3.1029 \(\int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=629 \[ \frac{2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{1287 b^3 d}+\frac{2 \sin (c+d x) \left (104 a^2 b B-48 a^3 C-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{9009 b^3 d}+\frac{2 \sin (c+d x) \left (-10 a^2 b^2 (143 A+124 C)+520 a^3 b B-240 a^4 C+4355 a b^3 B+539 b^4 (13 A+11 C)\right ) (a+b \cos (c+d x))^{3/2}}{45045 b^3 d}+\frac{2 \sin (c+d x) \left (-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+520 a^4 b B-240 a^5 C+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt{a+b \cos (c+d x)}}{45045 b^3 d}-\frac{2 \left (a^2-b^2\right ) \left (-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+520 a^4 b B-240 a^5 C+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{45045 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (-10 a^4 b^2 (143 A+76 C)+3 a^2 b^4 (13299 A+10223 C)+3315 a^3 b^3 B+520 a^5 b B-240 a^6 C+48165 a b^5 B+1617 b^6 (13 A+11 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{45045 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{143 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d} \]

[Out]

(2*(520*a^5*b*B + 3315*a^3*b^3*B + 48165*a*b^5*B - 240*a^6*C + 1617*b^6*(13*A + 11*C) - 10*a^4*b^2*(143*A + 76
*C) + 3*a^2*b^4*(13299*A + 10223*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(45045*b^
4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(520*a^4*b*B + 3705*a^2*b^3*B + 8775*b^5*B - 240*a^5*
C - 10*a^3*b^2*(143*A + 94*C) + 6*a*b^4*(2717*A + 2174*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d
*x)/2, (2*b)/(a + b)])/(45045*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(520*a^4*b*B + 3705*a^2*b^3*B + 8775*b^5*B
- 240*a^5*C - 10*a^3*b^2*(143*A + 94*C) + 6*a*b^4*(2717*A + 2174*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4
5045*b^3*d) + (2*(520*a^3*b*B + 4355*a*b^3*B - 240*a^4*C + 539*b^4*(13*A + 11*C) - 10*a^2*b^2*(143*A + 124*C))
*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45045*b^3*d) + (2*(104*a^2*b*B + 1053*b^3*B - 48*a^3*C - 2*a*b^2*(1
43*A + 166*C))*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9009*b^3*d) + (2*(143*A*b^2 - 52*a*b*B + 24*a^2*C + 1
21*b^2*C)*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(1287*b^3*d) + (2*(13*b*B - 6*a*C)*Cos[c + d*x]*(a + b*Cos[
c + d*x])^(7/2)*Sin[c + d*x])/(143*b^2*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(13*b
*d)

________________________________________________________________________________________

Rubi [A]  time = 1.51451, antiderivative size = 629, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{1287 b^3 d}+\frac{2 \sin (c+d x) \left (104 a^2 b B-48 a^3 C-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{9009 b^3 d}+\frac{2 \sin (c+d x) \left (-10 a^2 b^2 (143 A+124 C)+520 a^3 b B-240 a^4 C+4355 a b^3 B+539 b^4 (13 A+11 C)\right ) (a+b \cos (c+d x))^{3/2}}{45045 b^3 d}+\frac{2 \sin (c+d x) \left (-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+520 a^4 b B-240 a^5 C+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt{a+b \cos (c+d x)}}{45045 b^3 d}-\frac{2 \left (a^2-b^2\right ) \left (-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+520 a^4 b B-240 a^5 C+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{45045 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (-10 a^4 b^2 (143 A+76 C)+3 a^2 b^4 (13299 A+10223 C)+3315 a^3 b^3 B+520 a^5 b B-240 a^6 C+48165 a b^5 B+1617 b^6 (13 A+11 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{45045 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{143 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]

[Out]

