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

Optimal. Leaf size=243 \[ \frac{\left (52 a^2 b^2 C+15 a^3 b B-3 a^4 C+60 a b^3 B+16 b^4 C\right ) \sin (c+d x)}{30 b d}+\frac{\left (-3 a^2 C+15 a b B+16 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{60 b d}+\frac{\left (30 a^2 b B-6 a^3 C+71 a b^2 C+45 b^3 B\right ) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} x \left (12 a^2 b B+4 a^3 C+9 a b^2 C+3 b^3 B\right )+\frac{(5 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^3}{20 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 b d} \]

[Out]

((12*a^2*b*B + 3*b^3*B + 4*a^3*C + 9*a*b^2*C)*x)/8 + ((15*a^3*b*B + 60*a*b^3*B - 3*a^4*C + 52*a^2*b^2*C + 16*b
^4*C)*Sin[c + d*x])/(30*b*d) + ((30*a^2*b*B + 45*b^3*B - 6*a^3*C + 71*a*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(120
*d) + ((15*a*b*B - 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*b*d) + ((5*b*B - a*C)*(a + b*C
os[c + d*x])^3*Sin[c + d*x])/(20*b*d) + (C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*b*d)

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Rubi [A]  time = 0.293494, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {3023, 2753, 2734} \[ \frac{\left (52 a^2 b^2 C+15 a^3 b B-3 a^4 C+60 a b^3 B+16 b^4 C\right ) \sin (c+d x)}{30 b d}+\frac{\left (-3 a^2 C+15 a b B+16 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{60 b d}+\frac{\left (30 a^2 b B-6 a^3 C+71 a b^2 C+45 b^3 B\right ) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} x \left (12 a^2 b B+4 a^3 C+9 a b^2 C+3 b^3 B\right )+\frac{(5 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^3}{20 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((12*a^2*b*B + 3*b^3*B + 4*a^3*C + 9*a*b^2*C)*x)/8 + ((15*a^3*b*B + 60*a*b^3*B - 3*a^4*C + 52*a^2*b^2*C + 16*b
^4*C)*Sin[c + d*x])/(30*b*d) + ((30*a^2*b*B + 45*b^3*B - 6*a^3*C + 71*a*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(120
*d) + ((15*a*b*B - 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*b*d) + ((5*b*B - a*C)*(a + b*C
os[c + d*x])^3*Sin[c + d*x])/(20*b*d) + (C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*b*d)

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2753

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[
b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 2734

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((2*a*c
+ b*d)*x)/2, x] + (-Simp[((b*c + a*d)*Cos[e + f*x])/f, x] - Simp[(b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rubi steps

\begin{align*} \int (a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 b d}+\frac{\int (a+b \cos (c+d x))^3 (4 b C+(5 b B-a C) \cos (c+d x)) \, dx}{5 b}\\ &=\frac{(5 b B-a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{20 b d}+\frac{C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 b d}+\frac{\int (a+b \cos (c+d x))^2 \left (b (15 b B+13 a C)+\left (15 a b B-3 a^2 C+16 b^2 C\right ) \cos (c+d x)\right ) \, dx}{20 b}\\ &=\frac{\left (15 a b B-3 a^2 C+16 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{60 b d}+\frac{(5 b B-a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{20 b d}+\frac{C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 b d}+\frac{\int (a+b \cos (c+d x)) \left (b \left (75 a b B+33 a^2 C+32 b^2 C\right )+\left (30 a^2 b B+45 b^3 B-6 a^3 C+71 a b^2 C\right ) \cos (c+d x)\right ) \, dx}{60 b}\\ &=\frac{1}{8} \left (12 a^2 b B+3 b^3 B+4 a^3 C+9 a b^2 C\right ) x+\frac{\left (15 a^3 b B+60 a b^3 B-3 a^4 C+52 a^2 b^2 C+16 b^4 C\right ) \sin (c+d x)}{30 b d}+\frac{\left (30 a^2 b B+45 b^3 B-6 a^3 C+71 a b^2 C\right ) \cos (c+d x) \sin (c+d x)}{120 d}+\frac{\left (15 a b B-3 a^2 C+16 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{60 b d}+\frac{(5 b B-a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{20 b d}+\frac{C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 b d}\\ \end{align*}

