3.20 \(\int (a+a \cos (c+d x))^3 (A+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=148 \[ -\frac{a^3 (20 A+13 C) \sin ^3(c+d x)}{60 d}+\frac{a^3 (20 A+13 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (20 A+13 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (20 A+13 C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}-\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d} \]

[Out]

(a^3*(20*A + 13*C)*x)/8 + (a^3*(20*A + 13*C)*Sin[c + d*x])/(5*d) + (3*a^3*(20*A + 13*C)*Cos[c + d*x]*Sin[c + d
*x])/(40*d) - (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*a*d)
- (a^3*(20*A + 13*C)*Sin[c + d*x]^3)/(60*d)

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Rubi [A]  time = 0.198492, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {3024, 2751, 2645, 2637, 2635, 8, 2633} \[ -\frac{a^3 (20 A+13 C) \sin ^3(c+d x)}{60 d}+\frac{a^3 (20 A+13 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (20 A+13 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (20 A+13 C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}-\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(20*A + 13*C)*x)/8 + (a^3*(20*A + 13*C)*Sin[c + d*x])/(5*d) + (3*a^3*(20*A + 13*C)*Cos[c + d*x]*Sin[c + d
*x])/(40*d) - (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*a*d)
- (a^3*(20*A + 13*C)*Sin[c + d*x]^3)/(60*d)

Rule 3024

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (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) - a*C*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] &&  !LtQ[
m, -1]

Rule 2751

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[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*Sin
[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m,
-2^(-1)]

Rule 2645

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandTrig[(a + b*sin[c + d*x])^n, x], x] /;
FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}+\frac{\int (a+a \cos (c+d x))^3 (a (5 A+4 C)-a C \cos (c+d x)) \, dx}{5 a}\\ &=-\frac{C (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}+\frac{1}{20} (20 A+13 C) \int (a+a \cos (c+d x))^3 \, dx\\ &=-\frac{C (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}+\frac{1}{20} (20 A+13 C) \int \left (a^3+3 a^3 \cos (c+d x)+3 a^3 \cos ^2(c+d x)+a^3 \cos ^3(c+d x)\right ) \, dx\\ &=\frac{1}{20} a^3 (20 A+13 C) x-\frac{C (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}+\frac{1}{20} \left (a^3 (20 A+13 C)\right ) \int \cos ^3(c+d x) \, dx+\frac{1}{20} \left (3 a^3 (20 A+13 C)\right ) \int \cos (c+d x) \, dx+\frac{1}{20} \left (3 a^3 (20 A+13 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{1}{20} a^3 (20 A+13 C) x+\frac{3 a^3 (20 A+13 C) \sin (c+d x)}{20 d}+\frac{3 a^3 (20 A+13 C) \cos (c+d x) \sin (c+d x)}{40 d}-\frac{C (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}+\frac{1}{40} \left (3 a^3 (20 A+13 C)\right ) \int 1 \, dx-\frac{\left (a^3 (20 A+13 C)\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{20 d}\\ &=\frac{1}{8} a^3 (20 A+13 C) x+\frac{a^3 (20 A+13 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (20 A+13 C) \cos (c+d x) \sin (c+d x)}{40 d}-\frac{C (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}-\frac{a^3 (20 A+13 C) \sin ^3(c+d x)}{60 d}\\ \end{align*}

Mathematica [A]  time = 0.364388, size = 97, normalized size = 0.66 \[ \frac{a^3 (60 (30 A+23 C) \sin (c+d x)+120 (3 A+4 C) \sin (2 (c+d x))+40 A \sin (3 (c+d x))+1200 A d x+170 C \sin (3 (c+d x))+45 C \sin (4 (c+d x))+6 C \sin (5 (c+d x))+780 C d x)}{480 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(1200*A*d*x + 780*C*d*x + 60*(30*A + 23*C)*Sin[c + d*x] + 120*(3*A + 4*C)*Sin[2*(c + d*x)] + 40*A*Sin[3*(
c + d*x)] + 170*C*Sin[3*(c + d*x)] + 45*C*Sin[4*(c + d*x)] + 6*C*Sin[5*(c + d*x)]))/(480*d)

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Maple [A]  time = 0.025, size = 197, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{3}C\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 ) }+3\,{a}^{3}C \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 ) +{\frac{A{a}^{3} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+{a}^{3}C \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +3\,A{a}^{3} \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 ) +3\,A{a}^{3}\sin \left ( dx+c \right ) +A{a}^{3} \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02303, size = 257, normalized size = 1.74 \begin{align*} -\frac{160 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} - 360 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 480 \,{\left (d x + c\right )} A a^{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 a^{3} + 480 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} - 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^{3} - 120 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 1440 \, A 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+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