(2*(520*a^5*b*B + 3315*a^3*b^3*B + 48165*a*b^5*B - 240*a^6*C + 1617*b^6*(13*A + 11*C) - 10*a^4*b^2*(143*A + 76
*C) + 3*a^2*b^4*(13299*A + 10223*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(45045*b^
4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(520*a^4*b*B + 3705*a^2*b^3*B + 8775*b^5*B - 240*a^5*
C - 10*a^3*b^2*(143*A + 94*C) + 6*a*b^4*(2717*A + 2174*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d
*x)/2, (2*b)/(a + b)])/(45045*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(520*a^4*b*B + 3705*a^2*b^3*B + 8775*b^5*B
- 240*a^5*C - 10*a^3*b^2*(143*A + 94*C) + 6*a*b^4*(2717*A + 2174*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4
5045*b^3*d) + (2*(520*a^3*b*B + 4355*a*b^3*B - 240*a^4*C + 539*b^4*(13*A + 11*C) - 10*a^2*b^2*(143*A + 124*C))
*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45045*b^3*d) + (2*(104*a^2*b*B + 1053*b^3*B - 48*a^3*C - 2*a*b^2*(1
43*A + 166*C))*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9009*b^3*d) + (2*(143*A*b^2 - 52*a*b*B + 24*a^2*C + 1
21*b^2*C)*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(1287*b^3*d) + (2*(13*b*B - 6*a*C)*Cos[c + d*x]*(a + b*Cos[
c + d*x])^(7/2)*Sin[c + d*x])/(143*b^2*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(13*b
*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 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 2752

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

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

\begin{align*} \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac{2 \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (2 a C+\frac{1}{2} b (13 A+11 C) \cos (c+d x)+\frac{1}{2} (13 b B-6 a C) \cos ^2(c+d x)\right ) \, dx}{13 b}\\ &=\frac{2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac{2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac{4 \int (a+b \cos (c+d x))^{5/2} \left (\frac{1}{2} a (13 b B-6 a C)+\frac{1}{4} b (117 b B-10 a C) \cos (c+d x)+\frac{1}{4} \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx}{143 b^2}\\ &=\frac{2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac{2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac{2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac{8 \int (a+b \cos (c+d x))^{5/2} \left (\frac{1}{8} b \left (1001 A b^2-130 a b B+60 a^2 C+847 b^2 C\right )+\frac{1}{8} \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) \cos (c+d x)\right ) \, dx}{1287 b^3}\\ &=\frac{2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac{2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac{2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac{2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac{16 \int (a+b \cos (c+d x))^{3/2} \left (-\frac{3}{16} b \left (130 a^2 b B-1755 b^3 B-60 a^3 C-a b^2 (1859 A+1423 C)\right )+\frac{1}{16} \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) \cos (c+d x)\right ) \, dx}{9009 b^3}\\ &=\frac{2 \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}+\frac{2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac{2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac{2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac{2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac{32 \int \sqrt{a+b \cos (c+d x)} \left (-\frac{3}{32} b \left (130 a^3 b B-13130 a b^3 B-60 a^4 C-539 b^4 (13 A+11 C)-5 a^2 b^2 (1573 A+1175 C)\right )+\frac{3}{32} \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \cos (c+d x)\right ) \, dx}{45045 b^3}\\ &=\frac{2 \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{45045 b^3 d}+\frac{2 \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}+\frac{2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac{2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac{2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac{2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac{64 \int \frac{\frac{3}{64} b \left (130 a^4 b B+43095 a^2 b^3 B+8775 b^5 B-60 a^5 C+5 a^3 b^2 (4433 A+3337 C)+3 a b^4 (12441 A+10277 C)\right )+\frac{3}{64} \left (520 a^5 b B+3315 a^3 b^3 B+48165 a b^5 B-240 a^6 C+1617 b^6 (13 A+11 C)-10 a^4 b^2 (143 A+76 C)+3 a^2 b^4 (13299 A+10223 C)\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{135135 b^3}\\ &=\frac{2 \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{45045 b^3 d}+\frac{2 \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}+\frac{2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac{2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac{2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac{2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}-\frac{\left (\left (a^2-b^2\right ) \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right )\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{45045 b^4}+\frac{\left (520 a^5 b B+3315 a^3 b^3 B+48165 a b^5 B-240 a^6 C+1617 b^6 (13 A+11 C)-10 a^4 b^2 (143 A+76 C)+3 a^2 b^4 (13299 A+10223 C)\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{45045 b^4}\\ &=\frac{2 \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{45045 b^3 d}+\frac{2 \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}+\frac{2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac{2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac{2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac{2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac{\left (\left (520 a^5 b B+3315 a^3 b^3 B+48165 a b^5 B-240 a^6 C+1617 b^6 (13 A+11 C)-10 a^4 b^2 (143 A+76 C)+3 a^2 b^4 (13299 A+10223 C)\right ) \sqrt{a+b \cos (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \cos (c+d x)}{a+b}} \, dx}{45045 b^4 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left (\left (a^2-b^2\right ) \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \cos (c+d x)}{a+b}}} \, dx}{45045 b^4 \sqrt{a+b \cos (c+d x)}}\\ &=\frac{2 \left (520 a^5 b B+3315 a^3 b^3 B+48165 a b^5 B-240 a^6 C+1617 b^6 (13 A+11 C)-10 a^4 b^2 (143 A+76 C)+3 a^2 b^4 (13299 A+10223 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{45045 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 \left (a^2-b^2\right ) \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{45045 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{45045 b^3 d}+\frac{2 \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}+\frac{2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac{2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac{2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac{2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}\\ \end{align*}

