3.5.10 \(\int (a+b \cos (c+d x)) \, dx\) [410]

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

[Out]

a*x+b*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 = {2717} \begin {gather*} a x+\frac {b \sin (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a + b*Cos[c + d*x],x]

[Out]

a*x + (b*Sin[c + d*x])/d

Rule 2717

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*}

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Mathematica [A]
time = 0.01, size = 26, normalized size = 1.73 \begin {gather*} a x+\frac {b \cos (d x) \sin (c)}{d}+\frac {b \cos (c) \sin (d x)}{d} \end {gather*}

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 = 0.05, size = 16, normalized size = 1.07

method	result	size
default	\(a x +\frac {b \sin \left (d x +c \right )}{d}\)	\(16\)
risch	\(a x +\frac {b \sin \left (d x +c \right )}{d}\)	\(16\)
derivativedivides	\(\frac {a \left (d x +c \right )+b \sin \left (d x +c \right )}{d}\)	\(21\)
norman	\(\frac {a x +a x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 15, normalized size = 1.00 \begin {gather*} a x + \frac {b \sin \left (d x + c\right )}{d} \end {gather*}

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 = 0.41, size = 17, normalized size = 1.13 \begin {gather*} \frac {a d x + b \sin \left (d x + c\right )}{d} \end {gather*}

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 = 0.04, size = 17, normalized size = 1.13 \begin {gather*} 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 {gather*}

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 = 0.42, size = 15, normalized size = 1.00 \begin {gather*} a x + \frac {b \sin \left (d x + c\right )}{d} \end {gather*}

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

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Mupad [B]
time = 0.47, size = 17, normalized size = 1.13 \begin {gather*} \frac {b\,\sin \left (c+d\,x\right )+a\,d\,x}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*cos(c + d*x),x)

[Out]

(b*sin(c + d*x) + a*d*x)/d

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