Optimal. Leaf size=117 \[ \sqrt{c-\frac{c}{a^2 x^2}}-\frac{a x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{\sqrt{1-a x} \sqrt{a x+1}}-\frac{2 a x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{\sqrt{1-a x} \sqrt{a x+1}} \]
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Rubi [A] time = 0.525426, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6167, 6159, 6129, 98, 157, 41, 216, 92, 208} \[ \sqrt{c-\frac{c}{a^2 x^2}}-\frac{a x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{\sqrt{1-a x} \sqrt{a x+1}}-\frac{2 a x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{\sqrt{1-a x} \sqrt{a x+1}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6159
Rule 6129
Rule 98
Rule 157
Rule 41
Rule 216
Rule 92
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x} \, dx\\ &=-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{1-a x} \sqrt{1+a x}}{x^2} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{(1-a x)^{3/2}}{x^2 \sqrt{1+a x}} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=\sqrt{c-\frac{c}{a^2 x^2}}+\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{2 a-a^2 x}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=\sqrt{c-\frac{c}{a^2 x^2}}+\frac{\left (2 a \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{1}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}-\frac{\left (a^2 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=\sqrt{c-\frac{c}{a^2 x^2}}-\frac{\left (a^2 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}-\frac{\left (2 a^2 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \operatorname{Subst}\left (\int \frac{1}{a-a x^2} \, dx,x,\sqrt{1-a x} \sqrt{1+a x}\right )}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=\sqrt{c-\frac{c}{a^2 x^2}}-\frac{a \sqrt{c-\frac{c}{a^2 x^2}} x \sin ^{-1}(a x)}{\sqrt{1-a x} \sqrt{1+a x}}-\frac{2 a \sqrt{c-\frac{c}{a^2 x^2}} x \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{1+a x}\right )}{\sqrt{1-a x} \sqrt{1+a x}}\\ \end{align*}
Mathematica [A] time = 0.0766057, size = 82, normalized size = 0.7 \[ \frac{\sqrt{c-\frac{c}{a^2 x^2}} \left (\sqrt{a^2 x^2-1}+a x \log \left (\sqrt{a^2 x^2-1}+a x\right )+2 a x \tan ^{-1}\left (\frac{1}{\sqrt{a^2 x^2-1}}\right )\right )}{\sqrt{a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.186, size = 306, normalized size = 2.6 \begin{align*} -{\frac{1}{ac}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}} \left ( -\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{x}^{2}{a}^{3}c+{a}^{3} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{{\frac{3}{2}}}\sqrt{-{\frac{c}{{a}^{2}}}}+{c}^{{\frac{3}{2}}}\ln \left ( x\sqrt{c}+\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}} \right ) \sqrt{-{\frac{c}{{a}^{2}}}}xa-2\,{c}^{3/2}\sqrt{-{\frac{c}{{a}^{2}}}}\ln \left ({\frac{1}{\sqrt{c}} \left ( \sqrt{c}\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}+cx \right ) } \right ) xa+2\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}x{a}^{2}c-2\,\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}cx{a}^{2}\sqrt{-{\frac{c}{{a}^{2}}}}-2\,\ln \left ( 2\,{\frac{1}{{a}^{2}x} \left ( \sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{a}^{2}-c \right ) } \right ) x{c}^{2} \right ){\frac{1}{\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}}}{\frac{1}{\sqrt{-{\frac{c}{{a}^{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{{\left (a x - 1\right )} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{{\left (a x + 1\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74179, size = 554, normalized size = 4.74 \begin{align*} \left [-\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 ) + \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} - 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}, 2 \, \sqrt{c} \arctan \left (\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + \frac{1}{2} \, \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 ) + \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}\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{\sqrt{- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\right )} \left (a x - 1\right )}{x \left (a x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31971, size = 171, normalized size = 1.46 \begin{align*} -{\left (\frac{4 \, \sqrt{c} \arctan \left (-\frac{\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}}{\sqrt{c}}\right ) \mathrm{sgn}\left (x\right )}{a} + \frac{\sqrt{c} \log \left ({\left | -\sqrt{a^{2} c} x + \sqrt{a^{2} c x^{2} - c} \right |}\right ) \mathrm{sgn}\left (x\right )}{{\left | a \right |}} - \frac{2 \, c^{\frac{3}{2}} \mathrm{sgn}\left (x\right )}{{\left ({\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} + c\right )}{\left | a \right |}}\right )}{\left | a \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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