Mathematica [A]  time = 0.662376, size = 176, normalized size = 0.72 \[ \frac{60 (c+d x) \left (12 a^2 b B+4 a^3 C+9 a b^2 C+3 b^3 B\right )+10 b \left (12 a^2 C+12 a b B+5 b^2 C\right ) \sin (3 (c+d x))+60 \left (18 a^2 b C+8 a^3 B+18 a b^2 B+5 b^3 C\right ) \sin (c+d x)+120 \left (3 a^2 b B+a^3 C+3 a b^2 C+b^3 B\right ) \sin (2 (c+d x))+15 b^2 (3 a C+b B) \sin (4 (c+d x))+6 b^3 C \sin (5 (c+d x))}{480 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(60*(12*a^2*b*B + 3*b^3*B + 4*a^3*C + 9*a*b^2*C)*(c + d*x) + 60*(8*a^3*B + 18*a*b^2*B + 18*a^2*b*C + 5*b^3*C)*
Sin[c + d*x] + 120*(3*a^2*b*B + b^3*B + a^3*C + 3*a*b^2*C)*Sin[2*(c + d*x)] + 10*b*(12*a*b*B + 12*a^2*C + 5*b^
2*C)*Sin[3*(c + d*x)] + 15*b^2*(b*B + 3*a*C)*Sin[4*(c + d*x)] + 6*b^3*C*Sin[5*(c + d*x)])/(480*d)

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Maple [A]  time = 0.016, size = 227, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({\frac{C{b}^{3}\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+{b}^{3}B \left ({\frac{\sin \left ( dx+c \right ) }{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +3\,Ca{b}^{2} \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +a{b}^{2}B \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +{a}^{2}bC \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +3\,{a}^{2}bB \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +{a}^{3}C \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +{a}^{3}B\sin \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/5*C*b^3*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+b^3*B*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x
+c)+3/8*d*x+3/8*c)+3*C*a*b^2*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+a*b^2*B*(2+cos(d*x+c
)^2)*sin(d*x+c)+a^2*b*C*(2+cos(d*x+c)^2)*sin(d*x+c)+3*a^2*b*B*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+a^3*C*
(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+a^3*B*sin(d*x+c))

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Maxima [A]  time = 1.02662, size = 293, normalized size = 1.21 \begin{align*} \frac{120 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 360 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b - 480 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} b - 480 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a b^{2} + 45 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{2} + 15 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{3} + 32 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C b^{3} + 480 \, B a^{3} \sin \left (d x + c\right )}{480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/480*(120*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^3 + 360*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^2*b - 480*(sin(d*
x + c)^3 - 3*sin(d*x + c))*C*a^2*b - 480*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a*b^2 + 45*(12*d*x + 12*c + sin(4
*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a*b^2 + 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*b^3 +
32*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*b^3 + 480*B*a^3*sin(d*x + c))/d

