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

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

[Out]

-Sin[Cos[x]]

________________________________________________________________________________________

Rubi [A]  time = 0.0088673, 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, Rules used = {4335, 2637} \[ -\sin (\cos (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Sin[Cos[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])

Rule 2637

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

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*}

Mathematica [A]  time = 2.47877, size = 5, normalized size = 1. \[ -\sin (\cos (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Sin[Cos[x]]

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 6, normalized size = 1.2 \begin{align*} -\sin \left ( \cos \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(cos(x))*sin(x),x)

[Out]

-sin(cos(x))

________________________________________________________________________________________

Maxima [A]  time = 0.926392, size = 7, normalized size = 1.4 \begin{align*} -\sin \left (\cos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sin(cos(x))

________________________________________________________________________________________

Fricas [B]  time = 2.28435, size = 59, normalized size = 11.8 \begin{align*} \sin \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right ) \end{align*}

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))

________________________________________________________________________________________

Sympy [A]  time = 0.505605, size = 5, normalized size = 1. \begin{align*} - \sin{\left (\cos{\left (x \right )} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(cos(x))*sin(x),x)

[Out]

-sin(cos(x))

________________________________________________________________________________________

Giac [A]  time = 1.0521, size = 7, normalized size = 1.4 \begin{align*} -\sin \left (\cos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sin(cos(x))