Optimal. Leaf size=73 \[ -\frac{c^4 (1-a x)^9}{9 a}+\frac{3 c^4 (1-a x)^8}{4 a}-\frac{12 c^4 (1-a x)^7}{7 a}+\frac{4 c^4 (1-a x)^6}{3 a} \]
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Rubi [A] time = 0.0805593, antiderivative size = 73, 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} \[ -\frac{c^4 (1-a x)^9}{9 a}+\frac{3 c^4 (1-a x)^8}{4 a}-\frac{12 c^4 (1-a x)^7}{7 a}+\frac{4 c^4 (1-a x)^6}{3 a} \]
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 )^4 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx\\ &=-\left (c^4 \int (1-a x)^5 (1+a x)^3 \, dx\right )\\ &=-\left (c^4 \int \left (8 (1-a x)^5-12 (1-a x)^6+6 (1-a x)^7-(1-a x)^8\right ) \, dx\right )\\ &=\frac{4 c^4 (1-a x)^6}{3 a}-\frac{12 c^4 (1-a x)^7}{7 a}+\frac{3 c^4 (1-a x)^8}{4 a}-\frac{c^4 (1-a x)^9}{9 a}\\ \end{align*}
Mathematica [A] time = 0.0280838, size = 39, normalized size = 0.53 \[ \frac{c^4 (a x-1)^6 \left (28 a^3 x^3+105 a^2 x^2+138 a x+65\right )}{252 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 61, normalized size = 0.8 \begin{align*}{c}^{4} \left ({\frac{{x}^{9}{a}^{8}}{9}}-{\frac{{a}^{7}{x}^{8}}{4}}-{\frac{2\,{x}^{7}{a}^{6}}{7}}+{x}^{6}{a}^{5}-{\frac{3\,{x}^{4}{a}^{3}}{2}}+{\frac{2\,{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 = 1.1169, size = 108, normalized size = 1.48 \begin{align*} \frac{1}{9} \, a^{8} c^{4} x^{9} - \frac{1}{4} \, a^{7} c^{4} x^{8} - \frac{2}{7} \, a^{6} c^{4} x^{7} + a^{5} c^{4} x^{6} - \frac{3}{2} \, a^{3} c^{4} x^{4} + \frac{2}{3} \, a^{2} c^{4} x^{3} + a c^{4} x^{2} - c^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52778, size = 166, normalized size = 2.27 \begin{align*} \frac{1}{9} \, a^{8} c^{4} x^{9} - \frac{1}{4} \, a^{7} c^{4} x^{8} - \frac{2}{7} \, a^{6} c^{4} x^{7} + a^{5} c^{4} x^{6} - \frac{3}{2} \, a^{3} c^{4} x^{4} + \frac{2}{3} \, a^{2} c^{4} x^{3} + a c^{4} x^{2} - c^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.104245, size = 87, normalized size = 1.19 \begin{align*} \frac{a^{8} c^{4} x^{9}}{9} - \frac{a^{7} c^{4} x^{8}}{4} - \frac{2 a^{6} c^{4} x^{7}}{7} + a^{5} c^{4} x^{6} - \frac{3 a^{3} c^{4} x^{4}}{2} + \frac{2 a^{2} c^{4} x^{3}}{3} + a c^{4} x^{2} - c^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11704, size = 108, normalized size = 1.48 \begin{align*} \frac{1}{9} \, a^{8} c^{4} x^{9} - \frac{1}{4} \, a^{7} c^{4} x^{8} - \frac{2}{7} \, a^{6} c^{4} x^{7} + a^{5} c^{4} x^{6} - \frac{3}{2} \, a^{3} c^{4} x^{4} + \frac{2}{3} \, a^{2} c^{4} x^{3} + a c^{4} x^{2} - c^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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