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

Optimal. Leaf size=3 \[ \tan ^{-1}(\sin (x)) \]

[Out]

ArcTan[Sin[x]]

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Rubi [A]  time = 0.0278721, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \tan ^{-1}(\sin (x)) \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]/(1 + Sin[x]^2),x]

[Out]

ArcTan[Sin[x]]

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Rubi in Sympy [A]  time = 2.31605, size = 3, normalized size = 1. \[ \operatorname{atan}{\left (\sin{\left (x \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(sin(x))

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Mathematica [A]  time = 0.00816533, size = 3, normalized size = 1. \[ \tan ^{-1}(\sin (x)) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[Sin[x]]

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Maple [A]  time = 0.015, size = 4, normalized size = 1.3 \[ \arctan \left ( \sin \left ( x \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)/(1+sin(x)^2),x)

[Out]

arctan(sin(x))

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Maxima [A]  time = 1.48868, size = 4, normalized size = 1.33 \[ \arctan \left (\sin \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(sin(x))

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Fricas [A]  time = 0.219653, size = 4, normalized size = 1.33 \[ \arctan \left (\sin \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(sin(x))

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Sympy [A]  time = 0.278579, size = 3, normalized size = 1. \[ \operatorname{atan}{\left (\sin{\left (x \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(sin(x))

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GIAC/XCAS [A]  time = 0.226133, size = 4, normalized size = 1.33 \[ \arctan \left (\sin \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(sin(x))