3.1.21 \(\int \cos (3+2 x) \, dx\) [21]

Optimal. Leaf size=10 \[ \frac {1}{2} \sin (3+2 x) \]

[Out]

1/2*sin(3+2*x)

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Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2717} \begin {gather*} \frac {1}{2} \sin (2 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[3 + 2*x],x]

[Out]

Sin[3 + 2*x]/2

Rule 2717

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 (3+2 x) \, dx &=\frac {1}{2} \sin (3+2 x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 10, normalized size = 1.00 \begin {gather*} \frac {1}{2} \sin (3+2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[3 + 2*x],x]

[Out]

Sin[3 + 2*x]/2

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Mathics [A]
time = 1.77, size = 8, normalized size = 0.80 \begin {gather*} \frac {\text {Sin}\left [3+2 x\right ]}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Cos[2*x + 3],x]')

[Out]

Sin[3 + 2 x] / 2

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Maple [A]
time = 0.02, size = 9, normalized size = 0.90

method	result	size
derivativedivides	\(\frac {\sin \left (3+2 x \right )}{2}\)	\(9\)
default	\(\frac {\sin \left (3+2 x \right )}{2}\)	\(9\)
risch	\(\frac {\sin \left (3+2 x \right )}{2}\)	\(9\)
norman	\(\frac {\tan \left (\frac {3}{2}+x \right )}{1+\tan ^{2}\left (\frac {3}{2}+x \right )}\)	\(16\)
meijerg	\(\frac {\cos \left (3\right ) \sin \left (2 x \right )}{2}-\frac {\sin \left (3\right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (2 x \right )}{\sqrt {\pi }}\right )}{2}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(3+2*x),x,method=_RETURNVERBOSE)

[Out]

1/2*sin(3+2*x)

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Maxima [A]
time = 0.26, size = 8, normalized size = 0.80 \begin {gather*} \frac {1}{2} \, \sin \left (2 \, x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3+2*x),x, algorithm="maxima")

[Out]

1/2*sin(2*x + 3)

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Fricas [A]
time = 0.32, size = 8, normalized size = 0.80 \begin {gather*} \frac {1}{2} \, \sin \left (2 \, x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3+2*x),x, algorithm="fricas")

[Out]

1/2*sin(2*x + 3)

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Sympy [A]
time = 0.06, size = 7, normalized size = 0.70 \begin {gather*} \frac {\sin {\left (2 x + 3 \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3+2*x),x)

[Out]

sin(2*x + 3)/2

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Giac [A]
time = 0.00, size = 9, normalized size = 0.90 \begin {gather*} \frac {\sin \left (3+2 x\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3+2*x),x)

[Out]

1/2*sin(2*x + 3)

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Mupad [B]
time = 0.07, size = 8, normalized size = 0.80 \begin {gather*} \frac {\sin \left (2\,x+3\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(2*x + 3),x)

[Out]

sin(2*x + 3)/2

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