3.9 \(\int \tan ^{-1}\left (\sqrt{-1+\sec (x)}\right ) \sin (x) \, dx\)

Optimal. Leaf size=41 \[ \frac{1}{2} \cos (x) \sqrt{\sec (x)-1}+\frac{1}{2} \tan ^{-1}\left (\sqrt{\sec (x)-1}\right )-\cos (x) \tan ^{-1}\left (\sqrt{\sec (x)-1}\right ) \]

[Out]

ArcTan[Sqrt[-1 + Sec[x]]]/2 - ArcTan[Sqrt[-1 + Sec[x]]]*Cos[x] + (Cos[x]*Sqrt[-1
+ Sec[x]])/2

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Rubi [A]  time = 0.0481933, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583 \[ \frac{1}{2} \cos (x) \sqrt{\sec (x)-1}+\frac{1}{2} \tan ^{-1}\left (\sqrt{\sec (x)-1}\right )-\cos (x) \tan ^{-1}\left (\sqrt{\sec (x)-1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[ArcTan[Sqrt[-1 + Sec[x]]]*Sin[x],x]

[Out]

ArcTan[Sqrt[-1 + Sec[x]]]/2 - ArcTan[Sqrt[-1 + Sec[x]]]*Cos[x] + (Cos[x]*Sqrt[-1
+ Sec[x]])/2

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Rubi in Sympy [A]  time = 4.74659, size = 42, normalized size = 1.02 \[ \frac{\sqrt{-1 + \frac{1}{\cos{\left (x \right )}}} \cos{\left (x \right )}}{2} - \cos{\left (x \right )} \operatorname{atan}{\left (\sqrt{-1 + \frac{1}{\cos{\left (x \right )}}} \right )} + \frac{\operatorname{atan}{\left (\sqrt{-1 + \frac{1}{\cos{\left (x \right )}}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(atan((-1+sec(x))**(1/2))*sin(x),x)

[Out]

sqrt(-1 + 1/cos(x))*cos(x)/2 - cos(x)*atan(sqrt(-1 + 1/cos(x))) + atan(sqrt(-1 +
1/cos(x)))/2

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Mathematica [C]  time = 6.38502, size = 285, normalized size = 6.95 \[ \frac{1}{2} \cos (x) \sqrt{\sec (x)-1}-\cos (x) \tan ^{-1}\left (\sqrt{\sec (x)-1}\right )-\frac{1}{2} \left (-3-2 \sqrt{2}\right ) \left (\left (\sqrt{2}-2\right ) \cos \left (\frac{x}{2}\right )-\sqrt{2}+1\right ) \cos ^2\left (\frac{x}{4}\right ) \sqrt{-\tan ^2\left (\frac{x}{4}\right )-2 \sqrt{2}+3} \sqrt{\left (2 \sqrt{2}-3\right ) \tan ^2\left (\frac{x}{4}\right )+1} \cot \left (\frac{x}{4}\right ) \sqrt{\sec (x)-1} \sec (x) \sqrt{\left (\left (10-7 \sqrt{2}\right ) \cos \left (\frac{x}{2}\right )-5 \sqrt{2}+7\right ) \sec ^2\left (\frac{x}{4}\right )} \sqrt{\left (\left (2+\sqrt{2}\right ) \cos \left (\frac{x}{2}\right )-\sqrt{2}-1\right ) \sec ^2\left (\frac{x}{4}\right )} \left (F\left (\sin ^{-1}\left (\frac{\tan \left (\frac{x}{4}\right )}{\sqrt{3-2 \sqrt{2}}}\right )|17-12 \sqrt{2}\right )+2 \Pi \left (-3+2 \sqrt{2};-\sin ^{-1}\left (\frac{\tan \left (\frac{x}{4}\right )}{\sqrt{3-2 \sqrt{2}}}\right )|17-12 \sqrt{2}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[ArcTan[Sqrt[-1 + Sec[x]]]*Sin[x],x]

[Out]

-(ArcTan[Sqrt[-1 + Sec[x]]]*Cos[x]) + (Cos[x]*Sqrt[-1 + Sec[x]])/2 - ((-3 - 2*Sq
rt[2])*Cos[x/4]^2*(1 - Sqrt[2] + (-2 + Sqrt[2])*Cos[x/2])*Cot[x/4]*(EllipticF[Ar
cSin[Tan[x/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + 2*EllipticPi[-3 + 2*Sqrt[
2], -ArcSin[Tan[x/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]])*Sqrt[(7 - 5*Sqrt[2]
+ (10 - 7*Sqrt[2])*Cos[x/2])*Sec[x/4]^2]*Sqrt[(-1 - Sqrt[2] + (2 + Sqrt[2])*Cos
[x/2])*Sec[x/4]^2]*Sqrt[-1 + Sec[x]]*Sec[x]*Sqrt[3 - 2*Sqrt[2] - Tan[x/4]^2]*Sqr
t[1 + (-3 + 2*Sqrt[2])*Tan[x/4]^2])/2

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Maple [A]  time = 0.041, size = 42, normalized size = 1. \[ -{\frac{1}{\sec \left ( x \right ) }\arctan \left ( \sqrt{- \left ( \left ( \sec \left ( x \right ) \right ) ^{-1}-1 \right ) \sec \left ( x \right ) } \right ) }+{\frac{1}{2\,\sec \left ( x \right ) }\sqrt{-1+\sec \left ( x \right ) }}+{\frac{1}{2}\arctan \left ( \sqrt{-1+\sec \left ( x \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arctan((-1+sec(x))^(1/2))*sin(x),x)

[Out]

-1/sec(x)*arctan((-(1/sec(x)-1)*sec(x))^(1/2))+1/2*(-1+sec(x))^(1/2)/sec(x)+1/2*
arctan((-1+sec(x))^(1/2))

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Maxima [A]  time = 1.64201, size = 81, normalized size = 1.98 \[ -\arctan \left (\sqrt{-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right )}}\right ) \cos \left (x\right ) - \frac{\sqrt{-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right )}}}{2 \,{\left (\frac{\cos \left (x\right ) - 1}{\cos \left (x\right )} - 1\right )}} + \frac{1}{2} \, \arctan \left (\sqrt{-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(sqrt(sec(x) - 1))*sin(x),x, algorithm="maxima")

[Out]

-arctan(sqrt(-(cos(x) - 1)/cos(x)))*cos(x) - 1/2*sqrt(-(cos(x) - 1)/cos(x))/((co
s(x) - 1)/cos(x) - 1) + 1/2*arctan(sqrt(-(cos(x) - 1)/cos(x)))

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Fricas [A]  time = 0.243161, size = 43, normalized size = 1.05 \[ -\frac{1}{2} \,{\left (2 \, \cos \left (x\right ) - 1\right )} \arctan \left (\sqrt{\sec \left (x\right ) - 1}\right ) + \frac{1}{2} \, \sqrt{-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right )}} \cos \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(sqrt(sec(x) - 1))*sin(x),x, algorithm="fricas")

[Out]

-1/2*(2*cos(x) - 1)*arctan(sqrt(sec(x) - 1)) + 1/2*sqrt(-(cos(x) - 1)/cos(x))*co
s(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sin{\left (x \right )} \operatorname{atan}{\left (\sqrt{\sec{\left (x \right )} - 1} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(atan((-1+sec(x))**(1/2))*sin(x),x)

[Out]

Integral(sin(x)*atan(sqrt(sec(x) - 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(sqrt(sec(x) - 1))*sin(x),x, algorithm="giac")

[Out]

undef