Optimal. Leaf size=32 \[ \frac{\left (1-a^2 x^2\right )^{5/2}}{5 a c^2 (1-a x)^5} \]
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Rubi [A] time = 0.0367227, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6127, 651} \[ \frac{\left (1-a^2 x^2\right )^{5/2}}{5 a c^2 (1-a x)^5} \]
Antiderivative was successfully verified.
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Rule 6127
Rule 651
Rubi steps
\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)}}{(c-a c x)^2} \, dx &=c^3 \int \frac{\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^5} \, dx\\ &=\frac{\left (1-a^2 x^2\right )^{5/2}}{5 a c^2 (1-a x)^5}\\ \end{align*}
Mathematica [A] time = 0.0103136, size = 29, normalized size = 0.91 \[ \frac{(a x+1)^{5/2}}{5 a c^2 (1-a x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.031, size = 35, normalized size = 1.1 \begin{align*} -{\frac{ \left ( ax+1 \right ) ^{4}}{ \left ( 5\,ax-5 \right ){c}^{2}a} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02232, size = 197, normalized size = 6.16 \begin{align*} \frac{8}{5 \,{\left (\sqrt{-a^{2} x^{2} + 1} a^{3} c^{2} x^{2} - 2 \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{2} x + \sqrt{-a^{2} x^{2} + 1} a c^{2}\right )}} + \frac{12}{5 \,{\left (\sqrt{-a^{2} x^{2} + 1} a^{2} c^{2} x - \sqrt{-a^{2} x^{2} + 1} a c^{2}\right )}} + \frac{x}{5 \, \sqrt{-a^{2} x^{2} + 1} c^{2}} + \frac{1}{\sqrt{-a^{2} x^{2} + 1} a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.92553, size = 181, normalized size = 5.66 \begin{align*} \frac{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x -{\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt{-a^{2} x^{2} + 1} - 1}{5 \,{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{3 a x}{- a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{2}}{- a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{3}}{- a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (a c x - c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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