Optimal. Leaf size=111 \[ -\frac{(a x+1)^2}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{2 (1-a x) (a x+1)}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{2 \sqrt{1-a x} \sqrt{a x+1} \sin ^{-1}(a x)}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}} \]
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Rubi [A] time = 0.307312, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {6167, 6159, 6129, 78, 50, 41, 216} \[ -\frac{(a x+1)^2}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{2 (1-a x) (a x+1)}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{2 \sqrt{1-a x} \sqrt{a x+1} \sin ^{-1}(a x)}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6159
Rule 6129
Rule 78
Rule 50
Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a^2 x^2}}} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{\sqrt{c-\frac{c}{a^2 x^2}}} \, dx\\ &=-\frac{\left (\sqrt{1-a x} \sqrt{1+a x}\right ) \int \frac{e^{2 \tanh ^{-1}(a x)} x}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{\sqrt{c-\frac{c}{a^2 x^2}} x}\\ &=-\frac{\left (\sqrt{1-a x} \sqrt{1+a x}\right ) \int \frac{x \sqrt{1+a x}}{(1-a x)^{3/2}} \, dx}{\sqrt{c-\frac{c}{a^2 x^2}} x}\\ &=-\frac{(1+a x)^2}{a^2 \sqrt{c-\frac{c}{a^2 x^2}} x}+\frac{\left (2 \sqrt{1-a x} \sqrt{1+a x}\right ) \int \frac{\sqrt{1+a x}}{\sqrt{1-a x}} \, dx}{a \sqrt{c-\frac{c}{a^2 x^2}} x}\\ &=-\frac{2 (1-a x) (1+a x)}{a^2 \sqrt{c-\frac{c}{a^2 x^2}} x}-\frac{(1+a x)^2}{a^2 \sqrt{c-\frac{c}{a^2 x^2}} x}+\frac{\left (2 \sqrt{1-a x} \sqrt{1+a x}\right ) \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{a \sqrt{c-\frac{c}{a^2 x^2}} x}\\ &=-\frac{2 (1-a x) (1+a x)}{a^2 \sqrt{c-\frac{c}{a^2 x^2}} x}-\frac{(1+a x)^2}{a^2 \sqrt{c-\frac{c}{a^2 x^2}} x}+\frac{\left (2 \sqrt{1-a x} \sqrt{1+a x}\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a \sqrt{c-\frac{c}{a^2 x^2}} x}\\ &=-\frac{2 (1-a x) (1+a x)}{a^2 \sqrt{c-\frac{c}{a^2 x^2}} x}-\frac{(1+a x)^2}{a^2 \sqrt{c-\frac{c}{a^2 x^2}} x}+\frac{2 \sqrt{1-a x} \sqrt{1+a x} \sin ^{-1}(a x)}{a^2 \sqrt{c-\frac{c}{a^2 x^2}} x}\\ \end{align*}
Mathematica [A] time = 0.0708311, size = 68, normalized size = 0.61 \[ \frac{a^2 x^2+2 \sqrt{a^2 x^2-1} \log \left (\sqrt{a^2 x^2-1}+a x\right )-2 a x-3}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.184, size = 177, normalized size = 1.6 \begin{align*}{\frac{1}{x \left ( ax-1 \right ) a}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}} \left ( \sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}\sqrt{c}x{a}^{2}+2\,\ln \left ( x\sqrt{c}+\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}} \right ) xac-\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}a\sqrt{c}-2\,a\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}\sqrt{c}-2\,\ln \left ( x\sqrt{c}+\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}} \right ) c \right ){\frac{1}{\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}}}}{c}^{-{\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{a x + 1}{{\left (a x - 1\right )} \sqrt{c - \frac{c}{a^{2} x^{2}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76919, size = 448, normalized size = 4.04 \begin{align*} \left [\frac{{\left (a x - 1\right )} \sqrt{c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) +{\left (a^{2} x^{2} - 3 \, a x\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x - a c}, -\frac{2 \,{\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) -{\left (a^{2} x^{2} - 3 \, a x\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x - a c}\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{- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\right )} \left (a x - 1\right )}\, 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 x - 1\right )} \sqrt{c - \frac{c}{a^{2} x^{2}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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