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

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

[Out]

-sin(cos(x))

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Rubi [A]  time = 0.01, antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4335, 2637} \[ -\sin (\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cos[Cos[x]]*Sin[x],x]

[Out]

-Sin[Cos[x]]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 4335

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[d/(
b*c), Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a
+ b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])

Rubi steps

\begin {align*} \int \cos (\cos (x)) \sin (x) \, dx &=-\operatorname {Subst}(\int \cos (x) \, dx,x,\cos (x))\\ &=-\sin (\cos (x))\\ \end {align*}

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Mathematica [A]  time = 2.57, size = 5, normalized size = 1.00 \[ -\sin (\cos (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[Cos[x]]*Sin[x],x]

[Out]

-Sin[Cos[x]]

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fricas [B]  time = 0.79, size = 20, normalized size = 4.00 \[ \sin \left (\frac {\tan \left (\frac {1}{2} \, x\right )^{2} - 1}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \]

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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giac [A]  time = 0.14, size = 5, normalized size = 1.00 \[ -\sin \left (\cos \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sin(cos(x))

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maple [A]  time = 0.01, size = 6, normalized size = 1.20 \[ -\sin \left (\cos \relax (x )\right ) \]

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 = 0.55, size = 5, normalized size = 1.00 \[ -\sin \left (\cos \relax (x)\right ) \]

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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mupad [B]  time = 0.09, size = 5, normalized size = 1.00 \[ -\sin \left (\cos \relax (x)\right ) \]

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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sympy [A]  time = 0.44, size = 5, normalized size = 1.00 \[ - \sin {\left (\cos {\relax (x )} \right )} \]

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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