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

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

[Out]

Sin[a + b*x]/b

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Rubi [A]  time = 0.0035237, antiderivative size = 10, normalized size of antiderivative = 1., 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*}

Mathematica [B]  time = 0.0088529, size = 21, normalized size = 2.1 \[ \frac{\sin (a) \cos (b x)}{b}+\frac{\cos (a) \sin (b x)}{b} \]

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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Maple [A]  time = 0.005, size = 11, normalized size = 1.1 \begin{align*}{\frac{\sin \left ( bx+a \right ) }{b}} \end{align*}

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 = 0.956322, size = 14, normalized size = 1.4 \begin{align*} \frac{\sin \left (b x + a\right )}{b} \end{align*}

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 = 1.7505, size = 22, normalized size = 2.2 \begin{align*} \frac{\sin \left (b x + a\right )}{b} \end{align*}

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 = 0.129336, size = 12, normalized size = 1.2 \begin{align*} \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{align*}

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 = 1.08862, size = 14, normalized size = 1.4 \begin{align*} \frac{\sin \left (b x + a\right )}{b} \end{align*}

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