3.1087 \(\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^4 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=477 \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )}{231 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+468 a^3 b B+364 a b^3 B+7 b^4 (13 A+11 C)\right )}{195 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{1287 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left (2171 a^2 b B+192 a^3 C+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{9009 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (11 a^2 b^2 (637 A+491 C)+3458 a^3 b B+192 a^4 C+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{6435 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )}{231 d}+\frac{2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{143 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d} \]

[Out]

(2*(468*a^3*b*B + 364*a*b^3*B + 39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*EllipticE[(
c + d*x)/2, 2])/(195*d) + (2*(77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 44*a^3*b*(7*A + 5*C) + 20*a*b^3*(11*A + 9*
C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*(77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 44*a^3*b*(7*A + 5*C) + 20*a
*b^3*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(3458*a^3*b*B + 4004*a*b^3*B + 192*a^4*C + 77
*b^4*(13*A + 11*C) + 11*a^2*b^2*(637*A + 491*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(6435*d) + (2*b*(2171*a^2*b*
B + 1053*b^3*B + 192*a^3*C + 2*a*b^2*(1573*A + 1259*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d) + (2*(143*A*
b^2 + 221*a*b*B + 48*a^2*C + 121*b^2*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(1287*d) + (2*
(13*b*B + 8*a*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(143*d) + (2*C*Cos[c + d*x]^(3/2)*(a
+ b*Cos[c + d*x])^4*Sin[c + d*x])/(13*d)

________________________________________________________________________________________

Rubi [A]  time = 1.31604, antiderivative size = 477, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )}{231 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+468 a^3 b B+364 a b^3 B+7 b^4 (13 A+11 C)\right )}{195 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{1287 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left (2171 a^2 b B+192 a^3 C+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{9009 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (11 a^2 b^2 (637 A+491 C)+3458 a^3 b B+192 a^4 C+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{6435 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )}{231 d}+\frac{2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{143 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]

[Out]

(2*(468*a^3*b*B + 364*a*b^3*B + 39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*EllipticE[(
c + d*x)/2, 2])/(195*d) + (2*(77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 44*a^3*b*(7*A + 5*C) + 20*a*b^3*(11*A + 9*
C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*(77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 44*a^3*b*(7*A + 5*C) + 20*a
*b^3*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(3458*a^3*b*B + 4004*a*b^3*B + 192*a^4*C + 77
*b^4*(13*A + 11*C) + 11*a^2*b^2*(637*A + 491*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(6435*d) + (2*b*(2171*a^2*b*
B + 1053*b^3*B + 192*a^3*C + 2*a*b^2*(1573*A + 1259*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d) + (2*(143*A*
b^2 + 221*a*b*B + 48*a^2*C + 121*b^2*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(1287*d) + (2*
(13*b*B + 8*a*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(143*d) + (2*C*Cos[c + d*x]^(3/2)*(a
+ b*Cos[c + d*x])^4*Sin[c + d*x])/(13*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))^4 \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))^4 \sin (c+d x)}{13 d}+\frac{2}{13} \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 \left (\frac{1}{2} a (13 A+3 C)+\frac{1}{2} (13 A b+13 a B+11 b C) \cos (c+d x)+\frac{1}{2} (13 b B+8 a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 (13 b B+8 a C) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac{4}{143} \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \left (\frac{1}{4} a (143 a A+39 b B+57 a C)+\frac{1}{4} \left (286 a A b+143 a^2 B+117 b^2 B+226 a b C\right ) \cos (c+d x)+\frac{1}{4} \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac{2 (13 b B+8 a C) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac{8 \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x)) \left (\frac{3}{8} a \left (338 a b B+11 b^2 (13 A+11 C)+3 a^2 (143 A+73 C)\right )+\frac{1}{8} \left (1287 a^3 B+2951 a b^2 B+77 b^3 (13 A+11 C)+3 a^2 b (1287 A+961 C)\right ) \cos (c+d x)+\frac{1}{8} \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \cos ^2(c+d x)\right ) \, dx}{1287}\\ &=\frac{2 b \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac{2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac{2 (13 b B+8 a C) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac{16 \int \sqrt{\cos (c+d x)} \left (\frac{21}{16} a^2 \left (338 a b B+11 b^2 (13 A+11 C)+3 a^2 (143 A+73 C)\right )+\frac{117}{16} \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \cos (c+d x)+\frac{7}{16} \left (3458 a^3 b B+4004 a b^3 B+192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^2(c+d x)\right ) \, dx}{9009}\\ &=\frac{2 \left (3458 a^3 b B+4004 a b^3 B+192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{6435 d}+\frac{2 b \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac{2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac{2 (13 b B+8 a C) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac{32 \int \sqrt{\cos (c+d x)} \left (\frac{231}{32} \left (468 a^3 b B+364 a b^3 B+39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right )+\frac{585}{32} \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \cos (c+d x)\right ) \, dx}{45045}\\ &=\frac{2 \left (3458 a^3 b B+4004 a b^3 B+192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{6435 d}+\frac{2 b \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac{2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac{2 (13 b B+8 a C) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac{1}{77} \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{195} \left (468 a^3 b B+364 a b^3 B+39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (468 a^3 b B+364 a b^3 B+39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{195 d}+\frac{2 \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{2 \left (3458 a^3 b B+4004 a b^3 B+192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{6435 d}+\frac{2 b \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac{2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac{2 (13 b B+8 a C) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac{1}{231} \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (468 a^3 b B+364 a b^3 B+39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{195 d}+\frac{2 \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a 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^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{2 \left (3458 a^3 b B+4004 a b^3 B+192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{6435 d}+\frac{2 b \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac{2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac{2 (13 b B+8 a C) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}\\ \end{align*}

