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 = 0.06, antiderivative size = 66, normalized size of antiderivative = 1.00, 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 43
Rule 6140
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.45, size = 101, normalized size = 1.53 \[ -\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {2}{5} \, a^{9} c^{5} x^{10} - \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} + 2 \, a^{6} c^{5} x^{7} - \frac {14}{5} \, a^{4} c^{5} x^{5} - 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 \, a c^{5} x^{2} + c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 101, normalized size = 1.53 \[ -\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {2}{5} \, a^{9} c^{5} x^{10} - \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} + 2 \, a^{6} c^{5} x^{7} - \frac {14}{5} \, a^{4} c^{5} x^{5} - 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 \, a c^{5} x^{2} + c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 75, normalized size = 1.14 \[ c^{5} \left (-\frac {1}{11} a^{10} x^{11}-\frac {2}{5} a^{9} x^{10}-\frac {1}{3} x^{9} a^{8}+a^{7} x^{8}+2 x^{7} a^{6}-\frac {14}{5} a^{4} x^{5}-2 x^{4} a^{3}+x^{3} a^{2}+2 a \,x^{2}+x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 101, normalized size = 1.53 \[ -\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {2}{5} \, a^{9} c^{5} x^{10} - \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} + 2 \, a^{6} c^{5} x^{7} - \frac {14}{5} \, a^{4} c^{5} x^{5} - 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 \, a c^{5} x^{2} + c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 101, normalized size = 1.53 \[ -\frac {a^{10}\,c^5\,x^{11}}{11}-\frac {2\,a^9\,c^5\,x^{10}}{5}-\frac {a^8\,c^5\,x^9}{3}+a^7\,c^5\,x^8+2\,a^6\,c^5\,x^7-\frac {14\,a^4\,c^5\,x^5}{5}-2\,a^3\,c^5\,x^4+a^2\,c^5\,x^3+2\,a\,c^5\,x^2+c^5\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.12, size = 109, normalized size = 1.65 \[ - \frac {a^{10} c^{5} x^{11}}{11} - \frac {2 a^{9} c^{5} x^{10}}{5} - \frac {a^{8} c^{5} x^{9}}{3} + a^{7} c^{5} x^{8} + 2 a^{6} c^{5} x^{7} - \frac {14 a^{4} c^{5} x^{5}}{5} - 2 a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 a c^{5} x^{2} + c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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