3.2.82 \(\int \cos (x) \sin (\cos (\sin (x))) \sin (\sin (x)) \, dx\) [182]

Optimal. Leaf size=4 \[ \cos (\cos (\sin (x))) \]

[Out]

cos(cos(sin(x)))

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Rubi [A]
time = 0.01, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4419, 4420, 2718} \begin {gather*} \cos (\cos (\sin (x))) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Cos[Cos[Sin[x]]]

Rule 2718

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

Rule 4419

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

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 {gather*} \begin {aligned} \text {Integral} &=\text {Subst}(\int \sin (x) \sin (\cos (x)) \, dx,x,\sin (x))\\ &=-\text {Subst}(\int \sin (x) \, dx,x,\cos (\sin (x)))\\ &=\cos (\cos (\sin (x)))\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 6.25, size = 4, normalized size = 1.00 \begin {gather*} \cos (\cos (\sin (x))) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Cos[Cos[Sin[x]]]

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Maple [A]
time = 7.48, size = 5, normalized size = 1.25

method	result	size
derivativedivides	\(\cos \left (\cos \left (\sin \left (x \right )\right )\right )\)	\(5\)
default	\(\cos \left (\cos \left (\sin \left (x \right )\right )\right )\)	\(5\)
risch	\(\cos \left (\cos \left (\sin \left (x \right )\right )\right )\)	\(5\)
parallelrisch	\(-1+\cos \left (\cos \left (\sin \left (x \right )\right )\right )\)	\(7\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(cos(sin(x)))

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Maxima [A]
time = 0.39, size = 4, normalized size = 1.00 \begin {gather*} \cos \left (\cos \left (\sin \left (x\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(cos(sin(x)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (4) = 8\).
time = 0.62, size = 44, normalized size = 11.00 \begin {gather*} \cos \left (\frac {\tan \left (\frac {\tan \left (\frac {1}{2} \, x\right )}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right )^{2} - 1}{\tan \left (\frac {\tan \left (\frac {1}{2} \, x\right )}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right )^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 1.11, size = 5, normalized size = 1.25 \begin {gather*} \cos {\left (\cos {\left (\sin {\left (x \right )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(cos(sin(x)))

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Giac [A]
time = 0.44, size = 4, normalized size = 1.00 \begin {gather*} \cos \left (\cos \left (\sin \left (x\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(cos(sin(x)))

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Mupad [B]
time = 0.12, size = 4, normalized size = 1.00 \begin {gather*} \cos \left (\cos \left (\sin \left (x\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(cos(sin(x)))

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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

not solved

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