Optimal. Leaf size=91 \[ \frac{\sqrt{1-a^2 x^2}}{2 a c (1-a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{2 a c \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.0928384, antiderivative size = 91, 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 = {6143, 6140, 44, 207} \[ \frac{\sqrt{1-a^2 x^2}}{2 a c (1-a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{2 a c \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6143
Rule 6140
Rule 44
Rule 207
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{\tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{1}{(1-a x)^2 (1+a x)} \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \left (\frac{1}{2 (-1+a x)^2}-\frac{1}{2 \left (-1+a^2 x^2\right )}\right ) \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2}}{2 a c (1-a x) \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} \int \frac{1}{-1+a^2 x^2} \, dx}{2 c \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2}}{2 a c (1-a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{2 a c \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0296628, size = 60, normalized size = 0.66 \[ \frac{\sqrt{1-a^2 x^2} \left (\frac{1}{2 a (1-a x)}+\frac{\tanh ^{-1}(a x)}{2 a}\right )}{c \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.09, size = 88, normalized size = 1. \begin{align*} -{\frac{ax\ln \left ( ax+1 \right ) -\ln \left ( ax-1 \right ) xa-\ln \left ( ax+1 \right ) +\ln \left ( ax-1 \right ) -2}{ \left ( 4\,{a}^{2}{x}^{2}-4 \right ){c}^{2}a \left ( ax-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }} \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{a x + 1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01044, size = 705, normalized size = 7.75 \begin{align*} \left [\frac{4 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a x +{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \sqrt{c} \log \left (-\frac{a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \,{\left (a^{3} x^{3} + a x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right )}{8 \,{\left (a^{4} c^{2} x^{3} - a^{3} c^{2} x^{2} - a^{2} c^{2} x + a c^{2}\right )}}, \frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a x +{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \sqrt{-c} \arctan \left (\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a \sqrt{-c} x}{a^{4} c x^{4} - c}\right )}{4 \,{\left (a^{4} c^{2} x^{3} - a^{3} c^{2} x^{2} - a^{2} c^{2} x + a c^{2}\right )}}\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*} \int \frac{a x + 1}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \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{a x + 1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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