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 = 0.0553365, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {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 6140
Rule 43
Rubi steps
\begin{align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx &=c^4 \int (1-a x)^5 (1+a x)^3 \, dx\\ &=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\\ &=-\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 = 0.026989, 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 = 0.024, 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 = 0.963072, size = 109, normalized size = 1.49 \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 = 2.07214, size = 167, normalized size = 2.29 \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 = 0.112576, 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 = 1.13551, size = 105, normalized size = 1.44 \begin{align*} -\frac{{\left (28 \, c^{4} - \frac{315 \, c^{4}}{a x + 1} + \frac{1440 \, c^{4}}{{\left (a x + 1\right )}^{2}} - \frac{3360 \, c^{4}}{{\left (a x + 1\right )}^{3}} + \frac{4032 \, c^{4}}{{\left (a x + 1\right )}^{4}} - \frac{2016 \, c^{4}}{{\left (a x + 1\right )}^{5}}\right )}{\left (a x + 1\right )}^{9}}{252 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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