Optimal. Leaf size=37 \[ \frac{1}{3 a c^4 (1-a x)^3}-\frac{1}{2 a c^4 (1-a x)^4} \]
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Rubi [A] time = 0.0607729, antiderivative size = 37, normalized size of antiderivative = 1., 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} \[ \frac{1}{3 a c^4 (1-a x)^3}-\frac{1}{2 a c^4 (1-a x)^4} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6129
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{(c-a c x)^4} \, dx\\ &=-\frac{\int \frac{1+a x}{(1-a x)^5} \, dx}{c^4}\\ &=-\frac{\int \left (-\frac{2}{(-1+a x)^5}-\frac{1}{(-1+a x)^4}\right ) \, dx}{c^4}\\ &=-\frac{1}{2 a c^4 (1-a x)^4}+\frac{1}{3 a c^4 (1-a x)^3}\\ \end{align*}
Mathematica [A] time = 0.0162178, size = 23, normalized size = 0.62 \[ -\frac{2 a x+1}{6 a c^4 (a x-1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 30, normalized size = 0.8 \begin{align*}{\frac{1}{{c}^{4}} \left ( -{\frac{1}{3\,a \left ( ax-1 \right ) ^{3}}}-{\frac{1}{2\,a \left ( ax-1 \right ) ^{4}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02396, size = 77, normalized size = 2.08 \begin{align*} -\frac{2 \, a x + 1}{6 \,{\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47311, size = 116, normalized size = 3.14 \begin{align*} -\frac{2 \, a x + 1}{6 \,{\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.548321, size = 60, normalized size = 1.62 \begin{align*} - \frac{2 a x + 1}{6 a^{5} c^{4} x^{4} - 24 a^{4} c^{4} x^{3} + 36 a^{3} c^{4} x^{2} - 24 a^{2} c^{4} x + 6 a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16371, size = 28, normalized size = 0.76 \begin{align*} -\frac{2 \, a x + 1}{6 \,{\left (a x - 1\right )}^{4} a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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