-1/480*(160*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^3 - 360*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^3 - 480*(d*x +
c)*A*a^3 - 32*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*a^3 + 480*(sin(d*x + c)^3 - 3*sin(d*x
+ c))*C*a^3 - 45*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^3 - 120*(2*d*x + 2*c + sin(2*d*x
+ 2*c))*C*a^3 - 1440*A*a^3*sin(d*x + c))/d

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Fricas [A]  time = 1.42726, size = 266, normalized size = 1.8 \begin{align*} \frac{15 \,{\left (20 \, A + 13 \, C\right )} a^{3} d x +{\left (24 \, C a^{3} \cos \left (d x + c\right )^{4} + 90 \, C a^{3} \cos \left (d x + c\right )^{3} + 8 \,{\left (5 \, A + 19 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \,{\left (12 \, A + 13 \, C\right )} a^{3} \cos \left (d x + c\right ) + 8 \,{\left (55 \, A + 38 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(15*(20*A + 13*C)*a^3*d*x + (24*C*a^3*cos(d*x + c)^4 + 90*C*a^3*cos(d*x + c)^3 + 8*(5*A + 19*C)*a^3*cos(
d*x + c)^2 + 15*(12*A + 13*C)*a^3*cos(d*x + c) + 8*(55*A + 38*C)*a^3)*sin(d*x + c))/d

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Sympy [A]  time = 3.25116, size = 422, normalized size = 2.85 \begin{align*} \begin{cases} \frac{3 A a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{3 A a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + A a^{3} x + \frac{2 A a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{A a^{3} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{3 A a^{3} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{3 A a^{3} \sin{\left (c + d x \right )}}{d} + \frac{9 C a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{9 C a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{C a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{9 C a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{C a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{8 C a^{3} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{4 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{9 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{2 C a^{3} \sin ^{3}{\left (c + d x \right )}}{d} + \frac{C a^{3} \sin{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{15 C a^{3} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac{3 C a^{3} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{C a^{3} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\left (c \right )}\right ) \left (a \cos{\left (c \right )} + a\right )^{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((3*A*a**3*x*sin(c + d*x)**2/2 + 3*A*a**3*x*cos(c + d*x)**2/2 + A*a**3*x + 2*A*a**3*sin(c + d*x)**3/(
3*d) + A*a**3*sin(c + d*x)*cos(c + d*x)**2/d + 3*A*a**3*sin(c + d*x)*cos(c + d*x)/(2*d) + 3*A*a**3*sin(c + d*x
)/d + 9*C*a**3*x*sin(c + d*x)**4/8 + 9*C*a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + C*a**3*x*sin(c + d*x)**2/2
+ 9*C*a**3*x*cos(c + d*x)**4/8 + C*a**3*x*cos(c + d*x)**2/2 + 8*C*a**3*sin(c + d*x)**5/(15*d) + 4*C*a**3*sin(
c + d*x)**3*cos(c + d*x)**2/(3*d) + 9*C*a**3*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 2*C*a**3*sin(c + d*x)**3/d +
C*a**3*sin(c + d*x)*cos(c + d*x)**4/d + 15*C*a**3*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 3*C*a**3*sin(c + d*x)*
cos(c + d*x)**2/d + C*a**3*sin(c + d*x)*cos(c + d*x)/(2*d), Ne(d, 0)), (x*(A + C*cos(c)**2)*(a*cos(c) + a)**3,
True))

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Giac [A]  time = 1.22378, size = 177, normalized size = 1.2 \begin{align*} \frac{C a^{3} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{3 \, C a^{3} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac{1}{8} \,{\left (20 \, A a^{3} + 13 \, C a^{3}\right )} x + \frac{{\left (4 \, A a^{3} + 17 \, C a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{{\left (3 \, A a^{3} + 4 \, C a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{{\left (30 \, A a^{3} + 23 \, C a^{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+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

1/80*C*a^3*sin(5*d*x + 5*c)/d + 3/32*C*a^3*sin(4*d*x + 4*c)/d + 1/8*(20*A*a^3 + 13*C*a^3)*x + 1/48*(4*A*a^3 +
17*C*a^3)*sin(3*d*x + 3*c)/d + 1/4*(3*A*a^3 + 4*C*a^3)*sin(2*d*x + 2*c)/d + 1/8*(30*A*a^3 + 23*C*a^3)*sin(d*x
+ c)/d