3.311 \(\int \cos (\cos (x)) \sin (x) \, dx\)

Optimal. Leaf size=5 \[ -\sin (\cos (x)) \]

[Out]

-Sin[Cos[x]]

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Rubi [A]  time = 0.0147589, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\sin (\cos (x)) \]

Antiderivative was successfully verified.

[In]  Int[Cos[Cos[x]]*Sin[x],x]

[Out]

-Sin[Cos[x]]

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Rubi in Sympy [A]  time = 29.4721, size = 5, normalized size = 1. \[ - \sin{\left (\cos{\left (x \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(cos(x))*sin(x),x)

[Out]

-sin(cos(x))

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Mathematica [A]  time = 2.11923, size = 5, normalized size = 1. \[ -\sin (\cos (x)) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[Cos[x]]*Sin[x],x]

[Out]

-Sin[Cos[x]]

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Maple [A]  time = 0.012, size = 6, normalized size = 1.2 \[ -\sin \left ( \cos \left ( x \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(cos(x))*sin(x),x)

[Out]

-sin(cos(x))

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Maxima [A]  time = 1.32667, size = 7, normalized size = 1.4 \[ -\sin \left (\cos \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(cos(x))*sin(x),x, algorithm="maxima")

[Out]

-sin(cos(x))

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Fricas [A]  time = 0.235119, size = 27, normalized size = 5.4 \[ \sin \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(cos(x))*sin(x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.735445, size = 5, normalized size = 1. \[ - \sin{\left (\cos{\left (x \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(cos(x))*sin(x),x)

[Out]

-sin(cos(x))

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GIAC/XCAS [A]  time = 0.207089, size = 7, normalized size = 1.4 \[ -\sin \left (\cos \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(cos(x))*sin(x),x, algorithm="giac")

[Out]

-sin(cos(x))