Optimal. Leaf size=57 \[ \frac {1}{2 a c^3 (1-a x)}+\frac {5 \log (1-a x)}{4 a c^3}-\frac {\log (a x+1)}{4 a c^3}+\frac {x}{c^3} \]
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Rubi [A] time = 0.15, antiderivative size = 57, 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, 6131, 6129, 88} \[ \frac {1}{2 a c^3 (1-a x)}+\frac {5 \log (1-a x)}{4 a c^3}-\frac {\log (a x+1)}{4 a c^3}+\frac {x}{c^3} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6129
Rule 6131
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx\\ &=\frac {a^3 \int \frac {e^{-2 \tanh ^{-1}(a x)} x^3}{(1-a x)^3} \, dx}{c^3}\\ &=\frac {a^3 \int \frac {x^3}{(1-a x)^2 (1+a x)} \, dx}{c^3}\\ &=\frac {a^3 \int \left (\frac {1}{a^3}+\frac {1}{2 a^3 (-1+a x)^2}+\frac {5}{4 a^3 (-1+a x)}-\frac {1}{4 a^3 (1+a x)}\right ) \, dx}{c^3}\\ &=\frac {x}{c^3}+\frac {1}{2 a c^3 (1-a x)}+\frac {5 \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 = 0.13, size = 56, normalized size = 0.98 \[ -\frac {1}{2 a c^3 (a x-1)}+\frac {5 \log (1-a x)}{4 a c^3}-\frac {\log (a x+1)}{4 a c^3}+\frac {x}{c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 59, normalized size = 1.04 \[ \frac {4 \, a^{2} x^{2} - 4 \, a x - {\left (a x - 1\right )} \log \left (a x + 1\right ) + 5 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{4 \, {\left (a^{2} c^{3} x - a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 51, normalized size = 0.89 \[ \frac {x}{c^{3}} - \frac {\log \left ({\left | a x + 1 \right |}\right )}{4 \, a c^{3}} + \frac {5 \, \log \left ({\left | a x - 1 \right |}\right )}{4 \, a c^{3}} - \frac {1}{2 \, {\left (a x - 1\right )} a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 50, normalized size = 0.88 \[ \frac {x}{c^{3}}-\frac {1}{2 a \,c^{3} \left (a x -1\right )}+\frac {5 \ln \left (a x -1\right )}{4 a \,c^{3}}-\frac {\ln \left (a x +1\right )}{4 a \,c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 53, normalized size = 0.93 \[ -\frac {1}{2 \, {\left (a^{2} c^{3} x - a c^{3}\right )}} + \frac {x}{c^{3}} - \frac {\log \left (a x + 1\right )}{4 \, a c^{3}} + \frac {5 \, \log \left (a x - 1\right )}{4 \, a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 52, normalized size = 0.91 \[ \frac {x}{c^3}+\frac {1}{2\,a\,\left (c^3-a\,c^3\,x\right )}+\frac {5\,\ln \left (a\,x-1\right )}{4\,a\,c^3}-\frac {\ln \left (a\,x+1\right )}{4\,a\,c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 56, normalized size = 0.98 \[ a^{3} \left (- \frac {1}{2 a^{5} c^{3} x - 2 a^{4} c^{3}} + \frac {x}{a^{3} c^{3}} + \frac {\frac {5 \log {\left (x - \frac {1}{a} \right )}}{4} - \frac {\log {\left (x + \frac {1}{a} \right )}}{4}}{a^{4} c^{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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