3.3.43 \(\int (-\sin (x-\sin (x))+\sin (x+\sin (x))) \, dx\) [243]

Optimal. Leaf size=5 \[ -2 \cos (\sin (x)) \]

[Out]

-2*cos(sin(x))

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Rubi [F]
time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int (-\sin (x-\sin (x))+\sin (x+\sin (x))) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[-Sin[x - Sin[x]] + Sin[x + Sin[x]],x]

[Out]

-Defer[Int][Sin[x - Sin[x]], x] + Defer[Int][Sin[x + Sin[x]], x]

Rubi steps

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

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Mathematica [A]
time = 0.08, size = 5, normalized size = 1.00 \begin {gather*} -2 \cos (\sin (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-Sin[x - Sin[x]] + Sin[x + Sin[x]],x]

[Out]

-2*Cos[Sin[x]]

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Maple [A]
time = 10.26, size = 6, normalized size = 1.20

method	result	size
derivativedivides	\(-2 \cos \left (\sin \left (x \right )\right )\)	\(6\)
default	\(-2 \cos \left (\sin \left (x \right )\right )\)	\(6\)
risch	\(-2 \cos \left (\sin \left (x \right )\right )\)	\(6\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x+sin(x))-sin(x-sin(x)),x,method=_RETURNVERBOSE)

[Out]

-2*cos(sin(x))

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Maxima [A]
time = 0.61, size = 5, normalized size = 1.00 \begin {gather*} -2 \, \cos \left (\sin \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x+sin(x))-sin(x-sin(x)),x, algorithm="maxima")

[Out]

-2*cos(sin(x))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (5) = 10\).
time = 0.61, size = 65, normalized size = 13.00 \begin {gather*} -2 \, \cos \left (x\right ) \cos \left (\frac {x \tan \left (\frac {1}{2} \, x\right )^{2} + x + 2 \, \tan \left (\frac {1}{2} \, x\right )}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) - 2 \, \sin \left (x\right ) \sin \left (\frac {x \tan \left (\frac {1}{2} \, x\right )^{2} + x + 2 \, \tan \left (\frac {1}{2} \, x\right )}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x+sin(x))-sin(x-sin(x)),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- \sin {\left (x - \sin {\left (x \right )} \right )} + \sin {\left (x + \sin {\left (x \right )} \right )}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x+sin(x))-sin(x-sin(x)),x)

[Out]

Integral(-sin(x - sin(x)) + sin(x + sin(x)), x)

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Giac [A]
time = 0.44, size = 5, normalized size = 1.00 \begin {gather*} -2 \, \cos \left (\sin \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x+sin(x))-sin(x-sin(x)),x, algorithm="giac")

[Out]

-2*cos(sin(x))

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Mupad [B]
time = 0.37, size = 5, normalized size = 1.00 \begin {gather*} -2\,\cos \left (\sin \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x + sin(x)) - sin(x - sin(x)),x)

[Out]

-2*cos(sin(x))

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Chatgpt [F] Failed to verify
time = 1.00, size = 7, normalized size = 1.40 \begin {gather*} 2 \sin \left (x \right ) \sin \left (\sin \left (x \right )\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(sin(x+sin(x))-sin(x-sin(x)),x)

[Out]

2*sin(x)*sin(sin(x))

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