Mathematica [A]  time = 3.85216, size = 501, normalized size = 0.8 \[ \frac{b (a+b \cos (c+d x)) \left (4 \sin (c+d x) \left (10 a^3 b^2 (572 A+331 C)+121290 a^2 b^3 B-2080 a^4 b B+960 a^5 C+3 a b^4 (71214 A+60793 C)+84825 b^5 B\right )+b \left (\sin (2 (c+d x)) \left (120 a^2 b^2 (1430 A+1457 C)+3120 a^3 b B-1440 a^4 C+321880 a b^3 B+77 b^4 (1976 A+1897 C)\right )+5 b \left (2 \sin (3 (c+d x)) \left (5876 a^2 b B+60 a^3 C+a b^2 (10868 A+13939 C)+6669 b^3 B\right )+7 b \left (4 \sin (4 (c+d x)) \left (159 a^2 C+299 a b B+143 A b^2+220 b^2 C\right )+9 b ((54 a C+26 b B) \sin (5 (c+d x))+11 b C \sin (6 (c+d x)))\right )\right )\right )\right )+32 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left (b^2 \left (5 a^3 b^2 (4433 A+3337 C)+43095 a^2 b^3 B+130 a^4 b B-60 a^5 C+3 a b^4 (12441 A+10277 C)+8775 b^5 B\right ) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-\left (10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-3315 a^3 b^3 B-520 a^5 b B+240 a^6 C-48165 a b^5 B-1617 b^6 (13 A+11 C)\right ) \left ((a+b) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-a F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )\right )}{720720 b^4 d \sqrt{a+b \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]

[Out]

(32*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(130*a^4*b*B + 43095*a^2*b^3*B + 8775*b^5*B - 60*a^5*C + 5*a^3*b^2
*(4433*A + 3337*C) + 3*a*b^4*(12441*A + 10277*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (-520*a^5*b*B - 3315
*a^3*b^3*B - 48165*a*b^5*B + 240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299
*A + 10223*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*
(a + b*Cos[c + d*x])*(4*(-2080*a^4*b*B + 121290*a^2*b^3*B + 84825*b^5*B + 960*a^5*C + 10*a^3*b^2*(572*A + 331*
C) + 3*a*b^4*(71214*A + 60793*C))*Sin[c + d*x] + b*((3120*a^3*b*B + 321880*a*b^3*B - 1440*a^4*C + 120*a^2*b^2*
(1430*A + 1457*C) + 77*b^4*(1976*A + 1897*C))*Sin[2*(c + d*x)] + 5*b*(2*(5876*a^2*b*B + 6669*b^3*B + 60*a^3*C
+ a*b^2*(10868*A + 13939*C))*Sin[3*(c + d*x)] + 7*b*(4*(143*A*b^2 + 299*a*b*B + 159*a^2*C + 220*b^2*C)*Sin[4*(
c + d*x)] + 9*b*((26*b*B + 54*a*C)*Sin[5*(c + d*x)] + 11*b*C*Sin[6*(c + d*x)]))))))/(720720*b^4*d*Sqrt[a + b*C
os[c + d*x]])

________________________________________________________________________________________

Maple [B]  time = 1.168, size = 3165, normalized size = 5. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)

[Out]

