3.1524 \(\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)+C \cos ^2(c+d x))}{\sqrt{\sec (c+d x)}} \, dx\)

Optimal. Leaf size=894 \[ \frac{C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d \sqrt{\sec (c+d x)}}+\frac{(10 b B-3 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d \sqrt{\sec (c+d x)}}+\frac{\left (-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d \sqrt{\sec (c+d x)}}+\frac{\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d}+\frac{\left (-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{320 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left (45 C a^4-30 b (5 B+C) a^3-4 b^2 (660 A+295 B+423 C) a^2-8 b^3 (260 A+355 B+193 C) a-16 b^4 (80 A+45 B+64 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d \sqrt{\sec (c+d x)}} \]

[Out]

-((a - b)*Sqrt[a + b]*(150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 141*C
))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])]
, -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*a*b^2*d
*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(45*a^4*C - 30*a^3*b*(5*B + C) - 16*b^4*(80*A + 45*B + 64*C) - 8*a*b^3*(26
0*A + 355*B + 193*C) - 4*a^2*b^2*(660*A + 295*B + 423*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqr
t[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a +
b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(10*a^4*b*B - 240*a^2
*b^3*B - 96*b^5*B - 3*a^5*C - 40*a^3*b^2*(2*A + C) - 80*a*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*Ell
ipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt
[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(128*b^3*d*Sqrt[Sec[c + d*x]]) + ((50*a
^2*b*B + 120*b^3*B - 15*a^3*C + 4*a*b^2*(60*A + 43*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(320*b*d*Sqrt[Se
c[c + d*x]]) + ((80*A*b^2 + 50*a*b*B - 15*a^2*C + 64*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(240*b*d*
Sqrt[Sec[c + d*x]]) + ((10*b*B - 3*a*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(40*b*d*Sqrt[Sec[c + d*x]]) +
 (C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(5*b*d*Sqrt[Sec[c + d*x]]) + ((150*a^3*b*B + 2840*a*b^3*B - 45*a^
4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 141*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*
x])/(1920*b^2*d)

________________________________________________________________________________________

Rubi [A]  time = 4.05813, antiderivative size = 894, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.178, Rules used = {4221, 3049, 3061, 3053, 2809, 2998, 2816, 2994} \[ \frac{C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d \sqrt{\sec (c+d x)}}+\frac{(10 b B-3 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d \sqrt{\sec (c+d x)}}+\frac{\left (-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d \sqrt{\sec (c+d x)}}+\frac{\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d}+\frac{\left (-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{320 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left (45 C a^4-30 b (5 B+C) a^3-4 b^2 (660 A+295 B+423 C) a^2-8 b^3 (260 A+355 B+193 C) a-16 b^4 (80 A+45 B+64 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d \sqrt{\sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a - b)*Sqrt[a + b]*(150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 141*C
))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])]
, -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*a*b^2*d
*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(45*a^4*C - 30*a^3*b*(5*B + C) - 16*b^4*(80*A + 45*B + 64*C) - 8*a*b^3*(26
0*A + 355*B + 193*C) - 4*a^2*b^2*(660*A + 295*B + 423*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqr
t[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a +
b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(10*a^4*b*B - 240*a^2
*b^3*B - 96*b^5*B - 3*a^5*C - 40*a^3*b^2*(2*A + C) - 80*a*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*Ell
ipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt
[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(128*b^3*d*Sqrt[Sec[c + d*x]]) + ((50*a
^2*b*B + 120*b^3*B - 15*a^3*C + 4*a*b^2*(60*A + 43*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(320*b*d*Sqrt[Se
c[c + d*x]]) + ((80*A*b^2 + 50*a*b*B - 15*a^2*C + 64*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(240*b*d*
Sqrt[Sec[c + d*x]]) + ((10*b*B - 3*a*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(40*b*d*Sqrt[Sec[c + d*x]]) +
 (C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(5*b*d*Sqrt[Sec[c + d*x]]) + ((150*a^3*b*B + 2840*a*b^3*B - 45*a^
4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 141*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*
x])/(1920*b^2*d)

Rule 4221

Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Dist[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u,
 x]

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 3061

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

Rule 3053

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

Rule 2809

Int[Sqrt[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[(2*b*Tan
[e + f*x]*Rt[(c + d)/b, 2]*Sqrt[(c*(1 + Csc[e + f*x]))/(c - d)]*Sqrt[(c*(1 - Csc[e + f*x]))/(c + d)]*EllipticP
i[(c + d)/d, ArcSin[Sqrt[c + d*Sin[e + f*x]]/(Sqrt[b*Sin[e + f*x]]*Rt[(c + d)/b, 2])], -((c + d)/(c - d))])/(d
*f), x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/b]

Rule 2998

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*s
in[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[(A - B)/(a - b), Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e
+ f*x]]), x], x] - Dist[(A*b - a*B)/(a - b), Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin
[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2
 - d^2, 0] && NeQ[A, B]

