3.144 \(\int \frac{\sin ^2(x)}{1+\sin ^2(x)} \, dx\)

Optimal. Leaf size=36 \[ -\frac{x}{\sqrt{2}}+x-\frac{\tan ^{-1}\left (\frac{\sin (x) \cos (x)}{\sin ^2(x)+\sqrt{2}+1}\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0671648, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{x}{\sqrt{2}}+x-\frac{\tan ^{-1}\left (\frac{\sin (x) \cos (x)}{\sin ^2(x)+\sqrt{2}+1}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[Sin[x]^2/(1 + Sin[x]^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.66621, size = 24, normalized size = 0.67 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2}}{2 \tan{\left (x \right )}} \right )}}{2} - \operatorname{atan}{\left (\frac{1}{\tan{\left (x \right )}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*atan(sqrt(2)/(2*tan(x)))/2 - atan(1/tan(x))

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Mathematica [A]  time = 0.0209109, size = 18, normalized size = 0.5 \[ x-\frac{\tan ^{-1}\left (\sqrt{2} \tan (x)\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sin[x]^2/(1 + Sin[x]^2),x]

[Out]

x - ArcTan[Sqrt[2]*Tan[x]]/Sqrt[2]

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Maple [A]  time = 0.029, size = 15, normalized size = 0.4 \[ -{\frac{\sqrt{2}\arctan \left ( \tan \left ( x \right ) \sqrt{2} \right ) }{2}}+x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(x)^2/(1+sin(x)^2),x)

[Out]

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

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Maxima [A]  time = 1.52201, size = 19, normalized size = 0.53 \[ -\frac{1}{2} \, \sqrt{2} \arctan \left (\sqrt{2} \tan \left (x\right )\right ) + x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)^2/(sin(x)^2 + 1),x, algorithm="maxima")

[Out]

-1/2*sqrt(2)*arctan(sqrt(2)*tan(x)) + x

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Fricas [A]  time = 0.243118, size = 51, normalized size = 1.42 \[ \frac{1}{4} \, \sqrt{2}{\left (2 \, \sqrt{2} x + \arctan \left (\frac{3 \, \sqrt{2} \cos \left (x\right )^{2} - 2 \, \sqrt{2}}{4 \, \cos \left (x\right ) \sin \left (x\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)^2/(sin(x)^2 + 1),x, algorithm="fricas")

[Out]

1/4*sqrt(2)*(2*sqrt(2)*x + arctan(1/4*(3*sqrt(2)*cos(x)^2 - 2*sqrt(2))/(cos(x)*s
in(x))))

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Sympy [A]  time = 155.717, size = 416, normalized size = 11.56 \[ \frac{41 \sqrt{2} x \sqrt{- 2 \sqrt{2} + 3}}{41 \sqrt{2} \sqrt{- 2 \sqrt{2} + 3} + 58 \sqrt{- 2 \sqrt{2} + 3}} + \frac{58 x \sqrt{- 2 \sqrt{2} + 3}}{41 \sqrt{2} \sqrt{- 2 \sqrt{2} + 3} + 58 \sqrt{- 2 \sqrt{2} + 3}} - \frac{17 \left (\operatorname{atan}{\left (\frac{\tan{\left (\frac{x}{2} \right )}}{\sqrt{- 2 \sqrt{2} + 3}} \right )} + \pi \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\rfloor \right )}{41 \sqrt{2} \sqrt{- 2 \sqrt{2} + 3} + 58 \sqrt{- 2 \sqrt{2} + 3}} - \frac{12 \sqrt{2} \left (\operatorname{atan}{\left (\frac{\tan{\left (\frac{x}{2} \right )}}{\sqrt{- 2 \sqrt{2} + 3}} \right )} + \pi \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\rfloor \right )}{41 \sqrt{2} \sqrt{- 2 \sqrt{2} + 3} + 58 \sqrt{- 2 \sqrt{2} + 3}} - \frac{17 \sqrt{- 2 \sqrt{2} + 3} \sqrt{2 \sqrt{2} + 3} \left (\operatorname{atan}{\left (\frac{\tan{\left (\frac{x}{2} \right )}}{\sqrt{2 \sqrt{2} + 3}} \right )} + \pi \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\rfloor \right )}{41 \sqrt{2} \sqrt{- 2 \sqrt{2} + 3} + 58 \sqrt{- 2 \sqrt{2} + 3}} - \frac{12 \sqrt{2} \sqrt{- 2 \sqrt{2} + 3} \sqrt{2 \sqrt{2} + 3} \left (\operatorname{atan}{\left (\frac{\tan{\left (\frac{x}{2} \right )}}{\sqrt{2 \sqrt{2} + 3}} \right )} + \pi \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\rfloor \right )}{41 \sqrt{2} \sqrt{- 2 \sqrt{2} + 3} + 58 \sqrt{- 2 \sqrt{2} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

41*sqrt(2)*x*sqrt(-2*sqrt(2) + 3)/(41*sqrt(2)*sqrt(-2*sqrt(2) + 3) + 58*sqrt(-2*
sqrt(2) + 3)) + 58*x*sqrt(-2*sqrt(2) + 3)/(41*sqrt(2)*sqrt(-2*sqrt(2) + 3) + 58*
sqrt(-2*sqrt(2) + 3)) - 17*(atan(tan(x/2)/sqrt(-2*sqrt(2) + 3)) + pi*floor((x/2
- pi/2)/pi))/(41*sqrt(2)*sqrt(-2*sqrt(2) + 3) + 58*sqrt(-2*sqrt(2) + 3)) - 12*sq
rt(2)*(atan(tan(x/2)/sqrt(-2*sqrt(2) + 3)) + pi*floor((x/2 - pi/2)/pi))/(41*sqrt
(2)*sqrt(-2*sqrt(2) + 3) + 58*sqrt(-2*sqrt(2) + 3)) - 17*sqrt(-2*sqrt(2) + 3)*sq
rt(2*sqrt(2) + 3)*(atan(tan(x/2)/sqrt(2*sqrt(2) + 3)) + pi*floor((x/2 - pi/2)/pi
))/(41*sqrt(2)*sqrt(-2*sqrt(2) + 3) + 58*sqrt(-2*sqrt(2) + 3)) - 12*sqrt(2)*sqrt
(-2*sqrt(2) + 3)*sqrt(2*sqrt(2) + 3)*(atan(tan(x/2)/sqrt(2*sqrt(2) + 3)) + pi*fl
oor((x/2 - pi/2)/pi))/(41*sqrt(2)*sqrt(-2*sqrt(2) + 3) + 58*sqrt(-2*sqrt(2) + 3)
)

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GIAC/XCAS [A]  time = 0.227674, size = 65, normalized size = 1.81 \[ -\frac{1}{2} \, \sqrt{2}{\left (x + \arctan \left (-\frac{\sqrt{2} \sin \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )}{\sqrt{2} \cos \left (2 \, x\right ) + \sqrt{2} - 2 \, \cos \left (2 \, x\right ) + 2}\right )\right )} + x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)^2/(sin(x)^2 + 1),x, algorithm="giac")

[Out]

-1/2*sqrt(2)*(x + arctan(-(sqrt(2)*sin(2*x) - 2*sin(2*x))/(sqrt(2)*cos(2*x) + sq
rt(2) - 2*cos(2*x) + 2))) + x