Optimal. Leaf size=25 \[ \frac {(a x+1)^3}{6 a c^2 (1-a x)^3} \]
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Rubi [A] time = 0.05, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6167, 6129, 37} \[ \frac {(a x+1)^3}{6 a c^2 (1-a x)^3} \]
Antiderivative was successfully verified.
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Rule 37
Rule 6129
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{4 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx &=\int \frac {e^{4 \tanh ^{-1}(a x)}}{(c-a c x)^2} \, dx\\ &=\frac {\int \frac {(1+a x)^2}{(1-a x)^4} \, dx}{c^2}\\ &=\frac {(1+a x)^3}{6 a c^2 (1-a x)^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ \frac {(a x+1)^3}{6 a c^2 (1-a x)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 51, normalized size = 2.04 \[ -\frac {3 \, a^{2} x^{2} + 1}{3 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 50, normalized size = 2.00 \[ -\frac {2}{{\left (a c x - c\right )}^{2} a} - \frac {1}{{\left (a c x - c\right )} a c} - \frac {4 \, c}{3 \, {\left (a c x - c\right )}^{3} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 42, normalized size = 1.68 \[ \frac {-\frac {2}{a \left (a x -1\right )^{2}}-\frac {4}{3 a \left (a x -1\right )^{3}}-\frac {1}{a \left (a x -1\right )}}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 51, normalized size = 2.04 \[ -\frac {3 \, a^{2} x^{2} + 1}{3 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 25, normalized size = 1.00 \[ -\frac {3\,a^2\,x^2+1}{3\,a\,c^2\,{\left (a\,x-1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.24, size = 51, normalized size = 2.04 \[ \frac {- 3 a^{2} x^{2} - 1}{3 a^{4} c^{2} x^{3} - 9 a^{3} c^{2} x^{2} + 9 a^{2} c^{2} x - 3 a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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