3.1080 \(\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=361 \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (33 a^2 b (7 A+5 C)+77 a^3 B+165 a b^2 B+5 b^3 (11 A+9 C)\right )}{231 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right )}{15 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left (24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right )}{693 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (242 a^2 b B+24 a^3 C+33 a b^2 (9 A+7 C)+77 b^3 B\right )}{495 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (33 a^2 b (7 A+5 C)+77 a^3 B+165 a b^2 B+5 b^3 (11 A+9 C)\right )}{231 d}+\frac{2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d} \]

[Out]

(2*(27*a^2*b*B + 7*b^3*B + 3*a^3*(5*A + 3*C) + 3*a*b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(77
*a^3*B + 165*a*b^2*B + 33*a^2*b*(7*A + 5*C) + 5*b^3*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*(77*
a^3*B + 165*a*b^2*B + 33*a^2*b*(7*A + 5*C) + 5*b^3*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2
*(242*a^2*b*B + 77*b^3*B + 24*a^3*C + 33*a*b^2*(9*A + 7*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(495*d) + (2*b*(9
9*A*b^2 + 143*a*b*B + 24*a^2*C + 81*b^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*(11*b*B + 6*a*C)*Cos[
c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(99*d) + (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*Si
n[c + d*x])/(11*d)

________________________________________________________________________________________

Rubi [A]  time = 0.920162, antiderivative size = 361, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {3049, 3033, 3023, 2748, 2639, 2635, 2641} \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (33 a^2 b (7 A+5 C)+77 a^3 B+165 a b^2 B+5 b^3 (11 A+9 C)\right )}{231 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right )}{15 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left (24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right )}{693 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (242 a^2 b B+24 a^3 C+33 a b^2 (9 A+7 C)+77 b^3 B\right )}{495 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (33 a^2 b (7 A+5 C)+77 a^3 B+165 a b^2 B+5 b^3 (11 A+9 C)\right )}{231 d}+\frac{2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(27*a^2*b*B + 7*b^3*B + 3*a^3*(5*A + 3*C) + 3*a*b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(77
*a^3*B + 165*a*b^2*B + 33*a^2*b*(7*A + 5*C) + 5*b^3*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*(77*
a^3*B + 165*a*b^2*B + 33*a^2*b*(7*A + 5*C) + 5*b^3*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2
*(242*a^2*b*B + 77*b^3*B + 24*a^3*C + 33*a*b^2*(9*A + 7*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(495*d) + (2*b*(9
9*A*b^2 + 143*a*b*B + 24*a^2*C + 81*b^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*(11*b*B + 6*a*C)*Cos[
c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(99*d) + (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*Si
n[c + d*x])/(11*d)

Rule 3049

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((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*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x]
)^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2)
- C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2
, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rule 3033

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*d*Cos[e + f*x]*Sin[e + f*x]*(a + b
*Sin[e + f*x])^(m + 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*c*
(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e +
f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&
!LtQ[m, -1]

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 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/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 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac{2}{11} \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \left (\frac{1}{2} a (11 A+3 C)+\frac{1}{2} (11 A b+11 a B+9 b C) \cos (c+d x)+\frac{1}{2} (11 b B+6 a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 (11 b B+6 a C) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac{4}{99} \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x)) \left (\frac{3}{4} a (33 a A+11 b B+15 a C)+\frac{1}{4} \left (198 a A b+99 a^2 B+77 b^2 B+150 a b C\right ) \cos (c+d x)+\frac{1}{4} \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 (11 b B+6 a C) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac{8}{693} \int \sqrt{\cos (c+d x)} \left (\frac{21}{8} a^2 (33 a A+11 b B+15 a C)+\frac{9}{8} \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \cos (c+d x)+\frac{7}{8} \left (242 a^2 b B+77 b^3 B+24 a^3 C+33 a b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 \left (242 a^2 b B+77 b^3 B+24 a^3 C+33 a b^2 (9 A+7 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{495 d}+\frac{2 b \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 (11 b B+6 a C) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac{16 \int \sqrt{\cos (c+d x)} \left (\frac{231}{16} \left (27 a^2 b B+7 b^3 B+3 a^3 (5 A+3 C)+3 a b^2 (9 A+7 C)\right )+\frac{45}{16} \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \cos (c+d x)\right ) \, dx}{3465}\\ &=\frac{2 \left (242 a^2 b B+77 b^3 B+24 a^3 C+33 a b^2 (9 A+7 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{495 d}+\frac{2 b \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 (11 b B+6 a C) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac{1}{15} \left (27 a^2 b B+7 b^3 B+3 a^3 (5 A+3 C)+3 a b^2 (9 A+7 C)\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{77} \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 \left (27 a^2 b B+7 b^3 B+3 a^3 (5 A+3 C)+3 a b^2 (9 A+7 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{2 \left (242 a^2 b B+77 b^3 B+24 a^3 C+33 a b^2 (9 A+7 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{495 d}+\frac{2 b \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 (11 b B+6 a C) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac{1}{231} \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (27 a^2 b B+7 b^3 B+3 a^3 (5 A+3 C)+3 a b^2 (9 A+7 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{2 \left (242 a^2 b B+77 b^3 B+24 a^3 C+33 a b^2 (9 A+7 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{495 d}+\frac{2 b \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 (11 b B+6 a C) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d}\\ \end{align*}

