Optimal. Leaf size=53 \[ \frac{1}{3 a c^4 (1-a x)^3}-\frac{1}{a c^4 (1-a x)^4}+\frac{4}{5 a c^4 (1-a x)^5} \]
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Rubi [A] time = 0.0669647, antiderivative size = 53, 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} \[ \frac{1}{3 a c^4 (1-a x)^3}-\frac{1}{a c^4 (1-a x)^4}+\frac{4}{5 a c^4 (1-a x)^5} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6129
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{4 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx &=\int \frac{e^{4 \tanh ^{-1}(a x)}}{(c-a c x)^4} \, dx\\ &=\frac{\int \frac{(1+a x)^2}{(1-a x)^6} \, dx}{c^4}\\ &=\frac{\int \left (\frac{4}{(-1+a x)^6}+\frac{4}{(-1+a x)^5}+\frac{1}{(-1+a x)^4}\right ) \, dx}{c^4}\\ &=\frac{4}{5 a c^4 (1-a x)^5}-\frac{1}{a c^4 (1-a x)^4}+\frac{1}{3 a c^4 (1-a x)^3}\\ \end{align*}
Mathematica [A] time = 0.0187029, size = 31, normalized size = 0.58 \[ -\frac{5 a^2 x^2+5 a x+2}{15 a c^4 (a x-1)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 42, normalized size = 0.8 \begin{align*}{\frac{1}{{c}^{4}} \left ( -{\frac{1}{3\,a \left ( ax-1 \right ) ^{3}}}-{\frac{1}{a \left ( ax-1 \right ) ^{4}}}-{\frac{4}{5\,a \left ( ax-1 \right ) ^{5}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02298, size = 104, normalized size = 1.96 \begin{align*} -\frac{5 \, a^{2} x^{2} + 5 \, a x + 2}{15 \,{\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, 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 = 1.46517, size = 158, normalized size = 2.98 \begin{align*} -\frac{5 \, a^{2} x^{2} + 5 \, a x + 2}{15 \,{\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, 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 [A] time = 1.59646, size = 80, normalized size = 1.51 \begin{align*} - \frac{5 a^{2} x^{2} + 5 a x + 2}{15 a^{6} c^{4} x^{5} - 75 a^{5} c^{4} x^{4} + 150 a^{4} c^{4} x^{3} - 150 a^{3} c^{4} x^{2} + 75 a^{2} c^{4} x - 15 a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15388, size = 57, normalized size = 1.08 \begin{align*} -\frac{\frac{5}{{\left (a x - 1\right )}^{3} a} + \frac{15}{{\left (a x - 1\right )}^{4} a} + \frac{12}{{\left (a x - 1\right )}^{5} a}}{15 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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