Optimal. Leaf size=35 \[ \frac{c^3 (a x+1)^6}{3 a}-\frac{c^3 (a x+1)^7}{7 a} \]
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Rubi [A] time = 0.0628281, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6167, 6140, 43} \[ \frac{c^3 (a x+1)^6}{3 a}-\frac{c^3 (a x+1)^7}{7 a} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6140
Rule 43
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx\\ &=c^3 \int (1-a x) (1+a x)^5 \, dx\\ &=c^3 \int \left (2 (1+a x)^5-(1+a x)^6\right ) \, dx\\ &=\frac{c^3 (1+a x)^6}{3 a}-\frac{c^3 (1+a x)^7}{7 a}\\ \end{align*}
Mathematica [A] time = 0.0206693, size = 23, normalized size = 0.66 \[ -\frac{c^3 (a x+1)^6 (3 a x-4)}{21 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 45, normalized size = 1.3 \begin{align*}{c}^{3} \left ( -{\frac{{x}^{7}{a}^{6}}{7}}-{\frac{2\,{x}^{6}{a}^{5}}{3}}-{x}^{5}{a}^{4}+{\frac{5\,{x}^{3}{a}^{2}}{3}}+2\,a{x}^{2}+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980798, size = 80, normalized size = 2.29 \begin{align*} -\frac{1}{7} \, a^{6} c^{3} x^{7} - \frac{2}{3} \, a^{5} c^{3} x^{6} - a^{4} c^{3} x^{5} + \frac{5}{3} \, a^{2} c^{3} x^{3} + 2 \, a c^{3} x^{2} + c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51495, size = 122, normalized size = 3.49 \begin{align*} -\frac{1}{7} \, a^{6} c^{3} x^{7} - \frac{2}{3} \, a^{5} c^{3} x^{6} - a^{4} c^{3} x^{5} + \frac{5}{3} \, a^{2} c^{3} x^{3} + 2 \, a c^{3} x^{2} + c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.096159, size = 63, normalized size = 1.8 \begin{align*} - \frac{a^{6} c^{3} x^{7}}{7} - \frac{2 a^{5} c^{3} x^{6}}{3} - a^{4} c^{3} x^{5} + \frac{5 a^{2} c^{3} x^{3}}{3} + 2 a c^{3} x^{2} + c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15696, size = 105, normalized size = 3. \begin{align*} -\frac{{\left (3 \, c^{3} + \frac{35 \, c^{3}}{a x - 1} + \frac{168 \, c^{3}}{{\left (a x - 1\right )}^{2}} + \frac{420 \, c^{3}}{{\left (a x - 1\right )}^{3}} + \frac{560 \, c^{3}}{{\left (a x - 1\right )}^{4}} + \frac{336 \, c^{3}}{{\left (a x - 1\right )}^{5}}\right )}{\left (a x - 1\right )}^{7}}{21 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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