Optimal. Leaf size=52 \[ -\frac{c^3 (a x+1)^7}{7 a}+\frac{2 c^3 (a x+1)^6}{3 a}-\frac{4 c^3 (a x+1)^5}{5 a} \]
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Rubi [A] time = 0.076907, antiderivative size = 52, 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} \[ -\frac{c^3 (a x+1)^7}{7 a}+\frac{2 c^3 (a x+1)^6}{3 a}-\frac{4 c^3 (a x+1)^5}{5 a} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6140
Rule 43
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx\\ &=-\left (c^3 \int (1-a x)^2 (1+a x)^4 \, dx\right )\\ &=-\left (c^3 \int \left (4 (1+a x)^4-4 (1+a x)^5+(1+a x)^6\right ) \, dx\right )\\ &=-\frac{4 c^3 (1+a x)^5}{5 a}+\frac{2 c^3 (1+a x)^6}{3 a}-\frac{c^3 (1+a x)^7}{7 a}\\ \end{align*}
Mathematica [A] time = 0.0226959, size = 31, normalized size = 0.6 \[ -\frac{c^3 (a x+1)^5 \left (15 a^2 x^2-40 a x+29\right )}{105 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 54, normalized size = 1. \begin{align*}{c}^{3} \left ( -{\frac{{x}^{7}{a}^{6}}{7}}-{\frac{{x}^{6}{a}^{5}}{3}}+{\frac{{x}^{5}{a}^{4}}{5}}+{x}^{4}{a}^{3}+{\frac{{x}^{3}{a}^{2}}{3}}-a{x}^{2}-x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07407, size = 95, normalized size = 1.83 \begin{align*} -\frac{1}{7} \, a^{6} c^{3} x^{7} - \frac{1}{3} \, a^{5} c^{3} x^{6} + \frac{1}{5} \, a^{4} c^{3} x^{5} + a^{3} c^{3} x^{4} + \frac{1}{3} \, a^{2} c^{3} x^{3} - 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 = 1.53196, size = 143, normalized size = 2.75 \begin{align*} -\frac{1}{7} \, a^{6} c^{3} x^{7} - \frac{1}{3} \, a^{5} c^{3} x^{6} + \frac{1}{5} \, a^{4} c^{3} x^{5} + a^{3} c^{3} x^{4} + \frac{1}{3} \, a^{2} c^{3} x^{3} - 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 [A] time = 0.092369, size = 70, normalized size = 1.35 \begin{align*} - \frac{a^{6} c^{3} x^{7}}{7} - \frac{a^{5} c^{3} x^{6}}{3} + \frac{a^{4} c^{3} x^{5}}{5} + a^{3} c^{3} x^{4} + \frac{a^{2} c^{3} x^{3}}{3} - 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 [A] time = 1.12104, size = 95, normalized size = 1.83 \begin{align*} -\frac{1}{7} \, a^{6} c^{3} x^{7} - \frac{1}{3} \, a^{5} c^{3} x^{6} + \frac{1}{5} \, a^{4} c^{3} x^{5} + a^{3} c^{3} x^{4} + \frac{1}{3} \, a^{2} c^{3} x^{3} - a c^{3} x^{2} - c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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