Optimal. Leaf size=41 \[ \frac {1}{2} \cos (x) \sqrt {\sec (x)-1}+\frac {1}{2} \tan ^{-1}\left (\sqrt {\sec (x)-1}\right )-\cos (x) \tan ^{-1}\left (\sqrt {\sec (x)-1}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4335, 5203, 12, 242, 51, 63, 203} \[ \frac {1}{2} \cos (x) \sqrt {\sec (x)-1}+\frac {1}{2} \tan ^{-1}\left (\sqrt {\sec (x)-1}\right )-\cos (x) \tan ^{-1}\left (\sqrt {\sec (x)-1}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 51
Rule 63
Rule 203
Rule 242
Rule 4335
Rule 5203
Rubi steps
\begin {align*} \int \tan ^{-1}\left (\sqrt {-1+\sec (x)}\right ) \sin (x) \, dx &=-\operatorname {Subst}\left (\int \tan ^{-1}\left (\sqrt {-1+\frac {1}{x}}\right ) \, dx,x,\cos (x)\right )\\ &=-\tan ^{-1}\left (\sqrt {-1+\sec (x)}\right ) \cos (x)+\operatorname {Subst}\left (\int -\frac {1}{2 \sqrt {-1+\frac {1}{x}}} \, dx,x,\cos (x)\right )\\ &=-\tan ^{-1}\left (\sqrt {-1+\sec (x)}\right ) \cos (x)-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {1}{x}}} \, dx,x,\cos (x)\right )\\ &=-\tan ^{-1}\left (\sqrt {-1+\sec (x)}\right ) \cos (x)+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^2} \, dx,x,\sec (x)\right )\\ &=-\tan ^{-1}\left (\sqrt {-1+\sec (x)}\right ) \cos (x)+\frac {1}{2} \cos (x) \sqrt {-1+\sec (x)}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,\sec (x)\right )\\ &=-\tan ^{-1}\left (\sqrt {-1+\sec (x)}\right ) \cos (x)+\frac {1}{2} \cos (x) \sqrt {-1+\sec (x)}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+\sec (x)}\right )\\ &=\frac {1}{2} \tan ^{-1}\left (\sqrt {-1+\sec (x)}\right )-\tan ^{-1}\left (\sqrt {-1+\sec (x)}\right ) \cos (x)+\frac {1}{2} \cos (x) \sqrt {-1+\sec (x)}\\ \end {align*}
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Mathematica [C] time = 4.38, size = 283, normalized size = 6.90 \[ -\frac {1}{2} \left (-3-2 \sqrt {2}\right ) \left (\left (\sqrt {2}-2\right ) \cos \left (\frac {x}{2}\right )-\sqrt {2}+1\right ) \cos ^2\left (\frac {x}{4}\right ) \sqrt {-\tan ^2\left (\frac {x}{4}\right )-2 \sqrt {2}+3} \sqrt {\left (2 \sqrt {2}-3\right ) \tan ^2\left (\frac {x}{4}\right )+1} \cot \left (\frac {x}{4}\right ) \sqrt {\sec (x)-1} \sec (x) \sqrt {\left (\left (10-7 \sqrt {2}\right ) \cos \left (\frac {x}{2}\right )-5 \sqrt {2}+7\right ) \sec ^2\left (\frac {x}{4}\right )} \sqrt {\left (\left (2+\sqrt {2}\right ) \cos \left (\frac {x}{2}\right )-\sqrt {2}-1\right ) \sec ^2\left (\frac {x}{4}\right )} \left (\operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\tan \left (\frac {x}{4}\right )}{\sqrt {3-2 \sqrt {2}}}\right ),17-12 \sqrt {2}\right )-2 \Pi \left (-3+2 \sqrt {2};\sin ^{-1}\left (\frac {\tan \left (\frac {x}{4}\right )}{\sqrt {3-2 \sqrt {2}}}\right )|17-12 \sqrt {2}\right )\right )+\frac {1}{2} \cos (x) \sqrt {\sec (x)-1}-\cos (x) \tan ^{-1}\left (\sqrt {\sec (x)-1}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 32, normalized size = 0.78 \[ -\frac {1}{2} \, {\left (2 \, \cos \relax (x) - 1\right )} \arctan \left (\sqrt {\sec \relax (x) - 1}\right ) + \frac {1}{2} \, \sqrt {-\frac {\cos \relax (x) - 1}{\cos \relax (x)}} \cos \relax (x) \]
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 \[ \mathit {undef} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 42, normalized size = 1.02 \[ \frac {\arctan \left (\sqrt {\sec \relax (x )-1}\right )}{2}-\frac {\arctan \left (\sqrt {-\left (\frac {1}{\sec \relax (x )}-1\right ) \sec \relax (x )}\right )}{\sec \relax (x )}+\frac {\sqrt {\sec \relax (x )-1}}{2 \sec \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 60, normalized size = 1.46 \[ -\arctan \left (\sqrt {-\frac {\cos \relax (x) - 1}{\cos \relax (x)}}\right ) \cos \relax (x) - \frac {\sqrt {-\frac {\cos \relax (x) - 1}{\cos \relax (x)}}}{2 \, {\left (\frac {\cos \relax (x) - 1}{\cos \relax (x)} - 1\right )}} + \frac {1}{2} \, \arctan \left (\sqrt {-\frac {\cos \relax (x) - 1}{\cos \relax (x)}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 60, normalized size = 1.46 \[ -\mathrm {atan}\left (\sqrt {\frac {1}{\cos \relax (x)}-1}\right )\,\cos \relax (x)-\frac {\cos \relax (x)\,\left (\frac {3\,\mathrm {asin}\left (\sqrt {\cos \relax (x)}\right )}{2\,{\cos \relax (x)}^{3/2}}-\frac {3\,\sqrt {1-\cos \relax (x)}}{2\,\cos \relax (x)}\right )\,\sqrt {1-\cos \relax (x)}}{3\,\sqrt {\frac {1}{\cos \relax (x)}-1}} \]
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 \[ \int \sin {\relax (x )} \operatorname {atan}{\left (\sqrt {\sec {\relax (x )} - 1} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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