Optimal. Leaf size=69 \[ \frac{2 a (a x+1)}{c \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{c x}-\frac{2 a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c} \]
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Rubi [A] time = 0.193182, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {6128, 852, 1805, 807, 266, 63, 208} \[ \frac{2 a (a x+1)}{c \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{c x}-\frac{2 a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c} \]
Antiderivative was successfully verified.
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Rule 6128
Rule 852
Rule 1805
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x^2 (c-a c x)} \, dx &=c \int \frac{\sqrt{1-a^2 x^2}}{x^2 (c-a c x)^2} \, dx\\ &=\frac{\int \frac{(c+a c x)^2}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac{2 a (1+a x)}{c \sqrt{1-a^2 x^2}}-\frac{\int \frac{-c^2-2 a c^2 x}{x^2 \sqrt{1-a^2 x^2}} \, dx}{c^3}\\ &=\frac{2 a (1+a x)}{c \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{c x}+\frac{(2 a) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{c}\\ &=\frac{2 a (1+a x)}{c \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{c x}+\frac{a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{c}\\ &=\frac{2 a (1+a x)}{c \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{c x}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{a c}\\ &=\frac{2 a (1+a x)}{c \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{c x}-\frac{2 a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c}\\ \end{align*}
Mathematica [A] time = 0.0248545, size = 68, normalized size = 0.99 \[ \frac{3 a^2 x^2-2 a x \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+2 a x-1}{c x \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 77, normalized size = 1.1 \begin{align*} -{\frac{1}{c} \left ({\frac{1}{x}\sqrt{-{a}^{2}{x}^{2}+1}}+2\,a{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +2\,{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-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}{\sqrt{-a^{2} x^{2} + 1}{\left (a c x - c\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61137, size = 165, normalized size = 2.39 \begin{align*} \frac{2 \, a^{2} x^{2} - 2 \, a x + 2 \,{\left (a^{2} x^{2} - a x\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - \sqrt{-a^{2} x^{2} + 1}{\left (3 \, a x - 1\right )}}{a c x^{2} - c x} \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{a x}{a x^{3} \sqrt{- a^{2} x^{2} + 1} - x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a x^{3} \sqrt{- a^{2} x^{2} + 1} - x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25233, size = 215, normalized size = 3.12 \begin{align*} -\frac{2 \, a^{2} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{c{\left | a \right |}} - \frac{{\left (a^{2} - \frac{9 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{x}\right )} a^{2} x}{2 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{2 \, c x{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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