Optimal. Leaf size=116 \[ x \sqrt{c-\frac{c}{a^2 x^2}}-\frac{2 x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{\sqrt{1-a x} \sqrt{a x+1}}+\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{\sqrt{1-a x} \sqrt{a x+1}} \]
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Rubi [A] time = 0.360192, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6167, 6159, 6129, 102, 157, 41, 216, 92, 208} \[ x \sqrt{c-\frac{c}{a^2 x^2}}-\frac{2 x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{\sqrt{1-a x} \sqrt{a x+1}}+\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{\sqrt{1-a x} \sqrt{a x+1}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6159
Rule 6129
Rule 102
Rule 157
Rule 41
Rule 216
Rule 92
Rule 208
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, 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} \, 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 \sqrt{1-a x}} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=\sqrt{c-\frac{c}{a^2 x^2}} x+\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{-a-2 a^2 x}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{a \sqrt{1-a x} \sqrt{1+a x}}\\ &=\sqrt{c-\frac{c}{a^2 x^2}} x-\frac{\left (\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 (2 a \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}} x+\frac{\left (a \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}}-\frac{\left (2 a \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}}\\ &=\sqrt{c-\frac{c}{a^2 x^2}} x-\frac{2 \sqrt{c-\frac{c}{a^2 x^2}} x \sin ^{-1}(a x)}{\sqrt{1-a x} \sqrt{1+a x}}+\frac{\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 = 0.0780781, size = 80, normalized size = 0.69 \[ \frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left (\sqrt{a^2 x^2-1}+2 \log \left (\sqrt{a^2 x^2-1}+a x\right )-\tan ^{-1}\left (\frac{1}{\sqrt{a^2 x^2-1}}\right )\right )}{\sqrt{a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.191, size = 197, normalized size = 1.7 \begin{align*}{\frac{x}{{a}^{2}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}} \left ( 2\,\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}{a}^{2}\sqrt{-{\frac{c}{{a}^{2}}}}+2\,\sqrt{c}\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 ) a\sqrt{-{\frac{c}{{a}^{2}}}}-\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{a}^{2}-c\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 ) \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}}}}{a x - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7825, size = 576, normalized size = 4.97 \begin{align*} \left [\frac{2 \, a x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 4 \, \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 )}{2 \, a}, \frac{a x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{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 ) + \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 )}{a}\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 )}{a x - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18531, size = 207, normalized size = 1.78 \begin{align*}{\left (\frac{2 \, \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^{2}} - \frac{2 \, \sqrt{c} \log \left ({\left | -\sqrt{a^{2} c} x + \sqrt{a^{2} c x^{2} - c} \right |}\right ) \mathrm{sgn}\left (x\right )}{a{\left | a \right |}} + \frac{\sqrt{a^{2} c x^{2} - c} \mathrm{sgn}\left (x\right )}{a^{2}} - \frac{{\left (2 \, \sqrt{c}{\left | a \right |} \arctan \left (\frac{\sqrt{-c}}{\sqrt{c}}\right ) - a \sqrt{c} \log \left ({\left | c \right |}\right ) + \sqrt{-c}{\left | a \right |}\right )} \mathrm{sgn}\left (x\right )}{a^{2}{\left | a \right |}}\right )}{\left | a \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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