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

Optimal. Leaf size=28 \[ \frac{d \sin (a+b x)}{b^2}-\frac{(c+d x) \cos (a+b x)}{b} \]

[Out]

-(((c + d*x)*Cos[a + b*x])/b) + (d*Sin[a + b*x])/b^2

________________________________________________________________________________________

Rubi [A]  time = 0.0162482, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3296, 2637} \[ \frac{d \sin (a+b x)}{b^2}-\frac{(c+d x) \cos (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)*Sin[a + b*x],x]

[Out]

-(((c + d*x)*Cos[a + b*x])/b) + (d*Sin[a + b*x])/b^2

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

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 (c+d x) \sin (a+b x) \, dx &=-\frac{(c+d x) \cos (a+b x)}{b}+\frac{d \int \cos (a+b x) \, dx}{b}\\ &=-\frac{(c+d x) \cos (a+b x)}{b}+\frac{d \sin (a+b x)}{b^2}\\ \end{align*}

Mathematica [A]  time = 0.0703575, size = 27, normalized size = 0.96 \[ \frac{d \sin (a+b x)-b (c+d x) \cos (a+b x)}{b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)*Sin[a + b*x],x]

[Out]

(-(b*(c + d*x)*Cos[a + b*x]) + d*Sin[a + b*x])/b^2

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 52, normalized size = 1.9 \begin{align*}{\frac{1}{b} \left ({\frac{d \left ( \sin \left ( bx+a \right ) - \left ( bx+a \right ) \cos \left ( bx+a \right ) \right ) }{b}}+{\frac{da\cos \left ( bx+a \right ) }{b}}-c\cos \left ( bx+a \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(1/b*d*(sin(b*x+a)-(b*x+a)*cos(b*x+a))+1/b*d*a*cos(b*x+a)-c*cos(b*x+a))

________________________________________________________________________________________

Maxima [A]  time = 1.01222, size = 72, normalized size = 2.57 \begin{align*} -\frac{c \cos \left (b x + a\right ) - \frac{a d \cos \left (b x + a\right )}{b} + \frac{{\left ({\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} d}{b}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.66359, size = 70, normalized size = 2.5 \begin{align*} -\frac{{\left (b d x + b c\right )} \cos \left (b x + a\right ) - d \sin \left (b x + a\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((b*d*x + b*c)*cos(b*x + a) - d*sin(b*x + a))/b^2

________________________________________________________________________________________

Sympy [A]  time = 0.235433, size = 46, normalized size = 1.64 \begin{align*} \begin{cases} - \frac{c \cos{\left (a + b x \right )}}{b} - \frac{d x \cos{\left (a + b x \right )}}{b} + \frac{d \sin{\left (a + b x \right )}}{b^{2}} & \text{for}\: b \neq 0 \\\left (c x + \frac{d x^{2}}{2}\right ) \sin{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-c*cos(a + b*x)/b - d*x*cos(a + b*x)/b + d*sin(a + b*x)/b**2, Ne(b, 0)), ((c*x + d*x**2/2)*sin(a),
True))

________________________________________________________________________________________

Giac [A]  time = 1.1022, size = 42, normalized size = 1.5 \begin{align*} -\frac{{\left (b d x + b c\right )} \cos \left (b x + a\right )}{b^{2}} + \frac{d \sin \left (b x + a\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*d*x + b*c)*cos(b*x + a)/b^2 + d*sin(b*x + a)/b^2