Optimal. Leaf size=54 \[ \frac{c^2 (1-a x)^3}{3 a}+\frac{c^2 (1-a x)^2}{a}+\frac{8 c^2 \log (a x+1)}{a}-4 c^2 x \]
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Rubi [A] time = 0.0363179, antiderivative size = 54, 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} \[ \frac{c^2 (1-a x)^3}{3 a}+\frac{c^2 (1-a x)^2}{a}+\frac{8 c^2 \log (a x+1)}{a}-4 c^2 x \]
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)^2 \, dx &=c^2 \int \frac{(1-a x)^3}{1+a x} \, dx\\ &=c^2 \int \left (-4-2 (1-a x)-(1-a x)^2+\frac{8}{1+a x}\right ) \, dx\\ &=-4 c^2 x+\frac{c^2 (1-a x)^2}{a}+\frac{c^2 (1-a x)^3}{3 a}+\frac{8 c^2 \log (1+a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0134649, size = 39, normalized size = 0.72 \[ -\frac{c^2 \left (a^3 x^3-6 a^2 x^2+21 a x-24 \log (a x+1)-4\right )}{3 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 42, normalized size = 0.8 \begin{align*} -{\frac{{a}^{2}{c}^{2}{x}^{3}}{3}}+2\,{c}^{2}{x}^{2}a-7\,x{c}^{2}+8\,{\frac{{c}^{2}\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 = 0.944377, size = 55, normalized size = 1.02 \begin{align*} -\frac{1}{3} \, a^{2} c^{2} x^{3} + 2 \, a c^{2} x^{2} - 7 \, c^{2} x + \frac{8 \, c^{2} \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 = 1.66764, size = 99, normalized size = 1.83 \begin{align*} -\frac{a^{3} c^{2} x^{3} - 6 \, a^{2} c^{2} x^{2} + 21 \, a c^{2} x - 24 \, c^{2} \log \left (a x + 1\right )}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.3118, size = 41, normalized size = 0.76 \begin{align*} - \frac{a^{2} c^{2} x^{3}}{3} + 2 a c^{2} x^{2} - 7 c^{2} x + \frac{8 c^{2} \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 = 1.17303, size = 92, normalized size = 1.7 \begin{align*} -\frac{{\left (c^{2} - \frac{9 \, c^{2}}{a x + 1} + \frac{36 \, c^{2}}{{\left (a x + 1\right )}^{2}}\right )}{\left (a x + 1\right )}^{3}}{3 \, a} - \frac{8 \, c^{2} \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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