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

Optimal. Leaf size=443 \[ \frac{2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{231 b^3 d}-\frac{4 a \left (8 a^2 C+33 A b^2+34 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{1155 b^3 d}-\frac{2 \left (6 a^2 b^2 (11 A+8 C)+16 a^4 C-25 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1155 b^3 d}+\frac{2 \left (a^2-b^2\right ) \left (6 a^2 b^2 (11 A+8 C)+16 a^4 C-25 b^4 (11 A+9 C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{1155 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left (3 a^2 b^2 (11 A+6 C)+8 a^4 C-b^4 (451 A+348 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{1155 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{33 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d} \]

[Out]

(-4*a*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2,
(2*b)/(a + b)])/(1155*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(a^2 - b^2)*(16*a^4*C + 6*a^2*b^2*(11*A
+ 8*C) - 25*b^4*(11*A + 9*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(1155*
b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(16*a^4*C + 6*a^2*b^2*(11*A + 8*C) - 25*b^4*(11*A + 9*C))*Sqrt[a + b*Cos[
c + d*x]]*Sin[c + d*x])/(1155*b^3*d) - (4*a*(33*A*b^2 + 8*a^2*C + 34*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c +
d*x])/(1155*b^3*d) + (2*(8*a^2*C + 3*b^2*(11*A + 9*C))*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(231*b^3*d) -
(4*a*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(33*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 = 1.06326, antiderivative size = 443, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257, Rules used = {3050, 3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{231 b^3 d}-\frac{4 a \left (8 a^2 C+33 A b^2+34 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{1155 b^3 d}-\frac{2 \left (6 a^2 b^2 (11 A+8 C)+16 a^4 C-25 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1155 b^3 d}+\frac{2 \left (a^2-b^2\right ) \left (6 a^2 b^2 (11 A+8 C)+16 a^4 C-25 b^4 (11 A+9 C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{1155 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left (3 a^2 b^2 (11 A+6 C)+8 a^4 C-b^4 (451 A+348 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{1155 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{33 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2,
(2*b)/(a + b)])/(1155*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(a^2 - b^2)*(16*a^4*C + 6*a^2*b^2*(11*A
+ 8*C) - 25*b^4*(11*A + 9*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(1155*
b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(16*a^4*C + 6*a^2*b^2*(11*A + 8*C) - 25*b^4*(11*A + 9*C))*Sqrt[a + b*Cos[
c + d*x]]*Sin[c + d*x])/(1155*b^3*d) - (4*a*(33*A*b^2 + 8*a^2*C + 34*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c +
d*x])/(1155*b^3*d) + (2*(8*a^2*C + 3*b^2*(11*A + 9*C))*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(231*b^3*d) -
(4*a*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(33*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 3050

