Optimal. Leaf size=55 \[ -\frac{c^2 (1-a x)^3}{3 a}-\frac{c^2 (1-a x)^2}{a}-\frac{8 c^2 \log (a x+1)}{a}+4 c^2 x \]
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Rubi [A] time = 0.0582331, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6167, 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 6167
Rule 6129
Rule 43
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^2 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} (c-a c x)^2 \, dx\\ &=-\left (c^2 \int \frac{(1-a x)^3}{1+a x} \, dx\right )\\ &=-\left (c^2 \int \left (-4-2 (1-a x)-(1-a x)^2+\frac{8}{1+a x}\right ) \, dx\right )\\ &=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 = 0.014293, size = 39, normalized size = 0.71 \[ \frac{c^2 \left (a^3 x^3-6 a^2 x^2+21 a x-24 \log (a x+1)-4\right )}{3 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, 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 = 1.016, size = 55, normalized size = 1. \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 = 1.41193, size = 97, normalized size = 1.76 \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 = 0.644037, size = 41, normalized size = 0.75 \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 = 1.13615, size = 70, normalized size = 1.27 \begin{align*} -\frac{8 \, c^{2} \log \left ({\left | a x + 1 \right |}\right )}{a} + \frac{a^{5} c^{2} x^{3} - 6 \, a^{4} c^{2} x^{2} + 21 \, a^{3} c^{2} x}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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