Optimal. Leaf size=97 \[ \frac{2 \sqrt{1-a^2 x^2}}{15 a c^2 (1-a x)}+\frac{2 \sqrt{1-a^2 x^2}}{15 a c^2 (1-a x)^2}+\frac{\sqrt{1-a^2 x^2}}{5 a c^2 (1-a x)^3} \]
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Rubi [A] time = 0.0742634, antiderivative size = 97, 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 = {6138, 655, 659, 651} \[ \frac{2 \sqrt{1-a^2 x^2}}{15 a c^2 (1-a x)}+\frac{2 \sqrt{1-a^2 x^2}}{15 a c^2 (1-a x)^2}+\frac{\sqrt{1-a^2 x^2}}{5 a c^2 (1-a x)^3} \]
Antiderivative was successfully verified.
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Rule 6138
Rule 655
Rule 659
Rule 651
Rubi steps
\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac{\int \frac{(1+a x)^3}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^2}\\ &=\frac{\int \frac{1}{(1-a x)^3 \sqrt{1-a^2 x^2}} \, dx}{c^2}\\ &=\frac{\sqrt{1-a^2 x^2}}{5 a c^2 (1-a x)^3}+\frac{2 \int \frac{1}{(1-a x)^2 \sqrt{1-a^2 x^2}} \, dx}{5 c^2}\\ &=\frac{\sqrt{1-a^2 x^2}}{5 a c^2 (1-a x)^3}+\frac{2 \sqrt{1-a^2 x^2}}{15 a c^2 (1-a x)^2}+\frac{2 \int \frac{1}{(1-a x) \sqrt{1-a^2 x^2}} \, dx}{15 c^2}\\ &=\frac{\sqrt{1-a^2 x^2}}{5 a c^2 (1-a x)^3}+\frac{2 \sqrt{1-a^2 x^2}}{15 a c^2 (1-a x)^2}+\frac{2 \sqrt{1-a^2 x^2}}{15 a c^2 (1-a x)}\\ \end{align*}
Mathematica [A] time = 0.0179653, size = 43, normalized size = 0.44 \[ \frac{\sqrt{a x+1} \left (2 a^2 x^2-6 a x+7\right )}{15 a c^2 (1-a x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.03, size = 49, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,{a}^{2}{x}^{2}-6\,ax+7 \right ) \left ( ax+1 \right ) ^{2}}{ \left ( 15\,ax-15 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{3}}{{\left (a^{2} c x^{2} - c\right )}^{2}{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.59292, size = 190, normalized size = 1.96 \begin{align*} \frac{7 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 21 \, a x -{\left (2 \, a^{2} x^{2} - 6 \, a x + 7\right )} \sqrt{-a^{2} x^{2} + 1} - 7}{15 \,{\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^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{2}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{3}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \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 [A] time = 1.19143, size = 196, normalized size = 2.02 \begin{align*} -\frac{2 \,{\left (\frac{20 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{40 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac{30 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} - 7\right )}}{15 \, c^{2}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{5}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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