Optimal. Leaf size=71 \[ -\frac {1}{3 c x^3}-\frac {a}{c x^2}-\frac {3 a^2}{c x}+\frac {a^3}{c (1-a x)}+\frac {4 a^3 \log (x)}{c}-\frac {4 a^3 \log (1-a x)}{c} \]
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Rubi [A]
time = 0.08, 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 = {6285, 46}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 6285
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 = 0.03, size = 71, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 55, normalized size = 0.77
method | result | size |
default | \(\frac {-\frac {1}{3 x^{3}}-\frac {a}{x^{2}}-\frac {3 a^{2}}{x}+4 a^{3} \ln \left (x \right )-\frac {a^{3}}{a x -1}-4 a^{3} \ln \left (a x -1\right )}{c}\) | \(55\) |
risch | \(\frac {-4 a^{3} x^{3}+2 a^{2} x^{2}+\frac {2}{3} a x +\frac {1}{3}}{c \,x^{3} \left (a x -1\right )}+\frac {4 a^{3} \ln \left (-x \right )}{c}-\frac {4 a^{3} \ln \left (a x -1\right )}{c}\) | \(64\) |
norman | \(\frac {\frac {a x}{c}+\frac {1}{3 c}+\frac {8 a^{2} x^{2}}{3 c}-\frac {4 a^{4} x^{4}}{c}-\frac {2 a^{5} x^{5}}{c}}{\left (a^{2} x^{2}-1\right ) x^{3}}+\frac {4 a^{3} \ln \left (x \right )}{c}-\frac {4 a^{3} \ln \left (a x -1\right )}{c}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 64, normalized size = 0.90 \begin {gather*} -\frac {4 \, a^{3} \log \left (a x - 1\right )}{c} + \frac {4 \, a^{3} \log \left (x\right )}{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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 83, normalized size = 1.17 \begin {gather*} -\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 \left (x\right ) - 1}{3 \, {\left (a c x^{4} - c x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.16, size = 54, normalized size = 0.76 \begin {gather*} \frac {4 a^{3} \left (\log {\left (x \right )} - \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 64, normalized size = 0.90 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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