Optimal. Leaf size=52 \[ \frac{c^4 (a x+1)^9}{9 a}-\frac{c^4 (a x+1)^8}{2 a}+\frac{4 c^4 (a x+1)^7}{7 a} \]
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Rubi [A] time = 0.0788417, 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} \[ \frac{c^4 (a x+1)^9}{9 a}-\frac{c^4 (a x+1)^8}{2 a}+\frac{4 c^4 (a x+1)^7}{7 a} \]
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 )^4 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx\\ &=c^4 \int (1-a x)^2 (1+a x)^6 \, dx\\ &=c^4 \int \left (4 (1+a x)^6-4 (1+a x)^7+(1+a x)^8\right ) \, dx\\ &=\frac{4 c^4 (1+a x)^7}{7 a}-\frac{c^4 (1+a x)^8}{2 a}+\frac{c^4 (1+a x)^9}{9 a}\\ \end{align*}
Mathematica [A] time = 0.0300276, size = 31, normalized size = 0.6 \[ \frac{c^4 (a x+1)^7 \left (14 a^2 x^2-35 a x+23\right )}{126 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 69, normalized size = 1.3 \begin{align*}{c}^{4} \left ({\frac{{x}^{9}{a}^{8}}{9}}+{\frac{{a}^{7}{x}^{8}}{2}}+{\frac{4\,{x}^{7}{a}^{6}}{7}}-{\frac{2\,{x}^{6}{a}^{5}}{3}}-2\,{x}^{5}{a}^{4}-{x}^{4}{a}^{3}+{\frac{4\,{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 = 1.11268, size = 124, normalized size = 2.38 \begin{align*} \frac{1}{9} \, a^{8} c^{4} x^{9} + \frac{1}{2} \, a^{7} c^{4} x^{8} + \frac{4}{7} \, a^{6} c^{4} x^{7} - \frac{2}{3} \, a^{5} c^{4} x^{6} - 2 \, a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac{4}{3} \, a^{2} c^{4} x^{3} + 2 \, 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 = 1.47327, size = 190, normalized size = 3.65 \begin{align*} \frac{1}{9} \, a^{8} c^{4} x^{9} + \frac{1}{2} \, a^{7} c^{4} x^{8} + \frac{4}{7} \, a^{6} c^{4} x^{7} - \frac{2}{3} \, a^{5} c^{4} x^{6} - 2 \, a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac{4}{3} \, a^{2} c^{4} x^{3} + 2 \, 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 [B] time = 0.111963, size = 100, normalized size = 1.92 \begin{align*} \frac{a^{8} c^{4} x^{9}}{9} + \frac{a^{7} c^{4} x^{8}}{2} + \frac{4 a^{6} c^{4} x^{7}}{7} - \frac{2 a^{5} c^{4} x^{6}}{3} - 2 a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac{4 a^{2} c^{4} x^{3}}{3} + 2 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 = 1.16992, size = 122, normalized size = 2.35 \begin{align*} \frac{{\left (14 \, c^{4} + \frac{189 \, c^{4}}{a x - 1} + \frac{1080 \, c^{4}}{{\left (a x - 1\right )}^{2}} + \frac{3360 \, c^{4}}{{\left (a x - 1\right )}^{3}} + \frac{6048 \, c^{4}}{{\left (a x - 1\right )}^{4}} + \frac{6048 \, c^{4}}{{\left (a x - 1\right )}^{5}} + \frac{2688 \, c^{4}}{{\left (a x - 1\right )}^{6}}\right )}{\left (a x - 1\right )}^{9}}{126 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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