Rule 2816

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-2*
Tan[e + f*x]*Rt[(a + b)/d, 2]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*Ellipt
icF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[d*Sin[e + f*x]]*Rt[(a + b)/d, 2])], -((a + b)/(a - b))])/(a*f), x] /
; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]

Rule 2994

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]]), x_Symbol] :> Simp[(-2*A*(c - d)*Tan[e + f*x]*Rt[(c + d)/b, 2]*Sqrt[(c*(1 + Csc[e + f*x]))/(c
- d)]*Sqrt[(c*(1 - Csc[e + f*x]))/(c + d)]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/(Sqrt[b*Sin[e + f*x]]*Rt[
(c + d)/b, 2])], -((c + d)/(c - d))])/(f*b*c^2), x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] &&
 EqQ[A, B] && PosQ[(c + d)/b]

Rubi steps

\begin{align*} \int \frac{(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (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{C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d \sqrt{\sec (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^{5/2} \left (\frac{a C}{2}+b (5 A+4 C) \cos (c+d x)+\frac{1}{2} (10 b B-3 a C) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{5 b}\\ &=\frac{(10 b B-3 a C) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d \sqrt{\sec (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^{3/2} \left (\frac{5}{4} a (2 b B+a C)+\frac{1}{2} b (40 a A+30 b B+27 a C) \cos (c+d x)+\frac{1}{4} \left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{20 b}\\ &=\frac{\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt{\sec (c+d x)}}+\frac{(10 b B-3 a C) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d \sqrt{\sec (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+b \cos (c+d x)} \left (\frac{1}{8} a \left (80 A b^2+110 a b B+15 a^2 C+64 b^2 C\right )+\frac{1}{4} b \left (310 a b B+32 b^2 (5 A+4 C)+3 a^2 (80 A+49 C)\right ) \cos (c+d x)+\frac{3}{8} \left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{60 b}\\ &=\frac{\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d \sqrt{\sec (c+d x)}}+\frac{\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt{\sec (c+d x)}}+\frac{(10 b B-3 a C) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d \sqrt{\sec (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{16} a \left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right )+\frac{1}{8} b \left (1610 a^2 b B+360 b^3 B+4 a b^2 (380 A+289 C)+a^3 (960 A+573 C)\right ) \cos (c+d x)+\frac{1}{16} \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx}{120 b}\\ &=\frac{\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d \sqrt{\sec (c+d x)}}+\frac{\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt{\sec (c+d x)}}+\frac{(10 b B-3 a C) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d \sqrt{\sec (c+d x)}}+\frac{\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{1920 b^2 d}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{1}{16} a \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right )+\frac{1}{8} a b \left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right ) \cos (c+d x)-\frac{15}{16} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 (2 A+C)-80 a b^4 (4 A+3 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{240 b^2}\\ &=\frac{\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d \sqrt{\sec (c+d x)}}+\frac{\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt{\sec (c+d x)}}+\frac{(10 b B-3 a C) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d \sqrt{\sec (c+d x)}}+\frac{\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{1920 b^2 d}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{1}{16} a \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right )+\frac{1}{8} a b \left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right ) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{240 b^2}-\frac{\left (\left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 (2 A+C)-80 a b^4 (4 A+3 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx}{256 b^2}\\ &=\frac{\sqrt{a+b} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 (2 A+C)-80 a b^4 (4 A+3 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (1+\sec (c+d x))}{a-b}}}{128 b^3 d \sqrt{\sec (c+d x)}}+\frac{\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d \sqrt{\sec (c+d x)}}+\frac{\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt{\sec (c+d x)}}+\frac{(10 b B-3 a C) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d \sqrt{\sec (c+d x)}}+\frac{\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{1920 b^2 d}-\frac{\left (a \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{3840 b^2}-\frac{\left (a \left (45 a^4 C-30 a^3 b (5 B+C)-16 b^4 (80 A+45 B+64 C)-8 a b^3 (260 A+355 B+193 C)-4 a^2 b^2 (660 A+295 B+423 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx}{3840 b^2}\\ &=-\frac{(a-b) \sqrt{a+b} \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (1+\sec (c+d x))}{a-b}}}{1920 a b^2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left (45 a^4 C-30 a^3 b (5 B+C)-16 b^4 (80 A+45 B+64 C)-8 a b^3 (260 A+355 B+193 C)-4 a^2 b^2 (660 A+295 B+423 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (1+\sec (c+d x))}{a-b}}}{1920 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 (2 A+C)-80 a b^4 (4 A+3 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (1+\sec (c+d x))}{a-b}}}{128 b^3 d \sqrt{\sec (c+d x)}}+\frac{\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d \sqrt{\sec (c+d x)}}+\frac{\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt{\sec (c+d x)}}+\frac{(10 b B-3 a C) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d \sqrt{\sec (c+d x)}}+\frac{\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{1920 b^2 d}\\ \end{align*}