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Fricas [A]  time = 1.52349, size = 423, normalized size = 1.74 \begin{align*} \frac{15 \,{\left (4 \, C a^{3} + 12 \, B a^{2} b + 9 \, C a b^{2} + 3 \, B b^{3}\right )} d x +{\left (24 \, C b^{3} \cos \left (d x + c\right )^{4} + 120 \, B a^{3} + 240 \, C a^{2} b + 240 \, B a b^{2} + 64 \, C b^{3} + 30 \,{\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{3} + 8 \,{\left (15 \, C a^{2} b + 15 \, B a b^{2} + 4 \, C b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \,{\left (4 \, C a^{3} + 12 \, B a^{2} b + 9 \, C a b^{2} + 3 \, B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(15*(4*C*a^3 + 12*B*a^2*b + 9*C*a*b^2 + 3*B*b^3)*d*x + (24*C*b^3*cos(d*x + c)^4 + 120*B*a^3 + 240*C*a^2*
b + 240*B*a*b^2 + 64*C*b^3 + 30*(3*C*a*b^2 + B*b^3)*cos(d*x + c)^3 + 8*(15*C*a^2*b + 15*B*a*b^2 + 4*C*b^3)*cos
(d*x + c)^2 + 15*(4*C*a^3 + 12*B*a^2*b + 9*C*a*b^2 + 3*B*b^3)*cos(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 10.0765, size = 552, normalized size = 2.27 \begin{align*} \begin{cases} \frac{B a^{3} \sin{\left (c + d x \right )}}{d} + \frac{3 B a^{2} b x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{3 B a^{2} b x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{3 B a^{2} b \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{2 B a b^{2} \sin ^{3}{\left (c + d x \right )}}{d} + \frac{3 B a b^{2} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{3 B b^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{3 B b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{3 B b^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{3 B b^{3} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{5 B b^{3} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac{C a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{C a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{C a^{3} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{2 C a^{2} b \sin ^{3}{\left (c + d x \right )}}{d} + \frac{3 C a^{2} b \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{9 C a b^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{9 C a b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{9 C a b^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{9 C a b^{2} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{15 C a b^{2} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac{8 C b^{3} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{4 C b^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{C b^{3} \sin{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \cos{\left (c \right )}\right )^{3} \left (B \cos{\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((B*a**3*sin(c + d*x)/d + 3*B*a**2*b*x*sin(c + d*x)**2/2 + 3*B*a**2*b*x*cos(c + d*x)**2/2 + 3*B*a**2*
b*sin(c + d*x)*cos(c + d*x)/(2*d) + 2*B*a*b**2*sin(c + d*x)**3/d + 3*B*a*b**2*sin(c + d*x)*cos(c + d*x)**2/d +
3*B*b**3*x*sin(c + d*x)**4/8 + 3*B*b**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 3*B*b**3*x*cos(c + d*x)**4/8 +
3*B*b**3*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 5*B*b**3*sin(c + d*x)*cos(c + d*x)**3/(8*d) + C*a**3*x*sin(c + d
*x)**2/2 + C*a**3*x*cos(c + d*x)**2/2 + C*a**3*sin(c + d*x)*cos(c + d*x)/(2*d) + 2*C*a**2*b*sin(c + d*x)**3/d
+ 3*C*a**2*b*sin(c + d*x)*cos(c + d*x)**2/d + 9*C*a*b**2*x*sin(c + d*x)**4/8 + 9*C*a*b**2*x*sin(c + d*x)**2*co
s(c + d*x)**2/4 + 9*C*a*b**2*x*cos(c + d*x)**4/8 + 9*C*a*b**2*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 15*C*a*b**2
*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 8*C*b**3*sin(c + d*x)**5/(15*d) + 4*C*b**3*sin(c + d*x)**3*cos(c + d*x)*
*2/(3*d) + C*b**3*sin(c + d*x)*cos(c + d*x)**4/d, Ne(d, 0)), (x*(a + b*cos(c))**3*(B*cos(c) + C*cos(c)**2), Tr
ue))

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Giac [A]  time = 1.62097, size = 254, normalized size = 1.05 \begin{align*} \frac{C b^{3} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{1}{8} \,{\left (4 \, C a^{3} + 12 \, B a^{2} b + 9 \, C a b^{2} + 3 \, B b^{3}\right )} x + \frac{{\left (3 \, C a b^{2} + B b^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac{{\left (12 \, C a^{2} b + 12 \, B a b^{2} + 5 \, C b^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{{\left (C a^{3} + 3 \, B a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{{\left (8 \, B a^{3} + 18 \, C a^{2} b + 18 \, B a b^{2} + 5 \, C b^{3}\right )} \sin \left (d x + c\right )}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/80*C*b^3*sin(5*d*x + 5*c)/d + 1/8*(4*C*a^3 + 12*B*a^2*b + 9*C*a*b^2 + 3*B*b^3)*x + 1/32*(3*C*a*b^2 + B*b^3)*
sin(4*d*x + 4*c)/d + 1/48*(12*C*a^2*b + 12*B*a*b^2 + 5*C*b^3)*sin(3*d*x + 3*c)/d + 1/4*(C*a^3 + 3*B*a^2*b + 3*
C*a*b^2 + B*b^3)*sin(2*d*x + 2*c)/d + 1/8*(8*B*a^3 + 18*C*a^2*b + 18*B*a*b^2 + 5*C*b^3)*sin(d*x + c)/d