3.1021 \(\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=518 \[ \frac{2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{693 b^3 d}+\frac{2 \sin (c+d x) \left (88 a^2 b B-48 a^3 C-6 a b^2 (33 A+34 C)+539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{3465 b^3 d}+\frac{2 \sin (c+d x) \left (-18 a^2 b^2 (11 A+8 C)+88 a^3 b B-48 a^4 C+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt{a+b \cos (c+d x)}}{3465 b^3 d}-\frac{2 \left (a^2-b^2\right ) \left (-18 a^2 b^2 (11 A+8 C)+88 a^3 b B-48 a^4 C+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3465 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+88 a^4 b B-48 a^5 C+6 a b^4 (451 A+348 C)+1617 b^5 B\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3465 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{99 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d} \]

[Out]

(2*(88*a^4*b*B + 363*a^2*b^3*B + 1617*b^5*B - 48*a^5*C - 18*a^3*b^2*(11*A + 6*C) + 6*a*b^4*(451*A + 348*C))*Sq
rt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3465*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])
- (2*(a^2 - b^2)*(88*a^3*b*B + 429*a*b^3*B - 48*a^4*C - 18*a^2*b^2*(11*A + 8*C) + 75*b^4*(11*A + 9*C))*Sqrt[(a
+ b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3465*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*
(88*a^3*b*B + 429*a*b^3*B - 48*a^4*C - 18*a^2*b^2*(11*A + 8*C) + 75*b^4*(11*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]
*Sin[c + d*x])/(3465*b^3*d) + (2*(88*a^2*b*B + 539*b^3*B - 48*a^3*C - 6*a*b^2*(33*A + 34*C))*(a + b*Cos[c + d*
x])^(3/2)*Sin[c + d*x])/(3465*b^3*d) + (2*(99*A*b^2 - 44*a*b*B + 24*a^2*C + 81*b^2*C)*(a + b*Cos[c + d*x])^(5/
2)*Sin[c + d*x])/(693*b^3*d) + (2*(11*b*B - 6*a*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(99*b
^2*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*b*d)