Mathematica [C]  time = 20.6159, size = 667, normalized size = 0.75 \[ \frac{\sqrt{a+b \cos (c+d x)} \left (\frac{2 \sin (c+d x) \left (590 a^2 b B+15 a^3 C+16 a b^2 (65 A+64 C)+480 b^3 B\right )}{b}+2 \tan (c+d x) \left (93 a^2 C+170 a b B+80 A b^2+88 b^2 C\right )+\frac{\tan \left (\frac{1}{2} (c+d x)\right ) \left (12 a^2 b^2 (220 A+141 C)+150 a^3 b B-45 a^4 C+2840 a b^3 B+256 b^4 (5 A+4 C)\right )}{b^2}+2 \sin (3 (c+d x)) \sec (c+d x) \left (93 a^2 C+170 a b B+80 A b^2+100 b^2 C\right )-\frac{i \sqrt{\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+b \cos (c+d x))}{a+b}} \left (-2 (a-b) \left (-4 a^2 b^2 (180 A+185 B+129 C)-30 a^3 b (5 B-C)+45 a^4 C-8 a b^3 (220 A+45 B+161 C)-720 b^4 B\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{a-b}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{a+b}{a-b}\right )+(a-b) \left (-12 a^2 b^2 (220 A+141 C)-150 a^3 b B+45 a^4 C-2840 a b^3 B-256 b^4 (5 A+4 C)\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{a-b}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{a+b}{a-b}\right )+30 \left (40 a^3 b^2 (2 A+C)+240 a^2 b^3 B-10 a^4 b B+3 a^5 C+80 a b^4 (4 A+3 C)+96 b^5 B\right ) \Pi \left (\frac{a+b}{a-b};i \sinh ^{-1}\left (\sqrt{\frac{a-b}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{a+b}{a-b}\right )\right )}{b^2 \sqrt{\frac{a-b}{a+b}} \sqrt{\cos (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right )} (a+b \cos (c+d x))}+6 b (21 a C+10 b B) \sin (4 (c+d x)) \sec (c+d x)+24 b^2 C \sin (5 (c+d x)) \sec (c+d x)\right )}{1920 d \sqrt{\sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*Cos[c + d*x]]*(((-I)*((a - b)*(-150*a^3*b*B - 2840*a*b^3*B + 45*a^4*C - 256*b^4*(5*A + 4*C) - 12*a
^2*b^2*(220*A + 141*C))*EllipticE[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] - 2*(
a - b)*(-720*b^4*B - 30*a^3*b*(5*B - C) + 45*a^4*C - 4*a^2*b^2*(180*A + 185*B + 129*C) - 8*a*b^3*(220*A + 45*B
 + 161*C))*EllipticF[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] + 30*(-10*a^4*b*B
+ 240*a^2*b^3*B + 96*b^5*B + 3*a^5*C + 40*a^3*b^2*(2*A + C) + 80*a*b^4*(4*A + 3*C))*EllipticPi[(a + b)/(a - b)
, I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))])*Sqrt[((a + b*Cos[c + d*x])*Sec[(c +
d*x)/2]^2)/(a + b)])/(b^2*Sqrt[(a - b)/(a + b)]*(a + b*Cos[c + d*x])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]) +
(2*(590*a^2*b*B + 480*b^3*B + 15*a^3*C + 16*a*b^2*(65*A + 64*C))*Sin[c + d*x])/b + 2*(80*A*b^2 + 170*a*b*B + 9
3*a^2*C + 100*b^2*C)*Sec[c + d*x]*Sin[3*(c + d*x)] + 6*b*(10*b*B + 21*a*C)*Sec[c + d*x]*Sin[4*(c + d*x)] + 24*
b^2*C*Sec[c + d*x]*Sin[5*(c + d*x)] + ((150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b
^2*(220*A + 141*C))*Tan[(c + d*x)/2])/b^2 + 2*(80*A*b^2 + 170*a*b*B + 93*a^2*C + 88*b^2*C)*Tan[c + d*x]))/(192
0*d*Sqrt[Sec[c + d*x]])

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Maple [B]  time = 0.753, size = 7064, normalized size = 7.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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}}}{\sqrt{\sec \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))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(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)^(5/2)/sqrt(sec(d*x + c)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C b^{2} \cos \left (d x + c\right )^{4} +{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} + A a^{2} +{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )\right )} \sqrt{b \cos \left (d x + c\right ) + a}}{\sqrt{\sec \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))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

integral((C*b^2*cos(d*x + c)^4 + (2*C*a*b + B*b^2)*cos(d*x + c)^3 + A*a^2 + (C*a^2 + 2*B*a*b + A*b^2)*cos(d*x
+ c)^2 + (B*a^2 + 2*A*a*b)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)/sqrt(sec(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))**(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/sec(d*x+c)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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}}}{\sqrt{\sec \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))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x, algorithm="giac")

[Out]

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