Optimal. Leaf size=337 \[ \sqrt {2} \left (\sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {-2+2 \sqrt {2}} \left (-\sqrt {2}-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}\right )}{2 \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}}\right )-\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+2 \sqrt {2}} \left (-\sqrt {2}-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}\right )}{2 \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}}\right )-\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {-2+2 \sqrt {2}} \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}}{\sqrt {2}-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}\right )+\sqrt {-1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+2 \sqrt {2}} \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}}{\sqrt {2}-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}\right )\right ) \cot (x) \sqrt {-1+\sec (x)} \sqrt {1+\sec (x)} \]
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Rubi [F]
time = 0.58, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}} \, dx &=\int \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}} \, dx\\ \end {align*}
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Mathematica [A]
time = 1.48, size = 552, normalized size = 1.64 \begin {gather*} \frac {\sqrt [4]{2} \cos (x) \left (\sqrt {-1+\sec (x)}-\sqrt {1+\sec (x)}\right )^2 \left (2 \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )-\frac {\csc \left (\frac {\pi }{8}\right ) \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}}{\sqrt [4]{2}}\right ) \cos \left (\frac {\pi }{8}\right )-2 \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )+\frac {\csc \left (\frac {\pi }{8}\right ) \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}}{\sqrt [4]{2}}\right ) \cos \left (\frac {\pi }{8}\right )+\cos \left (\frac {\pi }{8}\right ) \log \left (2+\sqrt {2} \left (-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}\right )-2\ 2^{3/4} \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}} \sin \left (\frac {\pi }{8}\right )\right )-\cos \left (\frac {\pi }{8}\right ) \log \left (2+\sqrt {2} \left (-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}\right )+2\ 2^{3/4} \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}} \sin \left (\frac {\pi }{8}\right )\right )+2 \tan ^{-1}\left (\frac {\sec \left (\frac {\pi }{8}\right ) \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}}{\sqrt [4]{2}}-\tan \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )+2 \tan ^{-1}\left (\frac {\sec \left (\frac {\pi }{8}\right ) \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}}{\sqrt [4]{2}}+\tan \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )-\log \left (2-2\ 2^{3/4} \cos \left (\frac {\pi }{8}\right ) \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}+\sqrt {2} \left (-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}\right )\right ) \sin \left (\frac {\pi }{8}\right )+\log \left (2+\sqrt [4]{2} \csc \left (\frac {\pi }{8}\right ) \sqrt {-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}}+\sqrt {2} \left (-\sqrt {-1+\sec (x)}+\sqrt {1+\sec (x)}\right )\right ) \sin \left (\frac {\pi }{8}\right )\right ) \sin (x)}{-1+\cos (2 x)+2 \cos (x) \sqrt {-1+\sec (x)} \sqrt {1+\sec (x)}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \sqrt {-\sqrt {-1+\sec \left (x \right )}+\sqrt {1+\sec \left (x \right )}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- \sqrt {\sec {\left (x \right )} - 1} + \sqrt {\sec {\left (x \right )} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sqrt {\sqrt {\frac {1}{\cos \left (x\right )}+1}-\sqrt {\frac {1}{\cos \left (x\right )}-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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