________________________________________________________________________________________

Rubi [A]  time = 1.28733, antiderivative size = 518, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{693 b^3 d}+\frac{2 \sin (c+d x) \left (88 a^2 b B-48 a^3 C-6 a b^2 (33 A+34 C)+539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{3465 b^3 d}+\frac{2 \sin (c+d x) \left (-18 a^2 b^2 (11 A+8 C)+88 a^3 b B-48 a^4 C+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt{a+b \cos (c+d x)}}{3465 b^3 d}-\frac{2 \left (a^2-b^2\right ) \left (-18 a^2 b^2 (11 A+8 C)+88 a^3 b B-48 a^4 C+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3465 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+88 a^4 b B-48 a^5 C+6 a b^4 (451 A+348 C)+1617 b^5 B\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3465 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{99 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(88*a^4*b*B + 363*a^2*b^3*B + 1617*b^5*B - 48*a^5*C - 18*a^3*b^2*(11*A + 6*C) + 6*a*b^4*(451*A + 348*C))*Sq
rt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3465*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])
- (2*(a^2 - b^2)*(88*a^3*b*B + 429*a*b^3*B - 48*a^4*C - 18*a^2*b^2*(11*A + 8*C) + 75*b^4*(11*A + 9*C))*Sqrt[(a
+ b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3465*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*
(88*a^3*b*B + 429*a*b^3*B - 48*a^4*C - 18*a^2*b^2*(11*A + 8*C) + 75*b^4*(11*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]
*Sin[c + d*x])/(3465*b^3*d) + (2*(88*a^2*b*B + 539*b^3*B - 48*a^3*C - 6*a*b^2*(33*A + 34*C))*(a + b*Cos[c + d*
x])^(3/2)*Sin[c + d*x])/(3465*b^3*d) + (2*(99*A*b^2 - 44*a*b*B + 24*a^2*C + 81*b^2*C)*(a + b*Cos[c + d*x])^(5/
2)*Sin[c + d*x])/(693*b^3*d) + (2*(11*b*B - 6*a*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(99*b
^2*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*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))^{3/2} \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))^{5/2} \sin (c+d x)}{11 b d}+\frac{2 \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (2 a C+\frac{1}{2} b (11 A+9 C) \cos (c+d x)+\frac{1}{2} (11 b B-6 a C) \cos ^2(c+d x)\right ) \, dx}{11 b}\\ &=\frac{2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac{2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}+\frac{4 \int (a+b \cos (c+d x))^{3/2} \left (\frac{1}{2} a (11 b B-6 a C)+\frac{1}{4} b (77 b B-6 a C) \cos (c+d x)+\frac{1}{4} \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx}{99 b^2}\\ &=\frac{2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac{2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac{2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}+\frac{8 \int (a+b \cos (c+d x))^{3/2} \left (\frac{3}{8} b \left (165 A b^2-22 a b B+12 a^2 C+135 b^2 C\right )+\frac{1}{8} \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) \cos (c+d x)\right ) \, dx}{693 b^3}\\ &=\frac{2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^3 d}+\frac{2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac{2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac{2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}+\frac{16 \int \sqrt{a+b \cos (c+d x)} \left (-\frac{3}{16} b \left (22 a^2 b B-539 b^3 B-12 a^3 C-3 a b^2 (209 A+157 C)\right )+\frac{3}{16} \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \cos (c+d x)\right ) \, dx}{3465 b^3}\\ &=\frac{2 \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3465 b^3 d}+\frac{2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^3 d}+\frac{2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac{2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac{2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}+\frac{32 \int \frac{\frac{3}{32} b \left (22 a^3 b B+2046 a b^3 B-12 a^4 C+75 b^4 (11 A+9 C)+9 a^2 b^2 (187 A+141 C)\right )+\frac{3}{32} \left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-18 a^3 b^2 (11 A+6 C)+6 a b^4 (451 A+348 C)\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{10395 b^3}\\ &=\frac{2 \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3465 b^3 d}+\frac{2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^3 d}+\frac{2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac{2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac{2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}-\frac{\left (\left (a^2-b^2\right ) \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right )\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{3465 b^4}+\frac{\left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-18 a^3 b^2 (11 A+6 C)+6 a b^4 (451 A+348 C)\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{3465 b^4}\\ &=\frac{2 \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3465 b^3 d}+\frac{2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^3 d}+\frac{2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac{2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac{2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}+\frac{\left (\left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-18 a^3 b^2 (11 A+6 C)+6 a b^4 (451 A+348 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}{3465 b^4 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left (\left (a^2-b^2\right ) \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 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}{3465 b^4 \sqrt{a+b \cos (c+d x)}}\\ &=\frac{2 \left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-18 a^3 b^2 (11 A+6 C)+6 a b^4 (451 A+348 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3465 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 \left (a^2-b^2\right ) \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 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 )}{3465 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3465 b^3 d}+\frac{2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^3 d}+\frac{2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac{2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac{2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}\\ \end{align*}

