Optimal. Leaf size=58 \[ -\frac {x}{c^3}-\frac {1}{2 a c^3 (1-a x)}-\frac {5 \log (1-a x)}{4 a c^3}+\frac {\log (1+a x)}{4 a c^3} \]
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Rubi [A]
time = 0.08, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6266, 6264, 90}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 6264
Rule 6266
Rubi steps
\begin {align*} \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 = 0.05, size = 57, normalized size = 0.98 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 49, normalized size = 0.84
method | result | size |
default | \(\frac {a^{3} \left (-\frac {x}{a^{3}}+\frac {\ln \left (a x +1\right )}{4 a^{4}}-\frac {5 \ln \left (a x -1\right )}{4 a^{4}}+\frac {1}{2 a^{4} \left (a x -1\right )}\right )}{c^{3}}\) | \(49\) |
risch | \(-\frac {x}{c^{3}}+\frac {1}{2 a \,c^{3} \left (a x -1\right )}+\frac {\ln \left (-a x -1\right )}{4 a \,c^{3}}-\frac {5 \ln \left (a x -1\right )}{4 a \,c^{3}}\) | \(52\) |
norman | \(\frac {-\frac {a^{3} x^{4}}{c}-\frac {3 x}{2 c}+\frac {a \,x^{2}}{c}+\frac {3 a^{2} x^{3}}{2 c}}{\left (a x +1\right ) c^{2} \left (a x -1\right )^{2}}-\frac {5 \ln \left (a x -1\right )}{4 a \,c^{3}}+\frac {\ln \left (a x +1\right )}{4 a \,c^{3}}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 54, normalized size = 0.93 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 59, normalized size = 1.02 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.28, size = 58, normalized size = 1.00 \begin {gather*} - 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 85, normalized size = 1.47 \begin {gather*} -\frac {{\left (a x + 1\right )} {\left (\frac {9}{a x + 1} - 4\right )}}{4 \, a c^{3} {\left (\frac {2}{a x + 1} - 1\right )}} + \frac {\log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a c^{3}} - \frac {5 \, \log \left ({\left | -\frac {2}{a x + 1} + 1 \right |}\right )}{4 \, a c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 53, normalized size = 0.91 \begin {gather*} \frac {\ln \left (a\,x+1\right )}{4\,a\,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 {x}{c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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