Optimal. Leaf size=37 \[ \frac {2 c^3 (1-a x)^3}{3 a}-\frac {c^3 (1-a x)^4}{4 a} \]
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Rubi [A] time = 0.06, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6167, 6129, 43} \[ \frac {2 c^3 (1-a x)^3}{3 a}-\frac {c^3 (1-a x)^4}{4 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6129
Rule 6167
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} (c-a c x)^3 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} (c-a c x)^3 \, dx\\ &=-\left (c^3 \int (1-a x)^2 (1+a x) \, dx\right )\\ &=-\left (c^3 \int \left (2 (1-a x)^2-(1-a x)^3\right ) \, dx\right )\\ &=\frac {2 c^3 (1-a x)^3}{3 a}-\frac {c^3 (1-a x)^4}{4 a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 0.81 \[ -\frac {1}{12} c^3 x \left (3 a^3 x^3-4 a^2 x^2-6 a x+12\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 38, normalized size = 1.03 \[ -\frac {1}{4} \, a^{3} c^{3} x^{4} + \frac {1}{3} \, a^{2} c^{3} x^{3} + \frac {1}{2} \, a c^{3} x^{2} - c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 38, normalized size = 1.03 \[ -\frac {1}{4} \, a^{3} c^{3} x^{4} + \frac {1}{3} \, a^{2} c^{3} x^{3} + \frac {1}{2} \, a c^{3} x^{2} - c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 31, normalized size = 0.84 \[ c^{3} \left (-\frac {1}{4} x^{4} a^{3}+\frac {1}{3} x^{3} a^{2}+\frac {1}{2} a \,x^{2}-x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 38, normalized size = 1.03 \[ -\frac {1}{4} \, a^{3} c^{3} x^{4} + \frac {1}{3} \, a^{2} c^{3} x^{3} + \frac {1}{2} \, a c^{3} x^{2} - c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 38, normalized size = 1.03 \[ -\frac {a^3\,c^3\,x^4}{4}+\frac {a^2\,c^3\,x^3}{3}+\frac {a\,c^3\,x^2}{2}-c^3\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 37, normalized size = 1.00 \[ - \frac {a^{3} c^{3} x^{4}}{4} + \frac {a^{2} c^{3} x^{3}}{3} + \frac {a c^{3} x^{2}}{2} - c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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