a^3}{c (1-a x)}+\frac {4 a^3 \log (x)}{c}-\frac {4 a^3 \log (1-a x)}{c}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 71, normalized size = 1.00 \[ \frac {a^3}{c (1-a x)}+\frac {4 a^3 \log (x)}{c}-\frac {4 a^3 \log (1-a x)}{c}-\frac {3 a^2}{c x}-\frac {a}{c x^2}-\frac {1}{3 c x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 83, normalized size = 1.17 \[ -\frac {12 \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 2 \, a x + 12 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (a x - 1\right ) - 12 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \relax (x) - 1}{3 \, {\left (a c x^{4} - c x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 64, normalized size = 0.90 \[ -\frac {4 \, a^{3} \log \left ({\left | a x - 1 \right |}\right )}{c} + \frac {4 \, a^{3} \log \left ({\left | x \right |}\right )}{c} - \frac {12 \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 2 \, a x - 1}{3 \, {\left (a x - 1\right )} c x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 69, normalized size = 0.97 \[ -\frac {1}{3 c \,x^{3}}-\frac {a}{c \,x^{2}}-\frac {3 a^{2}}{c x}+\frac {4 a^{3} \ln \relax (x )}{c}-\frac {a^{3}}{c \left (a x -1\right )}-\frac {4 a^{3} \ln \left (a x -1\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 64, normalized size = 0.90 \[ -\frac {4 \, a^{3} \log \left (a x - 1\right )}{c} + \frac {4 \, a^{3} \log \relax (x)}{c} - \frac {12 \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 2 \, a x - 1}{3 \, {\left (a c x^{4} - c x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 55, normalized size = 0.77 \[ \frac {8\,a^3\,\mathrm {atanh}\left (2\,a\,x-1\right )}{c}-\frac {-4\,a^3\,x^3+2\,a^2\,x^2+\frac {2\,a\,x}{3}+\frac {1}{3}}{c\,x^3-a\,c\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 54, normalized size = 0.76 \[ \frac {4 a^{3} \left (\log {\relax (x )} - \log {\left (x - \frac {1}{a} \right )}\right )}{c} + \frac {- 12 a^{3} x^{3} + 6 a^{2} x^{2} + 2 a x + 1}{3 a c x^{4} - 3 c x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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