Optimal. Leaf size=73 \[ \frac {c^3 (1-a x)^4}{4 a}+\frac {2 c^3 (1-a x)^3}{3 a}+\frac {2 c^3 (1-a x)^2}{a}+\frac {16 c^3 \log (a x+1)}{a}-8 c^3 x \]
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Rubi [A] time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6129, 43} \[ \frac {c^3 (1-a x)^4}{4 a}+\frac {2 c^3 (1-a x)^3}{3 a}+\frac {2 c^3 (1-a x)^2}{a}+\frac {16 c^3 \log (a x+1)}{a}-8 c^3 x \]
Antiderivative was successfully verified.
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Rule 43
Rule 6129
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} (c-a c x)^3 \, dx &=c^3 \int \frac {(1-a x)^4}{1+a x} \, dx\\ &=c^3 \int \left (-8-4 (1-a x)-2 (1-a x)^2-(1-a x)^3+\frac {16}{1+a x}\right ) \, dx\\ &=-8 c^3 x+\frac {2 c^3 (1-a x)^2}{a}+\frac {2 c^3 (1-a x)^3}{3 a}+\frac {c^3 (1-a x)^4}{4 a}+\frac {16 c^3 \log (1+a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 0.66 \[ \frac {c^3 \left (3 a^4 x^4-20 a^3 x^3+66 a^2 x^2-180 a x+192 \log (a x+1)+35\right )}{12 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 57, normalized size = 0.78 \[ \frac {3 \, a^{4} c^{3} x^{4} - 20 \, a^{3} c^{3} x^{3} + 66 \, a^{2} c^{3} x^{2} - 180 \, a c^{3} x + 192 \, c^{3} \log \left (a x + 1\right )}{12 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 82, normalized size = 1.12 \[ \frac {{\left (3 \, c^{3} - \frac {32 \, c^{3}}{a x + 1} + \frac {144 \, c^{3}}{{\left (a x + 1\right )}^{2}} - \frac {384 \, c^{3}}{{\left (a x + 1\right )}^{3}}\right )} {\left (a x + 1\right )}^{4}}{12 \, a} - \frac {16 \, c^{3} \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 53, normalized size = 0.73 \[ \frac {c^{3} x^{4} a^{3}}{4}-\frac {5 a^{2} c^{3} x^{3}}{3}+\frac {11 c^{3} x^{2} a}{2}-15 c^{3} x +\frac {16 c^{3} \ln \left (a x +1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 52, normalized size = 0.71 \[ \frac {1}{4} \, a^{3} c^{3} x^{4} - \frac {5}{3} \, a^{2} c^{3} x^{3} + \frac {11}{2} \, a c^{3} x^{2} - 15 \, c^{3} x + \frac {16 \, c^{3} \log \left (a x + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.79, size = 52, normalized size = 0.71 \[ \frac {11\,a\,c^3\,x^2}{2}-15\,c^3\,x-\frac {5\,a^2\,c^3\,x^3}{3}+\frac {a^3\,c^3\,x^4}{4}+\frac {16\,c^3\,\ln \left (a\,x+1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 56, normalized size = 0.77 \[ \frac {a^{3} c^{3} x^{4}}{4} - \frac {5 a^{2} c^{3} x^{3}}{3} + \frac {11 a c^{3} x^{2}}{2} - 15 c^{3} x + \frac {16 c^{3} \log {\left (a x + 1 \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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