3.17 \(\int (c+d x) \cos (a+b x) \sin ^2(a+b x) \, dx\)

Optimal. Leaf size=51 \[ -\frac{d \cos ^3(a+b x)}{9 b^2}+\frac{d \cos (a+b x)}{3 b^2}+\frac{(c+d x) \sin ^3(a+b x)}{3 b} \]

[Out]

(d*Cos[a + b*x])/(3*b^2) - (d*Cos[a + b*x]^3)/(9*b^2) + ((c + d*x)*Sin[a + b*x]^3)/(3*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0332234, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4404, 2633} \[ -\frac{d \cos ^3(a+b x)}{9 b^2}+\frac{d \cos (a+b x)}{3 b^2}+\frac{(c+d x) \sin ^3(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)*Cos[a + b*x]*Sin[a + b*x]^2,x]

[Out]

(d*Cos[a + b*x])/(3*b^2) - (d*Cos[a + b*x]^3)/(9*b^2) + ((c + d*x)*Sin[a + b*x]^3)/(3*b)

Rule 4404

Int[Cos[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[((c +
d*x)^m*Sin[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Sin[a + b*x]^(n +
1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int (c+d x) \cos (a+b x) \sin ^2(a+b x) \, dx &=\frac{(c+d x) \sin ^3(a+b x)}{3 b}-\frac{d \int \sin ^3(a+b x) \, dx}{3 b}\\ &=\frac{(c+d x) \sin ^3(a+b x)}{3 b}+\frac{d \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (a+b x)\right )}{3 b^2}\\ &=\frac{d \cos (a+b x)}{3 b^2}-\frac{d \cos ^3(a+b x)}{9 b^2}+\frac{(c+d x) \sin ^3(a+b x)}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.181565, size = 44, normalized size = 0.86 \[ \frac{12 b (c+d x) \sin ^3(a+b x)+9 d \cos (a+b x)-d \cos (3 (a+b x))}{36 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)*Cos[a + b*x]*Sin[a + b*x]^2,x]

[Out]

(9*d*Cos[a + b*x] - d*Cos[3*(a + b*x)] + 12*b*(c + d*x)*Sin[a + b*x]^3)/(36*b^2)

________________________________________________________________________________________

Maple [A]  time = 0.018, size = 71, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ({\frac{d}{b} \left ({\frac{ \left ( bx+a \right ) \left ( \sin \left ( bx+a \right ) \right ) ^{3}}{3}}+{\frac{ \left ( 2+ \left ( \sin \left ( bx+a \right ) \right ) ^{2} \right ) \cos \left ( bx+a \right ) }{9}} \right ) }-{\frac{ad \left ( \sin \left ( bx+a \right ) \right ) ^{3}}{3\,b}}+{\frac{c \left ( \sin \left ( bx+a \right ) \right ) ^{3}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)*cos(b*x+a)*sin(b*x+a)^2,x)

[Out]

1/b*(d/b*(1/3*(b*x+a)*sin(b*x+a)^3+1/9*(2+sin(b*x+a)^2)*cos(b*x+a))-1/3/b*a*d*sin(b*x+a)^3+1/3*c*sin(b*x+a)^3)

________________________________________________________________________________________

Maxima [A]  time = 1.13164, size = 115, normalized size = 2.25 \begin{align*} \frac{12 \, c \sin \left (b x + a\right )^{3} - \frac{12 \, a d \sin \left (b x + a\right )^{3}}{b} - \frac{{\left (3 \,{\left (b x + a\right )} \sin \left (3 \, b x + 3 \, a\right ) - 9 \,{\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (3 \, b x + 3 \, a\right ) - 9 \, \cos \left (b x + a\right )\right )} d}{b}}{36 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*cos(b*x+a)*sin(b*x+a)^2,x, algorithm="maxima")

[Out]

1/36*(12*c*sin(b*x + a)^3 - 12*a*d*sin(b*x + a)^3/b - (3*(b*x + a)*sin(3*b*x + 3*a) - 9*(b*x + a)*sin(b*x + a)
+ cos(3*b*x + 3*a) - 9*cos(b*x + a))*d/b)/b

________________________________________________________________________________________

Fricas [A]  time = 0.471744, size = 149, normalized size = 2.92 \begin{align*} -\frac{d \cos \left (b x + a\right )^{3} - 3 \, d \cos \left (b x + a\right ) - 3 \,{\left (b d x -{\left (b d x + b c\right )} \cos \left (b x + a\right )^{2} + b c\right )} \sin \left (b x + a\right )}{9 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*cos(b*x+a)*sin(b*x+a)^2,x, algorithm="fricas")

[Out]

-1/9*(d*cos(b*x + a)^3 - 3*d*cos(b*x + a) - 3*(b*d*x - (b*d*x + b*c)*cos(b*x + a)^2 + b*c)*sin(b*x + a))/b^2

________________________________________________________________________________________

Sympy [A]  time = 1.17803, size = 85, normalized size = 1.67 \begin{align*} \begin{cases} \frac{c \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac{d x \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac{d \sin ^{2}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{3 b^{2}} + \frac{2 d \cos ^{3}{\left (a + b x \right )}}{9 b^{2}} & \text{for}\: b \neq 0 \\\left (c x + \frac{d x^{2}}{2}\right ) \sin ^{2}{\left (a \right )} \cos{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*cos(b*x+a)*sin(b*x+a)**2,x)

[Out]

Piecewise((c*sin(a + b*x)**3/(3*b) + d*x*sin(a + b*x)**3/(3*b) + d*sin(a + b*x)**2*cos(a + b*x)/(3*b**2) + 2*d
*cos(a + b*x)**3/(9*b**2), Ne(b, 0)), ((c*x + d*x**2/2)*sin(a)**2*cos(a), True))

________________________________________________________________________________________

Giac [A]  time = 1.10169, size = 93, normalized size = 1.82 \begin{align*} -\frac{d \cos \left (3 \, b x + 3 \, a\right )}{36 \, b^{2}} + \frac{d \cos \left (b x + a\right )}{4 \, b^{2}} - \frac{{\left (b d x + b c\right )} \sin \left (3 \, b x + 3 \, a\right )}{12 \, b^{2}} + \frac{{\left (b d x + b c\right )} \sin \left (b x + a\right )}{4 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*cos(b*x+a)*sin(b*x+a)^2,x, algorithm="giac")

[Out]

-1/36*d*cos(3*b*x + 3*a)/b^2 + 1/4*d*cos(b*x + a)/b^2 - 1/12*(b*d*x + b*c)*sin(3*b*x + 3*a)/b^2 + 1/4*(b*d*x +
b*c)*sin(b*x + a)/b^2