Optimal. Leaf size=66 \[ -\frac{c^5 (a x+1)^{11}}{11 a}+\frac{3 c^5 (a x+1)^{10}}{5 a}-\frac{4 c^5 (a x+1)^9}{3 a}+\frac{c^5 (a x+1)^8}{a} \]
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Rubi [A] time = 0.0643129, antiderivative size = 66, 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} \[ -\frac{c^5 (a x+1)^{11}}{11 a}+\frac{3 c^5 (a x+1)^{10}}{5 a}-\frac{4 c^5 (a x+1)^9}{3 a}+\frac{c^5 (a x+1)^8}{a} \]
Antiderivative was successfully verified.
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Rule 6140
Rule 43
Rubi steps
\begin{align*} \int e^{4 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx &=c^5 \int (1-a x)^3 (1+a x)^7 \, dx\\ &=c^5 \int \left (8 (1+a x)^7-12 (1+a x)^8+6 (1+a x)^9-(1+a x)^{10}\right ) \, dx\\ &=\frac{c^5 (1+a x)^8}{a}-\frac{4 c^5 (1+a x)^9}{3 a}+\frac{3 c^5 (1+a x)^{10}}{5 a}-\frac{c^5 (1+a x)^{11}}{11 a}\\ \end{align*}
Mathematica [A] time = 0.0322227, size = 39, normalized size = 0.59 \[ -\frac{c^5 (a x+1)^8 \left (15 a^3 x^3-54 a^2 x^2+67 a x-29\right )}{165 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 75, normalized size = 1.1 \begin{align*}{c}^{5} \left ( -{\frac{{a}^{10}{x}^{11}}{11}}-{\frac{2\,{a}^{9}{x}^{10}}{5}}-{\frac{{x}^{9}{a}^{8}}{3}}+{a}^{7}{x}^{8}+2\,{x}^{7}{a}^{6}-{\frac{14\,{x}^{5}{a}^{4}}{5}}-2\,{x}^{4}{a}^{3}+{x}^{3}{a}^{2}+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 = 0.970398, size = 136, normalized size = 2.06 \begin{align*} -\frac{1}{11} \, a^{10} c^{5} x^{11} - \frac{2}{5} \, a^{9} c^{5} x^{10} - \frac{1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} + 2 \, a^{6} c^{5} x^{7} - \frac{14}{5} \, a^{4} c^{5} x^{5} - 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 \, a c^{5} x^{2} + c^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.26109, size = 215, normalized size = 3.26 \begin{align*} -\frac{1}{11} \, a^{10} c^{5} x^{11} - \frac{2}{5} \, a^{9} c^{5} x^{10} - \frac{1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} + 2 \, a^{6} c^{5} x^{7} - \frac{14}{5} \, a^{4} c^{5} x^{5} - 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 \, a c^{5} x^{2} + c^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.122364, size = 109, normalized size = 1.65 \begin{align*} - \frac{a^{10} c^{5} x^{11}}{11} - \frac{2 a^{9} c^{5} x^{10}}{5} - \frac{a^{8} c^{5} x^{9}}{3} + a^{7} c^{5} x^{8} + 2 a^{6} c^{5} x^{7} - \frac{14 a^{4} c^{5} x^{5}}{5} - 2 a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 a c^{5} x^{2} + c^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14567, size = 136, normalized size = 2.06 \begin{align*} -\frac{1}{11} \, a^{10} c^{5} x^{11} - \frac{2}{5} \, a^{9} c^{5} x^{10} - \frac{1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} + 2 \, a^{6} c^{5} x^{7} - \frac{14}{5} \, a^{4} c^{5} x^{5} - 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 \, a c^{5} x^{2} + c^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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