3.108 \(\int \cos (a+b x) \, dx\)

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 = {2637} \[ \frac {\sin (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x],x]

[Out]

Sin[a + b*x]/b

Rule 2637

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]  time = 0.01, size = 21, normalized size = 2.10 \[ \frac {\sin (a) \cos (b x)}{b}+\frac {\cos (a) \sin (b x)}{b} \]

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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fricas [A]  time = 0.42, size = 10, normalized size = 1.00 \[ \frac {\sin \left (b x + a\right )}{b} \]

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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giac [A]  time = 0.92, size = 10, normalized size = 1.00 \[ \frac {\sin \left (b x + a\right )}{b} \]

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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maple [A]  time = 0.02, size = 11, normalized size = 1.10 \[ \frac {\sin \left (b x +a \right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(b*x+a)/b

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maxima [A]  time = 0.41, size = 10, normalized size = 1.00 \[ \frac {\sin \left (b x + a\right )}{b} \]

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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mupad [B]  time = 0.02, size = 10, normalized size = 1.00 \[ \frac {\sin \left (a+b\,x\right )}{b} \]

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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sympy [A]  time = 0.14, size = 12, normalized size = 1.20 \[ \begin {cases} \frac {\sin {\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \cos {\relax (a )} & \text {otherwise} \end {cases} \]

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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