Optimal. Leaf size=27 \[ \frac{\tanh ^{-1}\left (\frac{\sin (2 x)}{\sqrt{2} \sqrt{\cos (2 x)+1}}\right )}{\sqrt{2}} \]
[Out]
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Rubi [A] time = 0.0234976, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\tanh ^{-1}\left (\frac{\sin (2 x)}{\sqrt{2} \sqrt{\cos (2 x)+1}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[1 + Cos[2*x]],x]
[Out]
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Rubi in Sympy [A] time = 0.574884, size = 29, normalized size = 1.07 \[ \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sin{\left (2 x \right )}}{2 \sqrt{\cos{\left (2 x \right )} + 1}} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1+cos(2*x))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0257318, size = 47, normalized size = 1.74 \[ -\frac{\cos (x) \left (\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )\right )}{\sqrt{\cos (2 x)+1}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[1 + Cos[2*x]],x]
[Out]
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Maple [C] time = 0.033, size = 9, normalized size = 0.3 \[{\frac{\sqrt{2}{\it InverseJacobiAM} \left ( x,1 \right ) }{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1+cos(2*x))^(1/2),x)
[Out]
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Maxima [A] time = 1.73778, size = 26, normalized size = 0.96 \[ \frac{1}{2} \, \sqrt{2} \operatorname{arsinh}\left (\frac{\sin \left (2 \, x\right )}{\cos \left (2 \, x\right ) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(cos(2*x) + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.216912, size = 55, normalized size = 2.04 \[ \frac{1}{4} \, \sqrt{2} \log \left (-\frac{\sqrt{\cos \left (2 \, x\right ) + 1}{\left (\cos \left (2 \, x\right ) - 3\right )} - 2 \, \sqrt{2} \sin \left (2 \, x\right )}{{\left (\cos \left (2 \, x\right ) + 1\right )}^{\frac{3}{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(cos(2*x) + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\cos{\left (2 x \right )} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1+cos(2*x))**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\cos \left (2 \, x\right ) + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(cos(2*x) + 1),x, algorithm="giac")
[Out]