3.4.11 \(\int \cos (\cos (x)) \sin (x) \, dx\) [311]

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 = {4420, 2717} \begin {gather*} -\sin (\cos (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Sin[Cos[x]]

Rule 2717

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

Rule 4420

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

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Mathematica [A]
time = 1.67, size = 5, normalized size = 1.00 \begin {gather*} -\sin (\cos (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Sin[Cos[x]]

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Mathics [A]
time = 1.74, size = 5, normalized size = 1.00 \begin {gather*} -\text {Sin}\left [\text {Cos}\left [x\right ]\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Cos[Cos[x]]*Sin[x],x]')

[Out]

-Sin[Cos[x]]

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Maple [A]
time = 0.03, size = 6, normalized size = 1.20

method	result	size
derivativedivides	\(-\sin \left (\cos \left (x \right )\right )\)	\(6\)
default	\(-\sin \left (\cos \left (x \right )\right )\)	\(6\)
risch	\(-\sin \left (\cos \left (x \right )\right )\)	\(6\)
norman	\(\frac {-2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right ) \tan \left (\frac {1-\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2+2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}\right )-2 \tan \left (\frac {1-\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2+2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}\right )}{\left (1+\tan ^{2}\left (\frac {1-\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2 \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}\right )\right ) \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}\)	\(98\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(cos(x))*sin(x),x,method=_RETURNVERBOSE)

[Out]

-sin(cos(x))

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Maxima [A]
time = 0.27, size = 5, normalized size = 1.00 \begin {gather*} -\sin \left (\cos \left (x\right )\right ) \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (5) = 10\).
time = 0.38, size = 20, normalized size = 4.00 \begin {gather*} \sin \left (\frac {\tan \left (\frac {1}{2} \, x\right )^{2} - 1}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \end {gather*}

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 = 0.14, size = 5, normalized size = 1.00 \begin {gather*} - \sin {\left (\cos {\left (x \right )} \right )} \end {gather*}

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 [A]
time = 0.00, size = 4, normalized size = 0.80 \begin {gather*} -\sin \left (\cos x\right ) \end {gather*}

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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Mupad [B]
time = 0.22, size = 5, normalized size = 1.00 \begin {gather*} -\sin \left (\cos \left (x\right )\right ) \end {gather*}

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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