3.1.1 \(\int \cos (a+b x) \, dx\) [1]

Optimal. Leaf size=10 \[ \frac {\sin (a+b x)}{b} \]

[Out]

sin(b*x+a)/b

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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 {\sin (a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x],x]

[Out]

Sin[a + b*x]/b

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 (a+b x) \, dx &=\frac {\sin (a+b x)}{b}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(21\) vs. \(2(10)=20\).
time = 0.06, size = 21, normalized size = 2.10 \begin {gather*} \frac {\cos (b x) \sin (a)}{b}+\frac {\cos (a) \sin (b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x],x]

[Out]

(Cos[b*x]*Sin[a])/b + (Cos[a]*Sin[b*x])/b

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Maple [A]
time = 0.08, size = 11, normalized size = 1.10

method	result	size
derivativedivides	\(\frac {\sin \left (b x +a \right )}{b}\)	\(11\)
default	\(\frac {\sin \left (b x +a \right )}{b}\)	\(11\)
risch	\(\frac {\sin \left (b x +a \right )}{b}\)	\(11\)
norman	\(\frac {2 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\)	\(30\)
meijerg	\(\frac {\cos \left (a \right ) \sin \left (b x \right )}{b}-\frac {\sin \left (a \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (b x \right )}{\sqrt {\pi }}\right )}{b}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a),x,method=_RETURNVERBOSE)

[Out]

sin(b*x+a)/b

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Maxima [A]
time = 0.29, size = 10, normalized size = 1.00 \begin {gather*} \frac {\sin \left (b x + a\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a),x, algorithm="maxima")

[Out]

sin(b*x + a)/b

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Fricas [A]
time = 0.35, size = 10, normalized size = 1.00 \begin {gather*} \frac {\sin \left (b x + a\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a),x, algorithm="fricas")

[Out]

sin(b*x + a)/b

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Sympy [A]
time = 0.04, size = 12, normalized size = 1.20 \begin {gather*} \begin {cases} \frac {\sin {\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \cos {\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a),x)

[Out]

Piecewise((sin(a + b*x)/b, Ne(b, 0)), (x*cos(a), True))

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Giac [A]
time = 0.45, size = 10, normalized size = 1.00 \begin {gather*} \frac {\sin \left (b x + a\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a),x, algorithm="giac")

[Out]

sin(b*x + a)/b

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Mupad [B]
time = 0.25, size = 10, normalized size = 1.00 \begin {gather*} \frac {\sin \left (a+b\,x\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x),x)

[Out]

sin(a + b*x)/b

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