Mathematica [A]  time = 3.51933, size = 381, normalized size = 0.8 \[ \frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left (154 \cos (c+d x) \left (156 a^2 b^2 (36 A+43 C)+3744 a^3 b B+936 a^4 C+4472 a b^3 B+b^4 (1118 A+1171 C)\right )+5 \left (77 b^2 \cos (3 (c+d x)) \left (312 a^2 C+208 a b B+52 A b^2+89 b^2 C\right )+1872 b \cos (2 (c+d x)) \left (33 a^2 b B+22 a^3 C+2 a b^2 (11 A+16 C)+8 b^3 B\right )+78 \left (176 a^3 b (14 A+13 C)+3432 a^2 b^2 B+616 a^4 B+4 a b^3 (572 A+531 C)+531 b^4 B\right )+1638 b^3 (4 a C+b B) \cos (4 (c+d x))+693 b^4 C \cos (5 (c+d x))\right )\right )+48 \left (65 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )+77 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+468 a^3 b B+364 a b^3 B+7 b^4 (13 A+11 C)\right )\right )}{360360 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]

[Out]

(48*(77*(468*a^3*b*B + 364*a*b^3*B + 39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*Ellipt
icE[(c + d*x)/2, 2] + 65*(77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 44*a^3*b*(7*A + 5*C) + 20*a*b^3*(11*A + 9*C))*
EllipticF[(c + d*x)/2, 2]) + Sqrt[Cos[c + d*x]]*(154*(3744*a^3*b*B + 4472*a*b^3*B + 936*a^4*C + 156*a^2*b^2*(3
6*A + 43*C) + b^4*(1118*A + 1171*C))*Cos[c + d*x] + 5*(78*(616*a^4*B + 3432*a^2*b^2*B + 531*b^4*B + 176*a^3*b*
(14*A + 13*C) + 4*a*b^3*(572*A + 531*C)) + 1872*b*(33*a^2*b*B + 8*b^3*B + 22*a^3*C + 2*a*b^2*(11*A + 16*C))*Co
s[2*(c + d*x)] + 77*b^2*(52*A*b^2 + 208*a*b*B + 312*a^2*C + 89*b^2*C)*Cos[3*(c + d*x)] + 1638*b^3*(b*B + 4*a*C
)*Cos[4*(c + d*x)] + 693*b^4*C*Cos[5*(c + d*x)]))*Sin[c + d*x])/(360360*d)

