Optimal. Leaf size=71 \[ \frac{6}{a c^2 (1-a x)}-\frac{2}{a c^2 (1-a x)^2}+\frac{1}{3 a c^2 (1-a x)^3}+\frac{4 \log (1-a x)}{a c^2}+\frac{x}{c^2} \]
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Rubi [A] time = 0.172337, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6167, 6157, 6150, 43} \[ \frac{6}{a c^2 (1-a x)}-\frac{2}{a c^2 (1-a x)^2}+\frac{1}{3 a c^2 (1-a x)^3}+\frac{4 \log (1-a x)}{a c^2}+\frac{x}{c^2} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6157
Rule 6150
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{4 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^2} \, dx &=\int \frac{e^{4 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^2} \, dx\\ &=\frac{a^4 \int \frac{e^{4 \tanh ^{-1}(a x)} x^4}{\left (1-a^2 x^2\right )^2} \, dx}{c^2}\\ &=\frac{a^4 \int \frac{x^4}{(1-a x)^4} \, dx}{c^2}\\ &=\frac{a^4 \int \left (\frac{1}{a^4}+\frac{1}{a^4 (-1+a x)^4}+\frac{4}{a^4 (-1+a x)^3}+\frac{6}{a^4 (-1+a x)^2}+\frac{4}{a^4 (-1+a x)}\right ) \, dx}{c^2}\\ &=\frac{x}{c^2}+\frac{1}{3 a c^2 (1-a x)^3}-\frac{2}{a c^2 (1-a x)^2}+\frac{6}{a c^2 (1-a x)}+\frac{4 \log (1-a x)}{a c^2}\\ \end{align*}
Mathematica [A] time = 0.0379981, size = 63, normalized size = 0.89 \[ \frac{3 a^4 x^4-9 a^3 x^3-9 a^2 x^2+27 a x+12 (a x-1)^3 \log (1-a x)-13}{3 a c^2 (a x-1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 66, normalized size = 0.9 \begin{align*}{\frac{x}{{c}^{2}}}-2\,{\frac{1}{a{c}^{2} \left ( ax-1 \right ) ^{2}}}-{\frac{1}{3\,a{c}^{2} \left ( ax-1 \right ) ^{3}}}+4\,{\frac{\ln \left ( ax-1 \right ) }{a{c}^{2}}}-6\,{\frac{1}{a{c}^{2} \left ( ax-1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03598, size = 101, normalized size = 1.42 \begin{align*} -\frac{18 \, a^{2} x^{2} - 30 \, a x + 13}{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 )}} + \frac{x}{c^{2}} + \frac{4 \, \log \left (a x - 1\right )}{a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50434, size = 215, normalized size = 3.03 \begin{align*} \frac{3 \, a^{4} x^{4} - 9 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 27 \, a x + 12 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (a x - 1\right ) - 13}{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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.533337, size = 83, normalized size = 1.17 \begin{align*} a^{4} \left (- \frac{18 a^{2} x^{2} - 30 a x + 13}{3 a^{8} c^{2} x^{3} - 9 a^{7} c^{2} x^{2} + 9 a^{6} c^{2} x - 3 a^{5} c^{2}} + \frac{x}{a^{4} c^{2}} + \frac{4 \log{\left (a x - 1 \right )}}{a^{5} c^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12164, size = 126, normalized size = 1.77 \begin{align*} \frac{a x - 1}{a c^{2}} - \frac{4 \, \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a c^{2}} - \frac{\frac{18 \, a^{5} c^{4}}{a x - 1} + \frac{6 \, a^{5} c^{4}}{{\left (a x - 1\right )}^{2}} + \frac{a^{5} c^{4}}{{\left (a x - 1\right )}^{3}}}{3 \, a^{6} c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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