Optimal. Leaf size=84 \[ -\frac{1}{16 a c^3 (1-a x)}+\frac{3}{16 a c^3 (a x+1)}+\frac{1}{8 a c^3 (a x+1)^2}+\frac{1}{12 a c^3 (a x+1)^3}-\frac{\tanh ^{-1}(a x)}{4 a c^3} \]
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Rubi [A] time = 0.10043, antiderivative size = 84, 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, 6140, 44, 207} \[ -\frac{1}{16 a c^3 (1-a x)}+\frac{3}{16 a c^3 (a x+1)}+\frac{1}{8 a c^3 (a x+1)^2}+\frac{1}{12 a c^3 (a x+1)^3}-\frac{\tanh ^{-1}(a x)}{4 a c^3} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6140
Rule 44
Rule 207
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx\\ &=-\frac{\int \frac{1}{(1-a x)^2 (1+a x)^4} \, dx}{c^3}\\ &=-\frac{\int \left (\frac{1}{16 (-1+a x)^2}+\frac{1}{4 (1+a x)^4}+\frac{1}{4 (1+a x)^3}+\frac{3}{16 (1+a x)^2}-\frac{1}{4 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^3}\\ &=-\frac{1}{16 a c^3 (1-a x)}+\frac{1}{12 a c^3 (1+a x)^3}+\frac{1}{8 a c^3 (1+a x)^2}+\frac{3}{16 a c^3 (1+a x)}+\frac{\int \frac{1}{-1+a^2 x^2} \, dx}{4 c^3}\\ &=-\frac{1}{16 a c^3 (1-a x)}+\frac{1}{12 a c^3 (1+a x)^3}+\frac{1}{8 a c^3 (1+a x)^2}+\frac{3}{16 a c^3 (1+a x)}-\frac{\tanh ^{-1}(a x)}{4 a c^3}\\ \end{align*}
Mathematica [A] time = 0.0398514, size = 61, normalized size = 0.73 \[ \frac{3 a^3 x^3+6 a^2 x^2+a x-3 (a x-1) (a x+1)^3 \tanh ^{-1}(a x)-4}{12 a (a x-1) (a c x+c)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 90, normalized size = 1.1 \begin{align*}{\frac{1}{12\,a{c}^{3} \left ( ax+1 \right ) ^{3}}}+{\frac{1}{8\,a{c}^{3} \left ( ax+1 \right ) ^{2}}}+{\frac{3}{16\,a{c}^{3} \left ( ax+1 \right ) }}-{\frac{\ln \left ( ax+1 \right ) }{8\,a{c}^{3}}}+{\frac{1}{16\,a{c}^{3} \left ( ax-1 \right ) }}+{\frac{\ln \left ( ax-1 \right ) }{8\,a{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0935, size = 123, normalized size = 1.46 \begin{align*} \frac{3 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + a x - 4}{12 \,{\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}} - \frac{\log \left (a x + 1\right )}{8 \, a c^{3}} + \frac{\log \left (a x - 1\right )}{8 \, a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55461, size = 266, normalized size = 3.17 \begin{align*} \frac{6 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 2 \, a x - 3 \,{\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \log \left (a x + 1\right ) + 3 \,{\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \log \left (a x - 1\right ) - 8}{24 \,{\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.652583, size = 83, normalized size = 0.99 \begin{align*} \frac{3 a^{3} x^{3} + 6 a^{2} x^{2} + a x - 4}{12 a^{5} c^{3} x^{4} + 24 a^{4} c^{3} x^{3} - 24 a^{2} c^{3} x - 12 a c^{3}} - \frac{- \frac{\log{\left (x - \frac{1}{a} \right )}}{8} + \frac{\log{\left (x + \frac{1}{a} \right )}}{8}}{a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1132, size = 100, normalized size = 1.19 \begin{align*} -\frac{\log \left ({\left | a x + 1 \right |}\right )}{8 \, a c^{3}} + \frac{\log \left ({\left | a x - 1 \right |}\right )}{8 \, a c^{3}} + \frac{3 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + a x - 4}{12 \,{\left (a x + 1\right )}^{3}{\left (a x - 1\right )} a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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