Optimal. Leaf size=30 \[ -\frac{\tanh ^{-1}\left (\frac{\sin (2 x)}{\sqrt{2} \sqrt{1-\cos (2 x)}}\right )}{\sqrt{2}} \]
[Out]
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Rubi [A] time = 0.0277617, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{\tanh ^{-1}\left (\frac{\sin (2 x)}{\sqrt{2} \sqrt{1-\cos (2 x)}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[1 - Cos[2*x]],x]
[Out]
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Rubi in Sympy [A] time = 0.607716, size = 31, normalized size = 1.03 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sin{\left (2 x \right )}}{2 \sqrt{- \cos{\left (2 x \right )} + 1}} \right )}}{2} \]
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 = 0.0261692, size = 33, normalized size = 1.1 \[ -\frac{\sin (x) \left (\log \left (\cos \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )\right )\right )}{\sqrt{1-\cos (2 x)}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[1 - Cos[2*x]],x]
[Out]
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Maple [A] time = 0.055, size = 17, normalized size = 0.6 \[ -{\frac{\sin \left ( x \right ){\it Artanh} \left ( \cos \left ( x \right ) \right ) \sqrt{2}}{2}{\frac{1}{\sqrt{ \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \]
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 = 1.7707, size = 51, normalized size = 1.7 \[ -\frac{1}{4} \, \sqrt{2}{\left (\log \left (\frac{\sqrt{2}}{\sqrt{\cos \left (2 \, x\right ) + 1}} + 1\right ) - \log \left (\frac{\sqrt{2}}{\sqrt{\cos \left (2 \, x\right ) + 1}} - 1\right )\right )} \]
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 = 0.214218, size = 72, normalized size = 2.4 \[ \frac{1}{4} \, \sqrt{2} \log \left (-\frac{{\left (\cos \left (2 \, x\right ) + 3\right )} \sqrt{-\cos \left (2 \, x\right ) + 1} - 2 \, \sqrt{2} \sin \left (2 \, x\right )}{{\left (\cos \left (2 \, x\right ) - 1\right )} \sqrt{-\cos \left (2 \, x\right ) + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(-cos(2*x) + 1),x, algorithm="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]