Optimal. Leaf size=55 \[ \frac {c^3}{2 a^3 x^2}-\frac {5 c^3}{a^2 x}-\frac {11 c^3 \log (x)}{a}+\frac {16 c^3 \log (a x+1)}{a}+c^3 (-x) \]
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Rubi [A] time = 0.12, antiderivative size = 55, 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 = {6131, 6129, 88} \[ \frac {c^3}{2 a^3 x^2}-\frac {5 c^3}{a^2 x}-\frac {11 c^3 \log (x)}{a}+\frac {16 c^3 \log (a x+1)}{a}+c^3 (-x) \]
Antiderivative was successfully verified.
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Rule 88
Rule 6129
Rule 6131
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx &=-\frac {c^3 \int \frac {e^{-2 \tanh ^{-1}(a x)} (1-a x)^3}{x^3} \, dx}{a^3}\\ &=-\frac {c^3 \int \frac {(1-a x)^4}{x^3 (1+a x)} \, dx}{a^3}\\ &=-\frac {c^3 \int \left (a^3+\frac {1}{x^3}-\frac {5 a}{x^2}+\frac {11 a^2}{x}-\frac {16 a^3}{1+a x}\right ) \, dx}{a^3}\\ &=\frac {c^3}{2 a^3 x^2}-\frac {5 c^3}{a^2 x}-c^3 x-\frac {11 c^3 \log (x)}{a}+\frac {16 c^3 \log (1+a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 57, normalized size = 1.04 \[ \frac {c^3}{2 a^3 x^2}-\frac {5 c^3}{a^2 x}-\frac {11 c^3 \log (a x)}{a}+\frac {16 c^3 \log (a x+1)}{a}+c^3 (-x) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.42, size = 62, normalized size = 1.13 \[ -\frac {2 \, a^{3} c^{3} x^{3} - 32 \, a^{2} c^{3} x^{2} \log \left (a x + 1\right ) + 22 \, a^{2} c^{3} x^{2} \log \relax (x) + 10 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.79, size = 100, normalized size = 1.82 \[ -\frac {5 \, c^{3} \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a} - \frac {11 \, c^{3} \log \left ({\left | -\frac {1}{a x + 1} + 1 \right |}\right )}{a} - \frac {{\left (2 \, c^{3} + \frac {7 \, c^{3}}{a x + 1} - \frac {10 \, c^{3}}{{\left (a x + 1\right )}^{2}}\right )} {\left (a x + 1\right )}}{2 \, a {\left (\frac {1}{a x + 1} - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 54, normalized size = 0.98 \[ \frac {c^{3}}{2 x^{2} a^{3}}-\frac {5 c^{3}}{a^{2} x}-c^{3} x -\frac {11 c^{3} \ln \relax (x )}{a}+\frac {16 c^{3} \ln \left (a x +1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 52, normalized size = 0.95 \[ -c^{3} x + \frac {16 \, c^{3} \log \left (a x + 1\right )}{a} - \frac {11 \, c^{3} \log \relax (x)}{a} - \frac {10 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 51, normalized size = 0.93 \[ \frac {\frac {c^3}{2}-5\,a\,c^3\,x}{a^3\,x^2}-c^3\,x-\frac {11\,c^3\,\ln \relax (x)}{a}+\frac {16\,c^3\,\ln \left (a\,x+1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 44, normalized size = 0.80 \[ - c^{3} x - \frac {c^{3} \left (11 \log {\relax (x )} - 16 \log {\left (x + \frac {1}{a} \right )}\right )}{a} - \frac {10 a c^{3} x - c^{3}}{2 a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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