Optimal. Leaf size=18 \[ \frac {1}{3 a c^2 (1-a x)^3} \]
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Rubi [A] time = 0.06, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6167, 6140, 32} \[ \frac {1}{3 a c^2 (1-a x)^3} \]
Antiderivative was successfully verified.
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Rule 32
Rule 6140
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx &=\int \frac {e^{4 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx\\ &=\frac {\int \frac {1}{(1-a x)^4} \, dx}{c^2}\\ &=\frac {1}{3 a c^2 (1-a x)^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 17, normalized size = 0.94 \[ -\frac {1}{3 a c^2 (a x-1)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 41, normalized size = 2.28 \[ -\frac {1}{3 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 15, normalized size = 0.83 \[ -\frac {1}{3 \, {\left (a x - 1\right )}^{3} a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 16, normalized size = 0.89 \[ -\frac {1}{3 c^{2} a \left (a x -1\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.30, size = 41, normalized size = 2.28 \[ -\frac {1}{3 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 40, normalized size = 2.22 \[ \frac {1}{-3\,a^4\,c^2\,x^3+9\,a^3\,c^2\,x^2-9\,a^2\,c^2\,x+3\,a\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.23, size = 42, normalized size = 2.33 \[ - \frac {1}{3 a^{4} c^{2} x^{3} - 9 a^{3} c^{2} x^{2} + 9 a^{2} c^{2} x - 3 a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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