Mathematica [A]  time = 2.76781, size = 407, normalized size = 0.79 \[ \frac{b (a+b \cos (c+d x)) \left (2 \sin (c+d x) \left (18 a^2 b^2 (44 A+27 C)-352 a^3 b B+192 a^4 C+8844 a b^3 B+15 b^4 (506 A+435 C)\right )+b \left (4 \sin (2 (c+d x)) \left (66 a^2 b B-36 a^3 C+48 a b^2 (33 A+34 C)+1463 b^3 B\right )+5 b \left (\sin (3 (c+d x)) \left (12 a^2 C+440 a b B+396 A b^2+513 b^2 C\right )+7 b ((24 a C+22 b B) \sin (4 (c+d x))+9 b C \sin (5 (c+d x)))\right )\right )\right )+16 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left (b^2 \left (9 a^2 b^2 (187 A+141 C)+22 a^3 b B-12 a^4 C+2046 a b^3 B+75 b^4 (11 A+9 C)\right ) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-\left (18 a^3 b^2 (11 A+6 C)-363 a^2 b^3 B-88 a^4 b B+48 a^5 C-6 a b^4 (451 A+348 C)-1617 b^5 B\right ) \left ((a+b) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-a F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )\right )}{27720 b^4 d \sqrt{a+b \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(22*a^3*b*B + 2046*a*b^3*B - 12*a^4*C + 75*b^4*(11*A + 9*C) + 9*a^
2*b^2*(187*A + 141*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*
a^5*C + 18*a^3*b^2*(11*A + 6*C) - 6*a*b^4*(451*A + 348*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])*(2*(-352*a^3*b*B + 8844*a*b^3*B + 192*a^4*C +
18*a^2*b^2*(44*A + 27*C) + 15*b^4*(506*A + 435*C))*Sin[c + d*x] + b*(4*(66*a^2*b*B + 1463*b^3*B - 36*a^3*C +
48*a*b^2*(33*A + 34*C))*Sin[2*(c + d*x)] + 5*b*((396*A*b^2 + 440*a*b*B + 12*a^2*C + 513*b^2*C)*Sin[3*(c + d*x)
] + 7*b*((22*b*B + 24*a*C)*Sin[4*(c + d*x)] + 9*b*C*Sin[5*(c + d*x)])))))/(27720*b^4*d*Sqrt[a + b*Cos[c + d*x]
])

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Maple [B]  time = 1.13, size = 2603, normalized size = 5. \begin{align*} \text{result 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))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)

[Out]

-2/3465*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(108*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^3+2088*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^4-2088*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*b^5+96*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^4*b^2-819*a^2*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+198*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)*Ellip
ticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b^3+2706*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^4-2706*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^5+198*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^4*b^2-1023*A*a^2*(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+48*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-
108*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^2+20160*C*b^6*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12-198*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^2-1617*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(c
os(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^6+675*b^6*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))+825*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))-48*C*(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^6+48*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)*Ell
ipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^6-363*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^4+(7920*A*b^6+14960*B*a*b^5+
24640*B*b^6+6960*C*a^2*b^4+47040*C*a*b^5+56880*C*b^6)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-10296*A*a*b^5-
11880*A*b^6-4664*B*a^2*b^4-22440*B*a*b^5-22792*B*b^6+24*C*a^3*b^3-10440*C*a^2*b^4-43368*C*a*b^5-34920*C*b^6)*s
in(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(3564*A*a^2*b^4+10296*A*a*b^5+9240*A*b^6-44*B*a^3*b^3+4664*B*a^2*b^4+17
248*B*a*b^5+10472*B*b^6+24*C*a^4*b^2-24*C*a^3*b^3+7872*C*a^2*b^4+19848*C*a*b^5+13860*C*b^6)*sin(1/2*d*x+1/2*c)
^4*cos(1/2*d*x+1/2*c)+(-198*A*a^3*b^3-1782*A*a^2*b^4-4224*A*a*b^5-2640*A*b^6+88*B*a^4*b^2+22*B*a^3*b^3-3102*B*
a^2*b^4-4884*B*a*b^5-1848*B*b^6-48*C*a^5*b-12*C*a^4*b^2-108*C*a^3*b^3-2196*C*a^2*b^4-4842*C*a*b^5-2790*C*b^6)*
sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+(-12320*B*b^6-23520*C*a*b^5-50400*C*b^6)*sin(1/2*d*x+1/2*c)^10*cos(1/2
*d*x+1/2*c)-88*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(co
s(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b^2+88*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^5*b+1617*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^5-8
8*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^5*b-341*a^3*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))*b^3+429*a*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))+363*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^3)/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*s
in(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{3}{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))^(3/2)*(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)^(3/2)*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 \cos \left (d x + c\right )^{5} +{\left (C a + B b\right )} \cos \left (d x + c\right )^{4} + A a \cos \left (d x + c\right )^{2} +{\left (B 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))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [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{3}{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))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

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