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 = 0.07, antiderivative size = 53, 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} \[ \frac {1}{3 a c^4 (1-a x)^3}-\frac {1}{a c^4 (1-a x)^4}+\frac {4}{5 a c^4 (1-a x)^5} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6129
Rule 6167
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.69, size = 77, normalized size = 1.45 \[ -\frac {5 \, a^{2} x^{2} + 5 \, a x + 2}{15 \, {\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 42, normalized size = 0.79 \[ -\frac {\frac {5}{{\left (a x - 1\right )}^{3} a} + \frac {15}{{\left (a x - 1\right )}^{4} a} + \frac {12}{{\left (a x - 1\right )}^{5} a}}{15 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 42, normalized size = 0.79 \[ \frac {-\frac {4}{5 a \left (a x -1\right )^{5}}-\frac {1}{3 a \left (a x -1\right )^{3}}-\frac {1}{a \left (a x -1\right )^{4}}}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 77, normalized size = 1.45 \[ -\frac {5 \, a^{2} x^{2} + 5 \, a x + 2}{15 \, {\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 29, normalized size = 0.55 \[ -\frac {5\,a^2\,x^2+5\,a\,x+2}{15\,a\,c^4\,{\left (a\,x-1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 80, normalized size = 1.51 \[ \frac {- 5 a^{2} x^{2} - 5 a x - 2}{15 a^{6} c^{4} x^{5} - 75 a^{5} c^{4} x^{4} + 150 a^{4} c^{4} x^{3} - 150 a^{3} c^{4} x^{2} + 75 a^{2} c^{4} x - 15 a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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