3.410 \(\int (a+b \cos (c+d x)) \, dx\)

Optimal. Leaf size=15 \[ a x+\frac{b \sin (c+d x)}{d} \]

[Out]

a*x + (b*Sin[c + d*x])/d

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Rubi [A]  time = 0.0085139, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2637} \[ a x+\frac{b \sin (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[a + b*Cos[c + d*x],x]

[Out]

a*x + (b*Sin[c + d*x])/d

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

Mathematica [A]  time = 0.0063226, size = 26, normalized size = 1.73 \[ a x+\frac{b \sin (c) \cos (d x)}{d}+\frac{b \cos (c) \sin (d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*Cos[c + d*x],x]

[Out]

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

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Maple [A]  time = 0.023, size = 16, normalized size = 1.1 \begin{align*} ax+{\frac{b\sin \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x+b*sin(d*x+c)/d

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Maxima [A]  time = 0.947299, size = 20, normalized size = 1.33 \begin{align*} a x + \frac{b \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*sin(d*x + c)/d

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Fricas [A]  time = 1.86152, size = 38, normalized size = 2.53 \begin{align*} \frac{a d x + b \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*d*x + b*sin(d*x + c))/d

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Sympy [A]  time = 0.125019, size = 17, normalized size = 1.13 \begin{align*} a x + b \left (\begin{cases} \frac{\sin{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \cos{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.36589, size = 20, normalized size = 1.33 \begin{align*} a x + \frac{b \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*sin(d*x + c)/d