3.18 \(\int (c+d x)^2 \cos ^3(a+b x) \, dx\)

Optimal. Leaf size=123 \[ \frac{2 d (c+d x) \cos ^3(a+b x)}{9 b^2}+\frac{4 d (c+d x) \cos (a+b x)}{3 b^2}+\frac{2 d^2 \sin ^3(a+b x)}{27 b^3}-\frac{14 d^2 \sin (a+b x)}{9 b^3}+\frac{2 (c+d x)^2 \sin (a+b x)}{3 b}+\frac{(c+d x)^2 \sin (a+b x) \cos ^2(a+b x)}{3 b} \]

[Out]

(4*d*(c + d*x)*Cos[a + b*x])/(3*b^2) + (2*d*(c + d*x)*Cos[a + b*x]^3)/(9*b^2) - (14*d^2*Sin[a + b*x])/(9*b^3)
+ (2*(c + d*x)^2*Sin[a + b*x])/(3*b) + ((c + d*x)^2*Cos[a + b*x]^2*Sin[a + b*x])/(3*b) + (2*d^2*Sin[a + b*x]^3
)/(27*b^3)

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Rubi [A]  time = 0.0967742, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3311, 3296, 2637, 2633} \[ \frac{2 d (c+d x) \cos ^3(a+b x)}{9 b^2}+\frac{4 d (c+d x) \cos (a+b x)}{3 b^2}+\frac{2 d^2 \sin ^3(a+b x)}{27 b^3}-\frac{14 d^2 \sin (a+b x)}{9 b^3}+\frac{2 (c+d x)^2 \sin (a+b x)}{3 b}+\frac{(c+d x)^2 \sin (a+b x) \cos ^2(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2*Cos[a + b*x]^3,x]

[Out]

(4*d*(c + d*x)*Cos[a + b*x])/(3*b^2) + (2*d*(c + d*x)*Cos[a + b*x]^3)/(9*b^2) - (14*d^2*Sin[a + b*x])/(9*b^3)
+ (2*(c + d*x)^2*Sin[a + b*x])/(3*b) + ((c + d*x)^2*Cos[a + b*x]^2*Sin[a + b*x])/(3*b) + (2*d^2*Sin[a + b*x]^3
)/(27*b^3)

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

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]

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)^2 \cos ^3(a+b x) \, dx &=\frac{2 d (c+d x) \cos ^3(a+b x)}{9 b^2}+\frac{(c+d x)^2 \cos ^2(a+b x) \sin (a+b x)}{3 b}+\frac{2}{3} \int (c+d x)^2 \cos (a+b x) \, dx-\frac{\left (2 d^2\right ) \int \cos ^3(a+b x) \, dx}{9 b^2}\\ &=\frac{2 d (c+d x) \cos ^3(a+b x)}{9 b^2}+\frac{2 (c+d x)^2 \sin (a+b x)}{3 b}+\frac{(c+d x)^2 \cos ^2(a+b x) \sin (a+b x)}{3 b}-\frac{(4 d) \int (c+d x) \sin (a+b x) \, dx}{3 b}+\frac{\left (2 d^2\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (a+b x)\right )}{9 b^3}\\ &=\frac{4 d (c+d x) \cos (a+b x)}{3 b^2}+\frac{2 d (c+d x) \cos ^3(a+b x)}{9 b^2}-\frac{2 d^2 \sin (a+b x)}{9 b^3}+\frac{2 (c+d x)^2 \sin (a+b x)}{3 b}+\frac{(c+d x)^2 \cos ^2(a+b x) \sin (a+b x)}{3 b}+\frac{2 d^2 \sin ^3(a+b x)}{27 b^3}-\frac{\left (4 d^2\right ) \int \cos (a+b x) \, dx}{3 b^2}\\ &=\frac{4 d (c+d x) \cos (a+b x)}{3 b^2}+\frac{2 d (c+d x) \cos ^3(a+b x)}{9 b^2}-\frac{14 d^2 \sin (a+b x)}{9 b^3}+\frac{2 (c+d x)^2 \sin (a+b x)}{3 b}+\frac{(c+d x)^2 \cos ^2(a+b x) \sin (a+b x)}{3 b}+\frac{2 d^2 \sin ^3(a+b x)}{27 b^3}\\ \end{align*}

Mathematica [A]  time = 0.602845, size = 93, normalized size = 0.76 \[ \frac{2 \sin (a+b x) \left (\cos (2 (a+b x)) \left (9 b^2 (c+d x)^2-2 d^2\right )+45 b^2 (c+d x)^2-82 d^2\right )+162 b d (c+d x) \cos (a+b x)+6 b d (c+d x) \cos (3 (a+b x))}{108 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2*Cos[a + b*x]^3,x]

[Out]

