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 = 0.056424, antiderivative size = 69, 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} \[ -\frac{c^4 (a x+1)^9}{9 a}+\frac{3 c^4 (a x+1)^8}{4 a}-\frac{12 c^4 (a x+1)^7}{7 a}+\frac{4 c^4 (a x+1)^6}{3 a} \]
Antiderivative was successfully verified.
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Rule 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)^3 (1+a x)^5 \, 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 = 0.024709, 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 = 0.024, size = 59, 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 = 0.944406, size = 107, normalized size = 1.55 \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 = 1.92478, size = 167, normalized size = 2.42 \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 = 0.111312, 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 = 1.18944, size = 107, normalized size = 1.55 \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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