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

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

[Out]

a*x + (a*Sin[c + d*x])/d

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Rubi [A]  time = 0.0072543, 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} \[ \frac{a \sin (c+d x)}{d}+a x \]

Antiderivative was successfully verified.

[In]

Int[a + a*Cos[c + d*x],x]

[Out]

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

Mathematica [A]  time = 0.0054959, size = 26, normalized size = 1.73 \[ \frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d}+a x \]

Antiderivative was successfully verified.

[In]

Integrate[a + a*Cos[c + d*x],x]

[Out]

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

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Maple [A]  time = 0.025, size = 16, normalized size = 1.1 \begin{align*} ax+{\frac{a\sin \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.12463, size = 20, normalized size = 1.33 \begin{align*} a x + \frac{a \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + a*sin(d*x + c)/d

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Fricas [A]  time = 1.63994, size = 38, normalized size = 2.53 \begin{align*} \frac{a d x + a \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*d*x + a*sin(d*x + c))/d

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Sympy [A]  time = 0.132864, size = 17, normalized size = 1.13 \begin{align*} a x + a \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+a*cos(d*x+c),x)

[Out]

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

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Giac [A]  time = 1.32368, size = 20, normalized size = 1.33 \begin{align*} a x + \frac{a \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + a*sin(d*x + c)/d