Optimal. Leaf size=69 \[ \frac{c^4 (a x+1)^9}{9 a}-\frac{3 c^4 (a x+1)^8}{4 a}+\frac{12 c^4 (a x+1)^7}{7 a}-\frac{4 c^4 (a x+1)^6}{3 a} \]
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Rubi [A] time = 0.0850346, antiderivative size = 69, 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} \[ \frac{c^4 (a x+1)^9}{9 a}-\frac{3 c^4 (a x+1)^8}{4 a}+\frac{12 c^4 (a x+1)^7}{7 a}-\frac{4 c^4 (a x+1)^6}{3 a} \]
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)^3 (1+a x)^5 \, 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 = 0.0315883, size = 39, normalized size = 0.57 \[ \frac{c^4 (a x+1)^6 \left (28 a^3 x^3-105 a^2 x^2+138 a x-65\right )}{252 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 63, normalized size = 0.9 \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 = 1.21455, size = 111, normalized size = 1.61 \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 = 1.55152, size = 166, normalized size = 2.41 \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 = 0.100179, size = 87, normalized size = 1.26 \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 = 1.1178, size = 111, normalized size = 1.61 \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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