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(d*x+c)/d

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.75, size = 17, normalized size = 1.13 \[ \frac {a d x + a \sin \left (d x + c\right )}{d} \]

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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giac [A]  time = 0.40, size = 15, normalized size = 1.00 \[ a x + \frac {a \sin \left (d x + c\right )}{d} \]

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

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maple [A]  time = 0.02, size = 16, normalized size = 1.07 \[ a x +\frac {a \sin \left (d x +c \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.38, size = 15, normalized size = 1.00 \[ a x + \frac {a \sin \left (d x + c\right )}{d} \]

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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mupad [B]  time = 0.31, size = 15, normalized size = 1.00 \[ a\,x+\frac {a\,\sin \left (c+d\,x\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + a*cos(c + d*x),x)

[Out]

a*x + (a*sin(c + d*x))/d

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sympy [A]  time = 0.11, size = 17, normalized size = 1.13 \[ a x + a \left (\begin {cases} \frac {\sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \cos {\relax (c )} & \text {otherwise} \end {cases}\right ) \]

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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