3.1.76 \(\int \sin (x) \, dx\) [76]

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

[Out]

-cos(x)

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Rubi [A]
time = 0.00, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2718} \begin {gather*} -\cos (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x],x]

[Out]

-Cos[x]

Rule 2718

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 4, normalized size = 1.00 \begin {gather*} -\cos (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x],x]

[Out]

-Cos[x]

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Mathics [A]
time = 1.67, size = 4, normalized size = 1.00 \begin {gather*} -\text {Cos}\left [x\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Cos[x]

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Maple [A]
time = 0.01, size = 5, normalized size = 1.25

method	result	size
lookup	\(-\cos \left (x \right )\)	\(5\)
default	\(-\cos \left (x \right )\)	\(5\)
risch	\(-\cos \left (x \right )\)	\(5\)
norman	\(-\frac {2}{1+\tan ^{2}\left (\frac {x}{2}\right )}\)	\(13\)
meijerg	\(\sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (x \right )}{\sqrt {\pi }}\right )\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x)

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Maxima [A]
time = 0.26, size = 4, normalized size = 1.00 \begin {gather*} -\cos \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x)

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Fricas [A]
time = 0.34, size = 4, normalized size = 1.00 \begin {gather*} -\cos \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x)

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Sympy [A]
time = 0.03, size = 3, normalized size = 0.75 \begin {gather*} - \cos {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x),x)

[Out]

-cos(x)

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Giac [A]
time = 0.00, size = 3, normalized size = 0.75 \begin {gather*} -\cos x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x),x)

[Out]

-cos(x)

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Mupad [B]
time = 0.02, size = 4, normalized size = 1.00 \begin {gather*} -\cos \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x),x)

[Out]

-cos(x)

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