Optimal. Leaf size=69 \[ -\frac{1}{8 a c^5 (1-a x)}-\frac{1}{8 a c^5 (1-a x)^2}-\frac{1}{6 a c^5 (1-a x)^3}-\frac{\tanh ^{-1}(a x)}{8 a c^5} \]
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Rubi [A] time = 0.0796972, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6167, 6129, 44, 207} \[ -\frac{1}{8 a c^5 (1-a x)}-\frac{1}{8 a c^5 (1-a x)^2}-\frac{1}{6 a c^5 (1-a x)^3}-\frac{\tanh ^{-1}(a x)}{8 a c^5} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6129
Rule 44
Rule 207
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^5} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{(c-a c x)^5} \, dx\\ &=-\frac{\int \frac{1}{(1-a x)^4 (1+a x)} \, dx}{c^5}\\ &=-\frac{\int \left (\frac{1}{2 (-1+a x)^4}-\frac{1}{4 (-1+a x)^3}+\frac{1}{8 (-1+a x)^2}-\frac{1}{8 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^5}\\ &=-\frac{1}{6 a c^5 (1-a x)^3}-\frac{1}{8 a c^5 (1-a x)^2}-\frac{1}{8 a c^5 (1-a x)}+\frac{\int \frac{1}{-1+a^2 x^2} \, dx}{8 c^5}\\ &=-\frac{1}{6 a c^5 (1-a x)^3}-\frac{1}{8 a c^5 (1-a x)^2}-\frac{1}{8 a c^5 (1-a x)}-\frac{\tanh ^{-1}(a x)}{8 a c^5}\\ \end{align*}
Mathematica [A] time = 0.0312029, size = 44, normalized size = 0.64 \[ \frac{3 a^2 x^2-9 a x-3 (a x-1)^3 \tanh ^{-1}(a x)+10}{24 a c^5 (a x-1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 75, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( ax+1 \right ) }{16\,{c}^{5}a}}+{\frac{1}{6\,{c}^{5}a \left ( ax-1 \right ) ^{3}}}-{\frac{1}{8\,{c}^{5}a \left ( ax-1 \right ) ^{2}}}+{\frac{1}{8\,{c}^{5}a \left ( ax-1 \right ) }}+{\frac{\ln \left ( ax-1 \right ) }{16\,{c}^{5}a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03516, size = 113, normalized size = 1.64 \begin{align*} \frac{3 \, a^{2} x^{2} - 9 \, a x + 10}{24 \,{\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}} - \frac{\log \left (a x + 1\right )}{16 \, a c^{5}} + \frac{\log \left (a x - 1\right )}{16 \, a c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55361, size = 251, normalized size = 3.64 \begin{align*} \frac{6 \, a^{2} x^{2} - 18 \, a x - 3 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (a x + 1\right ) + 3 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (a x - 1\right ) + 20}{48 \,{\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.11456, size = 76, normalized size = 1.1 \begin{align*} \frac{3 a^{2} x^{2} - 9 a x + 10}{24 a^{4} c^{5} x^{3} - 72 a^{3} c^{5} x^{2} + 72 a^{2} c^{5} x - 24 a c^{5}} - \frac{- \frac{\log{\left (x - \frac{1}{a} \right )}}{16} + \frac{\log{\left (x + \frac{1}{a} \right )}}{16}}{a c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13845, size = 120, normalized size = 1.74 \begin{align*} -\frac{\log \left ({\left | -\frac{2 \, c}{a c x - c} - 1 \right |}\right )}{16 \, a c^{5}} + \frac{\frac{3 \, a^{2} c^{2}}{a c x - c} - \frac{3 \, a^{2} c^{3}}{{\left (a c x - c\right )}^{2}} + \frac{4 \, a^{2} c^{4}}{{\left (a c x - c\right )}^{3}}}{24 \, a^{3} c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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