Mathematica [A]  time = 1.87997, size = 285, normalized size = 0.79 \[ \frac{10 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (33 a^2 b (7 A+5 C)+77 a^3 B+165 a b^2 B+5 b^3 (11 A+9 C)\right )+154 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right )+\frac{1}{12} \sin (c+d x) \sqrt{\cos (c+d x)} \left (154 \cos (c+d x) \left (108 a^2 b B+36 a^3 C+3 a b^2 (36 A+43 C)+43 b^3 B\right )+5 \left (36 b \cos (2 (c+d x)) \left (33 a^2 C+33 a b B+11 A b^2+16 b^2 C\right )+396 a^2 b (14 A+13 C)+1848 a^3 B+154 b^2 (3 a C+b B) \cos (3 (c+d x))+5148 a b^2 B+3 b^3 (572 A+531 C)+63 b^3 C \cos (4 (c+d x))\right )\right )}{1155 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(154*(27*a^2*b*B + 7*b^3*B + 3*a^3*(5*A + 3*C) + 3*a*b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 10*(77*a^3*B
+ 165*a*b^2*B + 33*a^2*b*(7*A + 5*C) + 5*b^3*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(1
54*(108*a^2*b*B + 43*b^3*B + 36*a^3*C + 3*a*b^2*(36*A + 43*C))*Cos[c + d*x] + 5*(1848*a^3*B + 5148*a*b^2*B + 3
96*a^2*b*(14*A + 13*C) + 3*b^3*(572*A + 531*C) + 36*b*(11*A*b^2 + 33*a*b*B + 33*a^2*C + 16*b^2*C)*Cos[2*(c + d
*x)] + 154*b^2*(b*B + 3*a*C)*Cos[3*(c + d*x)] + 63*b^3*C*Cos[4*(c + d*x)]))*Sin[c + d*x])/12)/(1155*d)

________________________________________________________________________________________

Maple [B]  time = 0.889, size = 1082, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x)

[Out]

-2/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(20160*C*b^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/
2*c)^12+(-12320*B*b^3-36960*C*a*b^2-50400*C*b^3)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(7920*A*b^3+23760*B*
a*b^2+24640*B*b^3+23760*C*a^2*b+73920*C*a*b^2+56880*C*b^3)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-16632*A*a
*b^2-11880*A*b^3-16632*B*a^2*b-35640*B*a*b^2-22792*B*b^3-5544*C*a^3-35640*C*a^2*b-68376*C*a*b^2-34920*C*b^3)*s
in(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(13860*A*a^2*b+16632*A*a*b^2+9240*A*b^3+4620*B*a^3+16632*B*a^2*b+27720*
B*a*b^2+10472*B*b^3+5544*C*a^3+27720*C*a^2*b+31416*C*a*b^2+13860*C*b^3)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c
)+(-6930*A*a^2*b-4158*A*a*b^2-2640*A*b^3-2310*B*a^3-4158*B*a^2*b-7920*B*a*b^2-1848*B*b^3-1386*C*a^3-7920*C*a^2
*b-5544*C*a*b^2-2790*C*b^3)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-3465*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin
(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^3-6237*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*si
n(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^2+3465*A*a^2*b*(sin(1/2*d*x+1/2*c)^2)^(1
/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+825*A*b^3*(sin(1/2*d*x+1/2*c)^2)^(1
/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-6237*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)
*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2*b-1617*B*(sin(1/2*d*x+1/2*c)^2)^(1
/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^3+1155*a^3*B*(sin(1/2*d*x+1/2*c)^
2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+2475*a*b^2*B*(sin(1/2*d*x+1/2*
c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-2079*C*(sin(1/2*d*x+1/2*c)^
2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^3-4851*C*(sin(1/2*d*x+1/2*c)
^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^2+2475*a^2*b*C*(sin(1/2*d
*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+675*C*b^3*(sin(1/2*d
*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))/(-2*sin(1/2*d*x+1/2
*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sqrt{\cos \left (d x + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C b^{3} \cos \left (d x + c\right )^{5} +{\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{4} + A a^{3} +{\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{3} +{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\right )} \sqrt{\cos \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

integral((C*b^3*cos(d*x + c)^5 + (3*C*a*b^2 + B*b^3)*cos(d*x + c)^4 + A*a^3 + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*
cos(d*x + c)^3 + (C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*cos(d*x + c)^2 + (B*a^3 + 3*A*a^2*b)*cos(d*x + c))*sqrt(cos(d
*x + c)), x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*cos(d*x+c)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="giac")

[Out]

Timed out