(162*b*d*(c + d*x)*Cos[a + b*x] + 6*b*d*(c + d*x)*Cos[3*(a + b*x)] + 2*(-82*d^2 + 45*b^2*(c + d*x)^2 + (-2*d^2
+ 9*b^2*(c + d*x)^2)*Cos[2*(a + b*x)])*Sin[a + b*x])/(108*b^3)

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Maple [B]  time = 0.029, size = 265, normalized size = 2.2 \begin{align*}{\frac{1}{b} \left ({\frac{{d}^{2}}{{b}^{2}} \left ({\frac{ \left ( bx+a \right ) ^{2} \left ( 2+ \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \sin \left ( bx+a \right ) }{3}}-{\frac{4\,\sin \left ( bx+a \right ) }{3}}+{\frac{ \left ( 4\,bx+4\,a \right ) \cos \left ( bx+a \right ) }{3}}+{\frac{ \left ( 2\,bx+2\,a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{3}}{9}}-{\frac{ \left ( 4+2\, \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \sin \left ( bx+a \right ) }{27}} \right ) }-2\,{\frac{a{d}^{2} \left ( 1/3\, \left ( bx+a \right ) \left ( 2+ \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \sin \left ( bx+a \right ) +1/9\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}+2/3\,\cos \left ( bx+a \right ) \right ) }{{b}^{2}}}+2\,{\frac{cd \left ( 1/3\, \left ( bx+a \right ) \left ( 2+ \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \sin \left ( bx+a \right ) +1/9\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}+2/3\,\cos \left ( bx+a \right ) \right ) }{b}}+{\frac{{a}^{2}{d}^{2} \left ( 2+ \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \sin \left ( bx+a \right ) }{3\,{b}^{2}}}-{\frac{2\,acd \left ( 2+ \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \sin \left ( bx+a \right ) }{3\,b}}+{\frac{{c}^{2} \left ( 2+ \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \sin \left ( bx+a \right ) }{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^2*cos(b*x+a)^3,x)

[Out]

1/b*(1/b^2*d^2*(1/3*(b*x+a)^2*(2+cos(b*x+a)^2)*sin(b*x+a)-4/3*sin(b*x+a)+4/3*(b*x+a)*cos(b*x+a)+2/9*(b*x+a)*co
s(b*x+a)^3-2/27*(2+cos(b*x+a)^2)*sin(b*x+a))-2/b^2*a*d^2*(1/3*(b*x+a)*(2+cos(b*x+a)^2)*sin(b*x+a)+1/9*cos(b*x+
a)^3+2/3*cos(b*x+a))+2/b*c*d*(1/3*(b*x+a)*(2+cos(b*x+a)^2)*sin(b*x+a)+1/9*cos(b*x+a)^3+2/3*cos(b*x+a))+1/3/b^2
*a^2*d^2*(2+cos(b*x+a)^2)*sin(b*x+a)-2/3/b*a*c*d*(2+cos(b*x+a)^2)*sin(b*x+a)+1/3*c^2*(2+cos(b*x+a)^2)*sin(b*x+
a))

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Maxima [B]  time = 1.06168, size = 360, normalized size = 2.93 \begin{align*} -\frac{36 \,{\left (\sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )\right )} c^{2} - \frac{72 \,{\left (\sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )\right )} a c d}{b} + \frac{36 \,{\left (\sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )\right )} a^{2} d^{2}}{b^{2}} - \frac{6 \,{\left (3 \,{\left (b x + a\right )} \sin \left (3 \, b x + 3 \, a\right ) + 27 \,{\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (3 \, b x + 3 \, a\right ) + 27 \, \cos \left (b x + a\right )\right )} c d}{b} + \frac{6 \,{\left (3 \,{\left (b x + a\right )} \sin \left (3 \, b x + 3 \, a\right ) + 27 \,{\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (3 \, b x + 3 \, a\right ) + 27 \, \cos \left (b x + a\right )\right )} a d^{2}}{b^{2}} - \frac{{\left (6 \,{\left (b x + a\right )} \cos \left (3 \, b x + 3 \, a\right ) + 162 \,{\left (b x + a\right )} \cos \left (b x + a\right ) +{\left (9 \,{\left (b x + a\right )}^{2} - 2\right )} \sin \left (3 \, b x + 3 \, a\right ) + 81 \,{\left ({\left (b x + a\right )}^{2} - 2\right )} \sin \left (b x + a\right )\right )} d^{2}}{b^{2}}}{108 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2*cos(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/108*(36*(sin(b*x + a)^3 - 3*sin(b*x + a))*c^2 - 72*(sin(b*x + a)^3 - 3*sin(b*x + a))*a*c*d/b + 36*(sin(b*x
+ a)^3 - 3*sin(b*x + a))*a^2*d^2/b^2 - 6*(3*(b*x + a)*sin(3*b*x + 3*a) + 27*(b*x + a)*sin(b*x + a) + cos(3*b*x
+ 3*a) + 27*cos(b*x + a))*c*d/b + 6*(3*(b*x + a)*sin(3*b*x + 3*a) + 27*(b*x + a)*sin(b*x + a) + cos(3*b*x + 3
*a) + 27*cos(b*x + a))*a*d^2/b^2 - (6*(b*x + a)*cos(3*b*x + 3*a) + 162*(b*x + a)*cos(b*x + a) + (9*(b*x + a)^2
- 2)*sin(3*b*x + 3*a) + 81*((b*x + a)^2 - 2)*sin(b*x + a))*d^2/b^2)/b

