Optimal. Leaf size=40 \[ \sqrt{c x-1} \sqrt{c x+1}+\tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right ) \]
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Rubi [A] time = 0.0544183, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {460, 92, 205} \[ \sqrt{c x-1} \sqrt{c x+1}+\tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right ) \]
Antiderivative was successfully verified.
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Rule 460
Rule 92
Rule 205
Rubi steps
\begin{align*} \int \frac{1+c^2 x^2}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx &=\sqrt{-1+c x} \sqrt{1+c x}+\int \frac{1}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\sqrt{-1+c x} \sqrt{1+c x}+c \operatorname{Subst}\left (\int \frac{1}{c+c x^2} \, dx,x,\sqrt{-1+c x} \sqrt{1+c x}\right )\\ &=\sqrt{-1+c x} \sqrt{1+c x}+\tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )\\ \end{align*}
Mathematica [A] time = 0.0239749, size = 56, normalized size = 1.4 \[ \frac{c^2 x^2+\sqrt{c^2 x^2-1} \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )-1}{\sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.071, size = 53, normalized size = 1.3 \begin{align*}{ \left ( \sqrt{{c}^{2}{x}^{2}-1}-\arctan \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}} \right ) \right ) \sqrt{cx-1}\sqrt{cx+1}{\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.65547, size = 34, normalized size = 0.85 \begin{align*} \sqrt{c^{2} x^{2} - 1} - \arcsin \left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55084, size = 103, normalized size = 2.58 \begin{align*} \sqrt{c x + 1} \sqrt{c x - 1} + 2 \, \arctan \left (-c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 16.1366, size = 148, normalized size = 3.7 \begin{align*} \frac{{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13011, size = 54, normalized size = 1.35 \begin{align*} \sqrt{c x + 1} \sqrt{c x - 1} - 2 \, \arctan \left (\frac{1}{2} \,{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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