Optimal. Leaf size=70 \[ \frac{a x+1}{c x \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{1-a^2 x^2}}{c x}-\frac{a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c} \]
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Rubi [A] time = 0.116746, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {6148, 823, 807, 266, 63, 208} \[ \frac{a x+1}{c x \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{1-a^2 x^2}}{c x}-\frac{a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c} \]
Antiderivative was successfully verified.
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Rule 6148
Rule 823
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx &=\frac{\int \frac{1+a x}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c}\\ &=\frac{1+a x}{c x \sqrt{1-a^2 x^2}}+\frac{\int \frac{2 a^2+a^3 x}{x^2 \sqrt{1-a^2 x^2}} \, dx}{a^2 c}\\ &=\frac{1+a x}{c x \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{1-a^2 x^2}}{c x}+\frac{a \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{c}\\ &=\frac{1+a x}{c x \sqrt{1-a^2 x^2}}-\frac{2 \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 )}{2 c}\\ &=\frac{1+a x}{c x \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{1-a^2 x^2}}{c x}-\frac{\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{1+a x}{c x \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{1-a^2 x^2}}{c x}-\frac{a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c}\\ \end{align*}
Mathematica [A] time = 0.0259739, size = 67, normalized size = 0.96 \[ \frac{2 a^2 x^2-a x \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+a x-1}{c x \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 75, normalized size = 1.1 \begin{align*} -{\frac{1}{c} \left ({\frac{1}{x}\sqrt{-{a}^{2}{x}^{2}+1}}+a{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +{\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 [A] time = 1.00868, size = 140, normalized size = 2. \begin{align*} -\frac{\frac{a^{2} \log \left (\sqrt{-a^{2} x^{2} + 1} + 1\right )}{c} - \frac{a^{2} \log \left (\sqrt{-a^{2} x^{2} + 1} - 1\right )}{c} - \frac{2 \, a^{2}}{\sqrt{-a^{2} x^{2} + 1} c}}{2 \, a} + \frac{2 \, a^{2} x^{2} - 1}{\sqrt{a x + 1} \sqrt{-a x + 1} c x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58153, size = 157, normalized size = 2.24 \begin{align*} \frac{a^{2} x^{2} - a x +{\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 (2 \, 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}{- a^{2} x^{3} \sqrt{- a^{2} x^{2} + 1} + x \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{2} x^{4} \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 = 1.21332, size = 215, normalized size = 3.07 \begin{align*} -\frac{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{5 \,{\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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