-2/45045*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-443520*C*b^7*cos(1/2*d*x+1/2*c)*sin(1/2
*d*x+1/2*c)^14-520*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*Elliptic
E(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^5*b^2+(262080*B*b^7+766080*C*a*b^6+1330560*C*b^7)*sin(1/2*d*x+1/2*c
)^12*cos(1/2*d*x+1/2*c)+(-160160*A*b^7-465920*B*a*b^6-655200*B*b^7-450240*C*a^2*b^5-1915200*C*a*b^6-1798720*C*
b^7)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(297440*A*a*b^6+320320*A*b^7+284960*B*a^2*b^5+931840*B*a*b^6+739
440*B*b^7+90240*C*a^3*b^4+900480*C*a^2*b^5+2159680*C*a*b^6+1379840*C*b^7)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2
*c)+(-194480*A*a^2*b^5-446160*A*a*b^6-296296*A*b^7-60320*B*a^3*b^4-427440*B*a^2*b^5-860080*B*a*b^6-453960*B*b^
7+120*C*a^4*b^3-135360*C*a^3*b^4-828880*C*a^2*b^5-1324320*C*a*b^6-666512*C*b^7)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d
*x+1/2*c)+(45760*A*a^3*b^4+194480*A*a^2*b^5+344344*A*a*b^6+136136*A*b^7-260*B*a^4*b^3+60320*B*a^3*b^4+326560*B
*a^2*b^5+394160*B*a*b^6+180180*B*b^7+120*C*a^5*b^2-120*C*a^4*b^3+101840*C*a^3*b^4+378640*C*a^2*b^5+522368*C*a*
b^6+198352*C*b^7)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-1430*A*a^4*b^3-22880*A*a^3*b^4-95238*A*a^2*b^5-978
12*A*a*b^6-24024*A*b^7+520*B*a^5*b^2+130*B*a^4*b^3-41730*B*a^3*b^4-92040*B*a^2*b^5-86970*B*a*b^6-36270*B*b^7-2
40*C*a^6*b-60*C*a^5*b^2-760*C*a^4*b^3-28360*C*a^3*b^4-104466*C*a^2*b^5-104304*C*a*b^6-27258*C*b^7)*sin(1/2*d*x
+1/2*c)^2*cos(1/2*d*x+1/2*c)-240*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^
(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^7+8775*b^7*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)
*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-17787*C*(sin(1/2*d*x
+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))
^(1/2))*b^7-21021*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE
(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^7+240*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^
2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^7-17732*A*a^3*(sin(1/2*d*x+1/2*c)^2)^(
1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^4+
240*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1
/2*c),(-2*b/(a-b))^(1/2))*a^6*b-520*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b
))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^6*b+21021*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-
b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^6-39897*A*(sin
(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*
b/(a-b))^(1/2))*a^2*b^5+48165*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/
2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^5+13044*C*a*b^6*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(
a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-5070*a^2*b^5*B*(
sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(
-2*b/(a-b))^(1/2))-3185*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*Ell
ipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b^3+39897*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2
*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b^4+3315*B*(sin(1/2*d*x+
1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^
(1/2))*a^4*b^3+17787*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*Ellipt
icE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^6-48165*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+
1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^6+700*C*(sin(1/2*d*x+1/2*c)^2
)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a
^5*b^2-3315*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1
/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b^4-760*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+
(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^5*b^2+520*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)
*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^6*b+30
669*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1
/2*c),(-2*b/(a-b))^(1/2))*a^3*b^4+1430*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(
a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b^3+760*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/
(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b^3+1430*A*
(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),
(-2*b/(a-b))^(1/2))*a^5*b^2-30669*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))
^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^5-13984*a^3*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b
/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^4-1430*A*(si
n(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2
*b/(a-b))^(1/2))*a^5*b^2+16302*a*A*b^6*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-
b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2)))/b^4/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1
/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*b+a+b)^(1/2)/d

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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 )}^{\frac{5}{2}} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/2)*cos(d*x + c)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

integral((C*b^2*cos(d*x + c)^6 + (2*C*a*b + B*b^2)*cos(d*x + c)^5 + A*a^2*cos(d*x + c)^2 + (C*a^2 + 2*B*a*b +
A*b^2)*cos(d*x + c)^4 + (B*a^2 + 2*A*a*b)*cos(d*x + c)^3)*sqrt(b*cos(d*x + c) + a), 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(cos(d*x+c)**2*(a+b*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**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(cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

Timed out