Optimal. Leaf size=110 \[ \frac {39}{16 a c^3 (1-a x)}-\frac {1}{16 a c^3 (a x+1)}-\frac {5}{8 a c^3 (1-a x)^2}+\frac {1}{12 a c^3 (1-a x)^3}+\frac {9 \log (1-a x)}{4 a c^3}-\frac {\log (a x+1)}{4 a c^3}+\frac {x}{c^3} \]
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Rubi [A] time = 0.20, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6167, 6157, 6150, 88} \[ \frac {39}{16 a c^3 (1-a x)}-\frac {1}{16 a c^3 (a x+1)}-\frac {5}{8 a c^3 (1-a x)^2}+\frac {1}{12 a c^3 (1-a x)^3}+\frac {9 \log (1-a x)}{4 a c^3}-\frac {\log (a x+1)}{4 a c^3}+\frac {x}{c^3} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6157
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx\\ &=\frac {a^6 \int \frac {e^{2 \tanh ^{-1}(a x)} x^6}{\left (1-a^2 x^2\right )^3} \, dx}{c^3}\\ &=\frac {a^6 \int \frac {x^6}{(1-a x)^4 (1+a x)^2} \, dx}{c^3}\\ &=\frac {a^6 \int \left (\frac {1}{a^6}+\frac {1}{4 a^6 (-1+a x)^4}+\frac {5}{4 a^6 (-1+a x)^3}+\frac {39}{16 a^6 (-1+a x)^2}+\frac {9}{4 a^6 (-1+a x)}+\frac {1}{16 a^6 (1+a x)^2}-\frac {1}{4 a^6 (1+a x)}\right ) \, dx}{c^3}\\ &=\frac {x}{c^3}+\frac {1}{12 a c^3 (1-a x)^3}-\frac {5}{8 a c^3 (1-a x)^2}+\frac {39}{16 a c^3 (1-a x)}-\frac {1}{16 a c^3 (1+a x)}+\frac {9 \log (1-a x)}{4 a c^3}-\frac {\log (1+a x)}{4 a c^3}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 82, normalized size = 0.75 \[ \frac {\frac {2 \left (6 a^5 x^5-12 a^4 x^4-15 a^3 x^3+24 a^2 x^2+7 a x-11\right )}{(a x-1)^3 (a x+1)}+27 \log (1-a x)-3 \log (a x+1)}{12 a c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 137, normalized size = 1.25 \[ \frac {12 \, a^{5} x^{5} - 24 \, a^{4} x^{4} - 30 \, a^{3} x^{3} + 48 \, a^{2} x^{2} + 14 \, a x - 3 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x + 1\right ) + 27 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x - 1\right ) - 22}{12 \, {\left (a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 80, normalized size = 0.73 \[ \frac {x}{c^{3}} - \frac {\log \left ({\left | a x + 1 \right |}\right )}{4 \, a c^{3}} + \frac {9 \, \log \left ({\left | a x - 1 \right |}\right )}{4 \, a c^{3}} - \frac {15 \, a^{3} x^{3} - 12 \, a^{2} x^{2} - 13 \, a x + 11}{6 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{3} a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 95, normalized size = 0.86 \[ \frac {x}{c^{3}}-\frac {1}{12 c^{3} a \left (a x -1\right )^{3}}-\frac {5}{8 c^{3} a \left (a x -1\right )^{2}}-\frac {39}{16 a \,c^{3} \left (a x -1\right )}+\frac {9 \ln \left (a x -1\right )}{4 c^{3} a}-\frac {1}{16 a \,c^{3} \left (a x +1\right )}-\frac {\ln \left (a x +1\right )}{4 a \,c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 97, normalized size = 0.88 \[ -\frac {15 \, a^{3} x^{3} - 12 \, a^{2} x^{2} - 13 \, a x + 11}{6 \, {\left (a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{3}\right )}} + \frac {x}{c^{3}} - \frac {\log \left (a x + 1\right )}{4 \, a c^{3}} + \frac {9 \, \log \left (a x - 1\right )}{4 \, a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 94, normalized size = 0.85 \[ \frac {x}{c^3}-\frac {\frac {13\,x}{6}+2\,a\,x^2-\frac {11}{6\,a}-\frac {5\,a^2\,x^3}{2}}{-a^4\,c^3\,x^4+2\,a^3\,c^3\,x^3-2\,a\,c^3\,x+c^3}+\frac {9\,\ln \left (a\,x-1\right )}{4\,a\,c^3}-\frac {\ln \left (a\,x+1\right )}{4\,a\,c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.59, size = 102, normalized size = 0.93 \[ a^{6} \left (\frac {- 15 a^{3} x^{3} + 12 a^{2} x^{2} + 13 a x - 11}{6 a^{11} c^{3} x^{4} - 12 a^{10} c^{3} x^{3} + 12 a^{8} c^{3} x - 6 a^{7} c^{3}} + \frac {x}{a^{6} c^{3}} + \frac {\frac {9 \log {\left (x - \frac {1}{a} \right )}}{4} - \frac {\log {\left (x + \frac {1}{a} \right )}}{4}}{a^{7} c^{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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