Optimal. Leaf size=44 \[ \frac{a x+1}{c \sqrt{1-a^2 x^2}}-\frac{\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c} \]
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Rubi [A] time = 0.0993619, antiderivative size = 44, 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, 12, 266, 63, 208} \[ \frac{a x+1}{c \sqrt{1-a^2 x^2}}-\frac{\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c} \]
Antiderivative was successfully verified.
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Rule 6148
Rule 823
Rule 12
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )} \, dx &=\frac{\int \frac{1+a x}{x \left (1-a^2 x^2\right )^{3/2}} \, dx}{c}\\ &=\frac{1+a x}{c \sqrt{1-a^2 x^2}}+\frac{\int \frac{a^2}{x \sqrt{1-a^2 x^2}} \, dx}{a^2 c}\\ &=\frac{1+a x}{c \sqrt{1-a^2 x^2}}+\frac{\int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{c}\\ &=\frac{1+a x}{c \sqrt{1-a^2 x^2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{2 c}\\ &=\frac{1+a x}{c \sqrt{1-a^2 x^2}}-\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^2 c}\\ &=\frac{1+a x}{c \sqrt{1-a^2 x^2}}-\frac{\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c}\\ \end{align*}
Mathematica [A] time = 0.0213502, size = 55, normalized size = 1.25 \[ -\frac{a^3 (-x)-a^2}{a^2 c \sqrt{1-a^2 x^2}}-\frac{\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 60, normalized size = 1.4 \begin{align*} -{\frac{1}{c} \left ({\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +{\frac{1}{a}\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}{{\left (a^{2} c x^{2} - c\right )} \sqrt{-a^{2} x^{2} + 1} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56868, size = 119, normalized size = 2.7 \begin{align*} \frac{a x +{\left (a x - 1\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - \sqrt{-a^{2} x^{2} + 1} - 1}{a c x - c} \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^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{2} x^{3} \sqrt{- a^{2} x^{2} + 1} + x \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 [A] time = 1.22, size = 108, normalized size = 2.45 \begin{align*} -\frac{a \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{2 \, a}{c{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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