Optimal. Leaf size=17 \[ \frac{c^2 (a x+1)^3}{3 a} \]
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Rubi [A] time = 0.0464065, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6167, 6129, 32} \[ \frac{c^2 (a x+1)^3}{3 a} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6129
Rule 32
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} (c-a c x)^2 \, dx &=\int e^{4 \tanh ^{-1}(a x)} (c-a c x)^2 \, dx\\ &=c^2 \int (1+a x)^2 \, dx\\ &=\frac{c^2 (1+a x)^3}{3 a}\\ \end{align*}
Mathematica [A] time = 0.0125278, size = 21, normalized size = 1.24 \[ c^2 \left (\frac{a^2 x^3}{3}+a x^2+x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 16, normalized size = 0.9 \begin{align*}{\frac{{c}^{2} \left ( ax+1 \right ) ^{3}}{3\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0172, size = 34, normalized size = 2. \begin{align*} \frac{1}{3} \, a^{2} c^{2} x^{3} + a c^{2} x^{2} + c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48649, size = 50, normalized size = 2.94 \begin{align*} \frac{1}{3} \, a^{2} c^{2} x^{3} + a c^{2} x^{2} + c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.156255, size = 24, normalized size = 1.41 \begin{align*} \frac{a^{2} c^{2} x^{3}}{3} + a c^{2} x^{2} + c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17359, size = 54, normalized size = 3.18 \begin{align*} \frac{{\left (c^{2} + \frac{6 \, c^{2}}{a x - 1} + \frac{12 \, c^{2}}{{\left (a x - 1\right )}^{2}}\right )}{\left (a x - 1\right )}^{3}}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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