Optimal. Leaf size=84 \[ -\frac{c^5 (a x+1)^{11}}{11 a}+\frac{4 c^5 (a x+1)^{10}}{5 a}-\frac{8 c^5 (a x+1)^9}{3 a}+\frac{4 c^5 (a x+1)^8}{a}-\frac{16 c^5 (a x+1)^7}{7 a} \]
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Rubi [A] time = 0.0988688, antiderivative size = 84, 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} \[ -\frac{c^5 (a x+1)^{11}}{11 a}+\frac{4 c^5 (a x+1)^{10}}{5 a}-\frac{8 c^5 (a x+1)^9}{3 a}+\frac{4 c^5 (a x+1)^8}{a}-\frac{16 c^5 (a x+1)^7}{7 a} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6140
Rule 43
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx\\ &=-\left (c^5 \int (1-a x)^4 (1+a x)^6 \, dx\right )\\ &=-\left (c^5 \int \left (16 (1+a x)^6-32 (1+a x)^7+24 (1+a x)^8-8 (1+a x)^9+(1+a x)^{10}\right ) \, dx\right )\\ &=-\frac{16 c^5 (1+a x)^7}{7 a}+\frac{4 c^5 (1+a x)^8}{a}-\frac{8 c^5 (1+a x)^9}{3 a}+\frac{4 c^5 (1+a x)^{10}}{5 a}-\frac{c^5 (1+a x)^{11}}{11 a}\\ \end{align*}
Mathematica [A] time = 0.0411871, size = 47, normalized size = 0.56 \[ -\frac{c^5 (a x+1)^7 \left (105 a^4 x^4-504 a^3 x^3+938 a^2 x^2-812 a x+281\right )}{1155 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 85, normalized size = 1. \begin{align*}{c}^{5} \left ( -{\frac{{a}^{10}{x}^{11}}{11}}-{\frac{{a}^{9}{x}^{10}}{5}}+{\frac{{x}^{9}{a}^{8}}{3}}+{a}^{7}{x}^{8}-{\frac{2\,{x}^{7}{a}^{6}}{7}}-2\,{x}^{6}{a}^{5}-{\frac{2\,{x}^{5}{a}^{4}}{5}}+2\,{x}^{4}{a}^{3}+{x}^{3}{a}^{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 = 1.12469, size = 153, normalized size = 1.82 \begin{align*} -\frac{1}{11} \, a^{10} c^{5} x^{11} - \frac{1}{5} \, a^{9} c^{5} x^{10} + \frac{1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} - \frac{2}{7} \, a^{6} c^{5} x^{7} - 2 \, a^{5} c^{5} x^{6} - \frac{2}{5} \, a^{4} c^{5} x^{5} + 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37059, size = 235, normalized size = 2.8 \begin{align*} -\frac{1}{11} \, a^{10} c^{5} x^{11} - \frac{1}{5} \, a^{9} c^{5} x^{10} + \frac{1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} - \frac{2}{7} \, a^{6} c^{5} x^{7} - 2 \, a^{5} c^{5} x^{6} - \frac{2}{5} \, a^{4} c^{5} x^{5} + 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.111206, size = 119, normalized size = 1.42 \begin{align*} - \frac{a^{10} c^{5} x^{11}}{11} - \frac{a^{9} c^{5} x^{10}}{5} + \frac{a^{8} c^{5} x^{9}}{3} + a^{7} c^{5} x^{8} - \frac{2 a^{6} c^{5} x^{7}}{7} - 2 a^{5} c^{5} x^{6} - \frac{2 a^{4} c^{5} x^{5}}{5} + 2 a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15025, size = 153, normalized size = 1.82 \begin{align*} -\frac{1}{11} \, a^{10} c^{5} x^{11} - \frac{1}{5} \, a^{9} c^{5} x^{10} + \frac{1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} - \frac{2}{7} \, a^{6} c^{5} x^{7} - 2 \, a^{5} c^{5} x^{6} - \frac{2}{5} \, a^{4} c^{5} x^{5} + 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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