Optimal. Leaf size=53 \[ -\frac {c^5 (1-a x)^6}{6 a}+\frac {4 c^5 (1-a x)^5}{5 a}-\frac {c^5 (1-a x)^4}{a} \]
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Rubi [A] time = 0.04, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6129, 43} \[ -\frac {c^5 (1-a x)^6}{6 a}+\frac {4 c^5 (1-a x)^5}{5 a}-\frac {c^5 (1-a x)^4}{a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6129
Rubi steps
\begin {align*} \int e^{4 \tanh ^{-1}(a x)} (c-a c x)^5 \, dx &=c^5 \int (1-a x)^3 (1+a x)^2 \, dx\\ &=c^5 \int \left (4 (1-a x)^3-4 (1-a x)^4+(1-a x)^5\right ) \, dx\\ &=-\frac {c^5 (1-a x)^4}{a}+\frac {4 c^5 (1-a x)^5}{5 a}-\frac {c^5 (1-a x)^6}{6 a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 31, normalized size = 0.58 \[ -\frac {c^5 (a x-1)^4 \left (5 a^2 x^2+14 a x+11\right )}{30 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 59, normalized size = 1.11 \[ -\frac {1}{6} \, a^{5} c^{5} x^{6} + \frac {1}{5} \, a^{4} c^{5} x^{5} + \frac {1}{2} \, a^{3} c^{5} x^{4} - \frac {2}{3} \, a^{2} c^{5} x^{3} - \frac {1}{2} \, a c^{5} x^{2} + c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 59, normalized size = 1.11 \[ -\frac {1}{6} \, a^{5} c^{5} x^{6} + \frac {1}{5} \, a^{4} c^{5} x^{5} + \frac {1}{2} \, a^{3} c^{5} x^{4} - \frac {2}{3} \, a^{2} c^{5} x^{3} - \frac {1}{2} \, a c^{5} x^{2} + c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 45, normalized size = 0.85 \[ c^{5} \left (-\frac {1}{6} x^{6} a^{5}+\frac {1}{5} a^{4} x^{5}+\frac {1}{2} x^{4} a^{3}-\frac {2}{3} x^{3} a^{2}-\frac {1}{2} a \,x^{2}+x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 59, normalized size = 1.11 \[ -\frac {1}{6} \, a^{5} c^{5} x^{6} + \frac {1}{5} \, a^{4} c^{5} x^{5} + \frac {1}{2} \, a^{3} c^{5} x^{4} - \frac {2}{3} \, a^{2} c^{5} x^{3} - \frac {1}{2} \, a c^{5} x^{2} + c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.79, size = 59, normalized size = 1.11 \[ -\frac {a^5\,c^5\,x^6}{6}+\frac {a^4\,c^5\,x^5}{5}+\frac {a^3\,c^5\,x^4}{2}-\frac {2\,a^2\,c^5\,x^3}{3}-\frac {a\,c^5\,x^2}{2}+c^5\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 63, normalized size = 1.19 \[ - \frac {a^{5} c^{5} x^{6}}{6} + \frac {a^{4} c^{5} x^{5}}{5} + \frac {a^{3} c^{5} x^{4}}{2} - \frac {2 a^{2} c^{5} x^{3}}{3} - \frac {a c^{5} x^{2}}{2} + c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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