Optimal. Leaf size=35 \[ \frac {2 c^3 (a x+1)^3}{3 a}-\frac {c^3 (a x+1)^4}{4 a} \]
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Rubi [A] time = 0.06, antiderivative size = 35, 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} \[ \frac {2 c^3 (a x+1)^3}{3 a}-\frac {c^3 (a x+1)^4}{4 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6129
Rule 6167
Rubi steps
\begin {align*} \int e^{4 \coth ^{-1}(a x)} (c-a c x)^3 \, dx &=\int e^{4 \tanh ^{-1}(a x)} (c-a c x)^3 \, dx\\ &=c^3 \int (1-a x) (1+a x)^2 \, dx\\ &=c^3 \int \left (2 (1+a x)^2-(1+a x)^3\right ) \, dx\\ &=\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 = 0.86 \[ -\frac {1}{12} c^3 x \left (3 a^3 x^3+4 a^2 x^2-6 a x-12\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 37, normalized size = 1.06 \[ -\frac {1}{4} \, a^{3} c^{3} x^{4} - \frac {1}{3} \, a^{2} c^{3} x^{3} + \frac {1}{2} \, a c^{3} x^{2} + c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 42, normalized size = 1.20 \[ -\frac {{\left (3 \, c^{3} + \frac {16 \, c^{3}}{a x - 1} + \frac {24 \, c^{3}}{{\left (a x - 1\right )}^{2}}\right )} {\left (a x - 1\right )}^{4}}{12 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 29, normalized size = 0.83 \[ c^{3} \left (-\frac {1}{4} x^{4} a^{3}-\frac {1}{3} x^{3} a^{2}+\frac {1}{2} a \,x^{2}+x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 37, normalized size = 1.06 \[ -\frac {1}{4} \, a^{3} c^{3} x^{4} - \frac {1}{3} \, a^{2} c^{3} x^{3} + \frac {1}{2} \, a c^{3} x^{2} + c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 37, normalized size = 1.06 \[ -\frac {a^3\,c^3\,x^4}{4}-\frac {a^2\,c^3\,x^3}{3}+\frac {a\,c^3\,x^2}{2}+c^3\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 37, normalized size = 1.06 \[ - \frac {a^{3} c^{3} x^{4}}{4} - \frac {a^{2} c^{3} x^{3}}{3} + \frac {a c^{3} x^{2}}{2} + c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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