Optimal. Leaf size=337 \[ \sqrt{2} \cot (x) \sqrt{\sec (x)-1} \sqrt{\sec (x)+1} \left (\sqrt{\sqrt{2}-1} \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{2}-2} \left (-\sqrt{\sec (x)-1}+\sqrt{\sec (x)+1}-\sqrt{2}\right )}{2 \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}}\right )-\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+2 \sqrt{2}} \left (-\sqrt{\sec (x)-1}+\sqrt{\sec (x)+1}-\sqrt{2}\right )}{2 \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}}\right )-\sqrt{1+\sqrt{2}} \tanh ^{-1}\left (\frac{\sqrt{2 \sqrt{2}-2} \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}}{-\sqrt{\sec (x)-1}+\sqrt{\sec (x)+1}+\sqrt{2}}\right )+\sqrt{\sqrt{2}-1} \tanh ^{-1}\left (\frac{\sqrt{2+2 \sqrt{2}} \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}}{-\sqrt{\sec (x)-1}+\sqrt{\sec (x)+1}+\sqrt{2}}\right )\right ) \]
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Rubi [F] time = 1.37635, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\sqrt{-\sqrt{-1+\sec (x)}+\sqrt{1+\sec (x)}},x\right ) \]
Verification is Not applicable to the result.
[In] Int[Sqrt[-Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]]],x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-(-1+sec(x))**(1/2)+(1+sec(x))**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 2.8762, size = 552, normalized size = 1.64 \[ \frac{\sqrt [4]{2} \sin (x) \cos (x) \left (\sqrt{\sec (x)-1}-\sqrt{\sec (x)+1}\right )^2 \left (2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sec \left (\frac{\pi }{8}\right ) \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}}{\sqrt [4]{2}}-\tan \left (\frac{\pi }{8}\right )\right )+2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sec \left (\frac{\pi }{8}\right ) \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}}{\sqrt [4]{2}}+\tan \left (\frac{\pi }{8}\right )\right )+\cos \left (\frac{\pi }{8}\right ) \log \left (\sqrt{2} \left (\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}\right )-2\ 2^{3/4} \sin \left (\frac{\pi }{8}\right ) \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}+2\right )-\cos \left (\frac{\pi }{8}\right ) \log \left (\sqrt{2} \left (\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}\right )+2\ 2^{3/4} \sin \left (\frac{\pi }{8}\right ) \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}+2\right )-\sin \left (\frac{\pi }{8}\right ) \log \left (\sqrt{2} \left (\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}\right )-2\ 2^{3/4} \cos \left (\frac{\pi }{8}\right ) \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}+2\right )+\sin \left (\frac{\pi }{8}\right ) \log \left (\sqrt{2} \left (\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}\right )+\sqrt [4]{2} \csc \left (\frac{\pi }{8}\right ) \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}+2\right )+2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\csc \left (\frac{\pi }{8}\right ) \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}}{\sqrt [4]{2}}\right )-2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\csc \left (\frac{\pi }{8}\right ) \sqrt{\sqrt{\sec (x)+1}-\sqrt{\sec (x)-1}}}{\sqrt [4]{2}}+\cot \left (\frac{\pi }{8}\right )\right )\right )}{\cos (2 x)+2 \cos (x) \sqrt{\sec (x)-1} \sqrt{\sec (x)+1}-1} \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[-Sqrt[-1 + Sec[x]] + Sqrt[1 + Sec[x]]],x]
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Maple [F] time = 0.194, size = 0, normalized size = 0. \[ \int \sqrt{-\sqrt{-1+\sec \left ( x \right ) }+\sqrt{1+\sec \left ( x \right ) }}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-(-1+sec(x))^(1/2)+(1+sec(x))^(1/2))^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\sqrt{\sec \left (x\right ) + 1} - \sqrt{\sec \left (x\right ) - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(sec(x) + 1) - sqrt(sec(x) - 1)),x, algorithm="maxima")
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Fricas [A] time = 5.3774, size = 1690, normalized size = 5.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(sec(x) + 1) - sqrt(sec(x) - 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- \sqrt{\sec{\left (x \right )} - 1} + \sqrt{\sec{\left (x \right )} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-(-1+sec(x))**(1/2)+(1+sec(x))**(1/2))**(1/2),x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(sec(x) + 1) - sqrt(sec(x) - 1)),x, algorithm="giac")
[Out]