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 = 0.0434497, antiderivative size = 73, normalized size of antiderivative = 1., 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 6129
Rule 43
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*}
Mathematica [A] time = 0.0166093, 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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Maple [A] time = 0.029, size = 53, normalized size = 0.7 \begin{align*}{\frac{{c}^{3}{x}^{4}{a}^{3}}{4}}-{\frac{5\,{c}^{3}{x}^{3}{a}^{2}}{3}}+{\frac{11\,{c}^{3}{x}^{2}a}{2}}-15\,{c}^{3}x+16\,{\frac{{c}^{3}\ln \left ( ax+1 \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.945315, size = 70, normalized size = 0.96 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66449, size = 128, normalized size = 1.75 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.334794, size = 56, normalized size = 0.77 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18512, size = 111, normalized size = 1.52 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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