Optimal. Leaf size=37 \[ \frac {1}{2 a c^3 (1-a x)^2}-\frac {2}{3 a c^3 (1-a x)^3} \]
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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} \[ \frac {1}{2 a c^3 (1-a x)^2}-\frac {2}{3 a c^3 (1-a x)^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6129
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{(c-a c x)^3} \, dx\\ &=-\frac {\int \frac {1+a x}{(1-a x)^4} \, dx}{c^3}\\ &=-\frac {\int \left (\frac {2}{(-1+a x)^4}+\frac {1}{(-1+a x)^3}\right ) \, dx}{c^3}\\ &=-\frac {2}{3 a c^3 (1-a x)^3}+\frac {1}{2 a c^3 (1-a x)^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 23, normalized size = 0.62 \[ \frac {3 a x+1}{6 a c^3 (a x-1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 47, normalized size = 1.27 \[ \frac {3 \, a x + 1}{6 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 21, normalized size = 0.57 \[ \frac {3 \, a x + 1}{6 \, {\left (a x - 1\right )}^{3} a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 30, normalized size = 0.81 \[ \frac {\frac {1}{2 a \left (a x -1\right )^{2}}+\frac {2}{3 a \left (a x -1\right )^{3}}}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 47, normalized size = 1.27 \[ \frac {3 \, a x + 1}{6 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 46, normalized size = 1.24 \[ -\frac {\frac {x}{2}+\frac {1}{6\,a}}{-a^3\,c^3\,x^3+3\,a^2\,c^3\,x^2-3\,a\,c^3\,x+c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 49, normalized size = 1.32 \[ - \frac {- 3 a x - 1}{6 a^{4} c^{3} x^{3} - 18 a^{3} c^{3} x^{2} + 18 a^{2} c^{3} x - 6 a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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