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (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)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*
x] + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, 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 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+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)-3 a C \cos ^2(c+d x)\right ) \, dx}{11 b}\\ &=-\frac{4 a C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{33 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 (-3 a^2 C-\frac{3}{2} a b C \cos (c+d x)+\frac{3}{4} \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \cos ^2(c+d x)\right ) \, dx}{99 b^2}\\ &=\frac{2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{231 b^3 d}-\frac{4 a C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{33 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{9}{8} b \left (55 A b^2+4 a^2 C+45 b^2 C\right )-\frac{3}{4} a \left (33 A b^2+8 a^2 C+34 b^2 C\right ) \cos (c+d x)\right ) \, dx}{693 b^3}\\ &=-\frac{4 a \left (33 A b^2+8 a^2 C+34 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{1155 b^3 d}+\frac{2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{231 b^3 d}-\frac{4 a C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{33 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{9}{16} a b \left (209 A b^2+4 a^2 C+157 b^2 C\right )-\frac{9}{16} \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \cos (c+d x)\right ) \, dx}{3465 b^3}\\ &=-\frac{2 \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{1155 b^3 d}-\frac{4 a \left (33 A b^2+8 a^2 C+34 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{1155 b^3 d}+\frac{2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{231 b^3 d}-\frac{4 a C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{33 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{9}{32} b \left (4 a^4 C-25 b^4 (11 A+9 C)-3 a^2 b^2 (187 A+141 C)\right )-\frac{9}{16} a \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-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 (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{1155 b^3 d}-\frac{4 a \left (33 A b^2+8 a^2 C+34 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{1155 b^3 d}+\frac{2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{231 b^3 d}-\frac{4 a C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{33 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 (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right )\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{1155 b^4}-\frac{\left (2 a \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right )\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{1155 b^4}\\ &=-\frac{2 \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{1155 b^3 d}-\frac{4 a \left (33 A b^2+8 a^2 C+34 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{1155 b^3 d}+\frac{2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{231 b^3 d}-\frac{4 a C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{33 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 (2 a \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-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}{1155 b^4 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left (\left (a^2-b^2\right ) \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 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}{1155 b^4 \sqrt{a+b \cos (c+d x)}}\\ &=-\frac{4 a \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-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 )}{1155 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \left (a^2-b^2\right ) \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 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 )}{1155 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{1155 b^3 d}-\frac{4 a \left (33 A b^2+8 a^2 C+34 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{1155 b^3 d}+\frac{2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{231 b^3 d}-\frac{4 a C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{33 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 = 1.74965, size = 331, normalized size = 0.75 \[ \frac{b (a+b \cos (c+d x)) \left (2 \left (6 a^2 b^2 (44 A+27 C)+64 a^4 C+5 b^4 (506 A+435 C)\right ) \sin (c+d x)+b \left (16 a \left (-3 a^2 C+132 A b^2+136 b^2 C\right ) \sin (2 (c+d x))+5 b \left (\left (4 a^2 C+132 A b^2+171 b^2 C\right ) \sin (3 (c+d x))+7 b C (8 a \sin (4 (c+d x))+3 b \sin (5 (c+d x)))\right )\right )\right )+16 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left (b \left (3 a^2 b^3 (187 A+141 C)-4 a^4 b C+25 b^5 (11 A+9 C)\right ) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-2 a \left (3 a^2 b^2 (11 A+6 C)+8 a^4 C-b^4 (451 A+348 C)\right ) \left ((a+b) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-a F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )\right )}{9240 b^4 d \sqrt{a+b \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b*(-4*a^4*b*C + 25*b^5*(11*A + 9*C) + 3*a^2*b^3*(187*A + 141*C))*Ellip
ticF[(c + d*x)/2, (2*b)/(a + b)] - 2*a*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*((a + b)*Ellip
ticE[(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*(64*a
^4*C + 6*a^2*b^2*(44*A + 27*C) + 5*b^4*(506*A + 435*C))*Sin[c + d*x] + b*(16*a*(132*A*b^2 - 3*a^2*C + 136*b^2*
C)*Sin[2*(c + d*x)] + 5*b*((132*A*b^2 + 4*a^2*C + 171*b^2*C)*Sin[3*(c + d*x)] + 7*b*C*(8*a*Sin[4*(c + d*x)] +
3*b*Sin[5*(c + d*x)])))))/(9240*b^4*d*Sqrt[a + b*Cos[c + d*x]])

________________________________________________________________________________________

Maple [B]  time = 0.428, size = 1791, normalized size = 4. \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+C*cos(d*x+c)^2),x)

[Out]

-2/1155*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(6720*C*b^6*cos(1/2*d*x+1/2*c)*sin(1/2*d*x
+1/2*c)^12+(-7840*C*a*b^5-16800*C*b^6)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(2640*A*b^6+2320*C*a^2*b^4+156
80*C*a*b^5+18960*C*b^6)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-3432*A*a*b^5-3960*A*b^6+8*C*a^3*b^3-3480*C*a
^2*b^4-14456*C*a*b^5-11640*C*b^6)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(1188*A*a^2*b^4+3432*A*a*b^5+3080*A*
b^6+8*C*a^4*b^2-8*C*a^3*b^3+2624*C*a^2*b^4+6616*C*a*b^5+4620*C*b^6)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-
66*A*a^3*b^3-594*A*a^2*b^4-1408*A*a*b^5-880*A*b^6-16*C*a^5*b-4*C*a^4*b^2-36*C*a^3*b^3-732*C*a^2*b^4-1614*C*a*b
^5-930*C*b^6)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-66*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+66*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^3+902*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^4-902*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+66*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^2-3
41*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+275*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))-16*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*s
in(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+16*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-36*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(c
os(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b^2+36*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+696*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
^4-696*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+16*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^6+32*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^2-273*a^2*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)*EllipticF(cos(1/2*d*x+1/2*c),(-2
*b/(a-b))^(1/2))*b^4+225*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)))/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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + 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+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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

[Out]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + 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+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

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