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 = 0.198525, 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 \[ \sqrt{c x-1} \sqrt{c x+1}+\tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 + c^2*x^2)/(x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]
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Rubi in Sympy [A] time = 9.2897, size = 34, normalized size = 0.85 \[ \sqrt{c x - 1} \sqrt{c x + 1} + \operatorname{atan}{\left (\sqrt{c x - 1} \sqrt{c x + 1} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c**2*x**2+1)/x/(c*x-1)**(1/2)/(c*x+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0540141, size = 42, normalized size = 1.05 \[ \sqrt{c x-1} \sqrt{c x+1}-\tan ^{-1}\left (\frac{1}{\sqrt{c x-1} \sqrt{c x+1}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 + c^2*x^2)/(x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]
[Out]
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Maple [A] time = 0.024, size = 53, normalized size = 1.3 \[{1\sqrt{cx-1}\sqrt{cx+1} \left ( \sqrt{{c}^{2}{x}^{2}-1}-\arctan \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}} \right ) \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c^2*x^2+1)/x/(c*x-1)^(1/2)/(c*x+1)^(1/2),x)
[Out]
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Maxima [A] time = 1.52933, size = 34, normalized size = 0.85 \[ \sqrt{c^{2} x^{2} - 1} - \arcsin \left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c^2*x^2 + 1)/(sqrt(c*x + 1)*sqrt(c*x - 1)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.241092, size = 127, normalized size = 3.18 \[ -\frac{c^{2} x^{2} - \sqrt{c x + 1} \sqrt{c x - 1} c x - 2 \,{\left (c x - \sqrt{c x + 1} \sqrt{c x - 1}\right )} \arctan \left (-c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) - 1}{c x - \sqrt{c x + 1} \sqrt{c x - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c^2*x^2 + 1)/(sqrt(c*x + 1)*sqrt(c*x - 1)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 31.06, size = 148, normalized size = 3.7 \[ \frac{{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c**2*x**2+1)/x/(c*x-1)**(1/2)/(c*x+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.219661, size = 54, normalized size = 1.35 \[ \sqrt{c x + 1} \sqrt{c x - 1} - 2 \, \arctan \left (\frac{1}{2} \,{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c^2*x^2 + 1)/(sqrt(c*x + 1)*sqrt(c*x - 1)*x),x, algorithm="giac")
[Out]