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

Optimal. Leaf size=24 \[ \frac{a \sin (c+d x)}{d}-\frac{b \cos (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.0142294, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2637, 2638} \[ \frac{a \sin (c+d x)}{d}-\frac{b \cos (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[a*Cos[c + d*x] + b*Sin[c + d*x],x]

[Out]

-((b*Cos[c + d*x])/d) + (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]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0121349, size = 46, normalized size = 1.92 \[ \frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d}+\frac{b \sin (c) \sin (d x)}{d}-\frac{b \cos (c) \cos (d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[a*Cos[c + d*x] + b*Sin[c + d*x],x]

[Out]

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

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Maple [A]  time = 0.008, size = 25, normalized size = 1. \begin{align*} -{\frac{b\cos \left ( dx+c \right ) }{d}}+{\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)+b*sin(d*x+c),x)

[Out]

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

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Maxima [A]  time = 0.98092, size = 32, normalized size = 1.33 \begin{align*} -\frac{b \cos \left (d x + c\right )}{d} + \frac{a \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*cos(d*x + c)/d + a*sin(d*x + c)/d

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Fricas [A]  time = 1.92775, size = 51, normalized size = 2.12 \begin{align*} -\frac{b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*cos(d*x + c) - a*sin(d*x + c))/d

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Sympy [A]  time = 0.179612, size = 31, normalized size = 1.29 \begin{align*} 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 ) + b \left (\begin{cases} - \frac{\cos{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \sin{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13581, size = 32, normalized size = 1.33 \begin{align*} -\frac{b \cos \left (d x + c\right )}{d} + \frac{a \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*cos(d*x + c)/d + a*sin(d*x + c)/d