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Maple [B]  time = 1.152, size = 1407, 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))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x)

[Out]

-2/45045*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-443520*C*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x
+1/2*c)^14+(262080*B*b^4+1048320*C*a*b^3+1330560*C*b^4)*sin(1/2*d*x+1/2*c)^12*cos(1/2*d*x+1/2*c)+(-160160*A*b^
4-640640*B*a*b^3-655200*B*b^4-960960*C*a^2*b^2-2620800*C*a*b^3-1798720*C*b^4)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*
x+1/2*c)+(411840*A*a*b^3+320320*A*b^4+617760*B*a^2*b^2+1281280*B*a*b^3+739440*B*b^4+411840*C*a^3*b+1921920*C*a
^2*b^2+2957760*C*a*b^3+1379840*C*b^4)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-432432*A*a^2*b^2-617760*A*a*b^
3-296296*A*b^4-288288*B*a^3*b-926640*B*a^2*b^2-1185184*B*a*b^3-453960*B*b^4-72072*C*a^4-617760*C*a^3*b-1777776
*C*a^2*b^2-1815840*C*a*b^3-666512*C*b^4)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(240240*A*a^3*b+432432*A*a^2*
b^2+480480*A*a*b^3+136136*A*b^4+60060*B*a^4+288288*B*a^3*b+720720*B*a^2*b^2+544544*B*a*b^3+180180*B*b^4+72072*
C*a^4+480480*C*a^3*b+816816*C*a^2*b^2+720720*C*a*b^3+198352*C*b^4)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-1
20120*A*a^3*b-108108*A*a^2*b^2-137280*A*a*b^3-24024*A*b^4-30030*B*a^4-72072*B*a^3*b-205920*B*a^2*b^2-96096*B*a
*b^3-36270*B*b^4-18018*C*a^4-137280*C*a^3*b-144144*C*a^2*b^2-145080*C*a*b^3-27258*C*b^4)*sin(1/2*d*x+1/2*c)^2*
cos(1/2*d*x+1/2*c)-45045*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^4-162162*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^2*b^2-21021*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(co
s(1/2*d*x+1/2*c),2^(1/2))*b^4+60060*A*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)*Elli
pticF(cos(1/2*d*x+1/2*c),2^(1/2))+42900*a*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))-108108*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)*E
llipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^3*b-84084*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*b^3+15015*a^4*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))+64350*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))*a^2*b^2+8775*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))*b^4-27027*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^4-126126*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^2*b^2-17787*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))*b^4+42900*a^3*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))+35100*C*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)))/(-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

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Maxima [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))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C b^{4} \cos \left (d x + c\right )^{6} +{\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{5} + A a^{4} +{\left (6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (2 \, C a^{3} b + 3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} +{\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{4} + 4 \, A a^{3} 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))^4*(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^4*cos(d*x + c)^6 + (4*C*a*b^3 + B*b^4)*cos(d*x + c)^5 + A*a^4 + (6*C*a^2*b^2 + 4*B*a*b^3 + A*b^4
)*cos(d*x + c)^4 + 2*(2*C*a^3*b + 3*B*a^2*b^2 + 2*A*a*b^3)*cos(d*x + c)^3 + (C*a^4 + 4*B*a^3*b + 6*A*a^2*b^2)*
cos(d*x + c)^2 + (B*a^4 + 4*A*a^3*b)*cos(d*x + c))*sqrt(cos(d*x + c)), x)

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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))**4*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*cos(d*x+c)**(1/2),x)

[Out]

Timed out

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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))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="giac")

[Out]

Timed out