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Fricas [A]  time = 1.47013, size = 297, normalized size = 2.41 \begin{align*} \frac{6 \,{\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{3} + 36 \,{\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) +{\left (18 \, b^{2} d^{2} x^{2} + 36 \, b^{2} c d x + 18 \, b^{2} c^{2} +{\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} - 2 \, d^{2}\right )} \cos \left (b x + a\right )^{2} - 40 \, d^{2}\right )} \sin \left (b x + a\right )}{27 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2*cos(b*x+a)^3,x, algorithm="fricas")

[Out]

1/27*(6*(b*d^2*x + b*c*d)*cos(b*x + a)^3 + 36*(b*d^2*x + b*c*d)*cos(b*x + a) + (18*b^2*d^2*x^2 + 36*b^2*c*d*x
+ 18*b^2*c^2 + (9*b^2*d^2*x^2 + 18*b^2*c*d*x + 9*b^2*c^2 - 2*d^2)*cos(b*x + a)^2 - 40*d^2)*sin(b*x + a))/b^3

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Sympy [A]  time = 3.02884, size = 284, normalized size = 2.31 \begin{align*} \begin{cases} \frac{2 c^{2} \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac{c^{2} \sin{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b} + \frac{4 c d x \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac{2 c d x \sin{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b} + \frac{2 d^{2} x^{2} \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac{d^{2} x^{2} \sin{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b} + \frac{4 c d \sin ^{2}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{3 b^{2}} + \frac{14 c d \cos ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac{4 d^{2} x \sin ^{2}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{3 b^{2}} + \frac{14 d^{2} x \cos ^{3}{\left (a + b x \right )}}{9 b^{2}} - \frac{40 d^{2} \sin ^{3}{\left (a + b x \right )}}{27 b^{3}} - \frac{14 d^{2} \sin{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{9 b^{3}} & \text{for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac{d^{2} x^{3}}{3}\right ) \cos ^{3}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2*cos(b*x+a)**3,x)

[Out]

Piecewise((2*c**2*sin(a + b*x)**3/(3*b) + c**2*sin(a + b*x)*cos(a + b*x)**2/b + 4*c*d*x*sin(a + b*x)**3/(3*b)
+ 2*c*d*x*sin(a + b*x)*cos(a + b*x)**2/b + 2*d**2*x**2*sin(a + b*x)**3/(3*b) + d**2*x**2*sin(a + b*x)*cos(a +
b*x)**2/b + 4*c*d*sin(a + b*x)**2*cos(a + b*x)/(3*b**2) + 14*c*d*cos(a + b*x)**3/(9*b**2) + 4*d**2*x*sin(a + b
*x)**2*cos(a + b*x)/(3*b**2) + 14*d**2*x*cos(a + b*x)**3/(9*b**2) - 40*d**2*sin(a + b*x)**3/(27*b**3) - 14*d**
2*sin(a + b*x)*cos(a + b*x)**2/(9*b**3), Ne(b, 0)), ((c**2*x + c*d*x**2 + d**2*x**3/3)*cos(a)**3, True))

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Giac [A]  time = 1.20051, size = 185, normalized size = 1.5 \begin{align*} \frac{{\left (b d^{2} x + b c d\right )} \cos \left (3 \, b x + 3 \, a\right )}{18 \, b^{3}} + \frac{3 \,{\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )}{2 \, b^{3}} + \frac{{\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} - 2 \, d^{2}\right )} \sin \left (3 \, b x + 3 \, a\right )}{108 \, b^{3}} + \frac{3 \,{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, d^{2}\right )} \sin \left (b x + a\right )}{4 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2*cos(b*x+a)^3,x, algorithm="giac")

[Out]

1/18*(b*d^2*x + b*c*d)*cos(3*b*x + 3*a)/b^3 + 3/2*(b*d^2*x + b*c*d)*cos(b*x + a)/b^3 + 1/108*(9*b^2*d^2*x^2 +
18*b^2*c*d*x + 9*b^2*c^2 - 2*d^2)*sin(3*b*x + 3*a)/b^3 + 3/4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*sin
(b*x + a)/b^3