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 = 0.06, 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} \[ \frac {1}{3 a c^4 (1-a x)^3}-\frac {1}{2 a c^4 (1-a x)^4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6129
Rule 6167
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.46, size = 57, normalized size = 1.54 \[ -\frac {2 \, a x + 1}{6 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \]
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 \[ -\frac {2 \, a x + 1}{6 \, {\left (a x - 1\right )}^{4} a c^{4}} \]
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 \[ \frac {-\frac {1}{3 a \left (a x -1\right )^{3}}-\frac {1}{2 a \left (a x -1\right )^{4}}}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 57, normalized size = 1.54 \[ -\frac {2 \, a x + 1}{6 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 56, normalized size = 1.51 \[ -\frac {\frac {x}{3}+\frac {1}{6\,a}}{a^4\,c^4\,x^4-4\,a^3\,c^4\,x^3+6\,a^2\,c^4\,x^2-4\,a\,c^4\,x+c^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.28, size = 60, normalized size = 1.62 \[ \frac {- 2 a x - 1}{6 a^{5} c^{4} x^{4} - 24 a^{4} c^{4} x^{3} + 36 a^{3} c^{4} x^{2} - 24 a^{2} c^{4} x + 6 a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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