3.1509 \(\int \frac{\sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)+C \cos ^2(c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=766 \[ -\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left (24 a^2 b B-15 a^3 C-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt{a+b \cos (c+d x)}}{192 b^3 d}+\frac{\sin (c+d x) \left (5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right ) \sqrt{a+b \cos (c+d x)}}{32 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (-2 a^2 b (12 B+5 C)+15 a^3 C+4 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 C)\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{192 b^3 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (24 a^2 b B-15 a^3 C-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{192 a b^3 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (-8 a^2 b^2 (2 A+C)+8 a^3 b B-5 a^4 C+32 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{64 b^4 d \sqrt{\sec (c+d x)}}+\frac{(8 b B-5 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d \sec ^{\frac{3}{2}}(c+d x)} \]

[Out]

((a - b)*Sqrt[a + b]*(24*a^2*b*B - 128*b^3*B - 15*a^3*C - 4*a*b^2*(12*A + 7*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)])/(192*a*b^3*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b
]*(15*a^3*C - 2*a^2*b*(12*B + 5*C) + 4*a*b^2*(12*A + 4*B + 7*C) + 8*b^3*(12*A + 16*B + 9*C))*Sqrt[Cos[c + d*x]
]*Csc[c + d*x]*EllipticF[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)])/(192*b^3*d*Sqrt[Sec[c + d*x]]) -
(Sqrt[a + b]*(8*a^3*b*B + 32*a*b^3*B - 5*a^4*C - 8*a^2*b^2*(2*A + C) + 16*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*
Csc[c + d*x]*EllipticPi[(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)])/(64*b^4*d*Sqrt[Sec[c +
d*x]]) + (C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*b*d*Sec[c + d*x]^(3/2)) + ((16*A*b^2 - 8*a*b*B + 5*a^2
*C + 12*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b^2*d*Sqrt[Sec[c + d*x]]) + ((8*b*B - 5*a*C)*(a + b*
Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*b^2*d*Sqrt[Sec[c + d*x]]) - ((24*a^2*b*B - 128*b^3*B - 15*a^3*C - 4*a*b^
2*(12*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*b^3*d)

________________________________________________________________________________________

Rubi [A]  time = 2.65667, antiderivative size = 766, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ -\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left (24 a^2 b B-15 a^3 C-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt{a+b \cos (c+d x)}}{192 b^3 d}+\frac{\sin (c+d x) \left (5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right ) \sqrt{a+b \cos (c+d x)}}{32 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (-2 a^2 b (12 B+5 C)+15 a^3 C+4 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 C)\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{192 b^3 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (24 a^2 b B-15 a^3 C-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{192 a b^3 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (-8 a^2 b^2 (2 A+C)+8 a^3 b B-5 a^4 C+32 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{64 b^4 d \sqrt{\sec (c+d x)}}+\frac{(8 b B-5 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d \sec ^{\frac{3}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a - b)*Sqrt[a + b]*(24*a^2*b*B - 128*b^3*B - 15*a^3*C - 4*a*b^2*(12*A + 7*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)])/(192*a*b^3*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b
]*(15*a^3*C - 2*a^2*b*(12*B + 5*C) + 4*a*b^2*(12*A + 4*B + 7*C) + 8*b^3*(12*A + 16*B + 9*C))*Sqrt[Cos[c + d*x]
]*Csc[c + d*x]*EllipticF[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)])/(192*b^3*d*Sqrt[Sec[c + d*x]]) -
(Sqrt[a + b]*(8*a^3*b*B + 32*a*b^3*B - 5*a^4*C - 8*a^2*b^2*(2*A + C) + 16*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*
Csc[c + d*x]*EllipticPi[(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)])/(64*b^4*d*Sqrt[Sec[c +
d*x]]) + (C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*b*d*Sec[c + d*x]^(3/2)) + ((16*A*b^2 - 8*a*b*B + 5*a^2
*C + 12*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b^2*d*Sqrt[Sec[c + d*x]]) + ((8*b*B - 5*a*C)*(a + b*
Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*b^2*d*Sqrt[Sec[c + d*x]]) - ((24*a^2*b*B - 128*b^3*B - 15*a^3*C - 4*a*b^
2*(12*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*b^3*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{\sqrt{a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac{3}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 b d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left (\frac{3 a C}{2}+b (4 A+3 C) \cos (c+d x)+\frac{1}{2} (8 b B-5 a C) \cos ^2(c+d x)\right ) \, dx}{4 b}\\ &=\frac{C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 b d \sec ^{\frac{3}{2}}(c+d x)}+\frac{(8 b B-5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b^2 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}{4} a (8 b B-5 a C)+\frac{1}{2} b (16 b B-a C) \cos (c+d x)+\frac{3}{4} \left (16 A b^2-8 a b B+5 a^2 C+12 b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{12 b^2}\\ &=\frac{C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 b d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (16 A b^2-8 a b B+5 a^2 C+12 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 b^2 d \sqrt{\sec (c+d x)}}+\frac{(8 b B-5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b^2 d \sqrt{\sec (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{8} a \left (48 A b^2+8 a b B-5 a^2 C+36 b^2 C\right )+\frac{1}{4} b \left (48 A b^2+56 a b B+a^2 C+36 b^2 C\right ) \cos (c+d x)-\frac{1}{8} \left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx}{24 b^2}\\ &=\frac{C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 b d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (16 A b^2-8 a b B+5 a^2 C+12 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 b^2 d \sqrt{\sec (c+d x)}}+\frac{(8 b B-5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b^2 d \sqrt{\sec (c+d x)}}-\frac{\left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{192 b^3 d}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{8} a \left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right )+\frac{1}{4} a b \left (48 A b^2+8 a b B-5 a^2 C+36 b^2 C\right ) \cos (c+d x)+\frac{3}{8} \left (8 a^3 b B+32 a b^3 B-5 a^4 C-8 a^2 b^2 (2 A+C)+16 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}{48 b^3}\\ &=\frac{C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 b d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (16 A b^2-8 a b B+5 a^2 C+12 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 b^2 d \sqrt{\sec (c+d x)}}+\frac{(8 b B-5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b^2 d \sqrt{\sec (c+d x)}}-\frac{\left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{192 b^3 d}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{8} a \left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right )+\frac{1}{4} a b \left (48 A b^2+8 a b B-5 a^2 C+36 b^2 C\right ) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{48 b^3}+\frac{\left (\left (8 a^3 b B+32 a b^3 B-5 a^4 C-8 a^2 b^2 (2 A+C)+16 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}{128 b^3}\\ &=-\frac{\sqrt{a+b} \left (8 a^3 b B+32 a b^3 B-5 a^4 C-8 a^2 b^2 (2 A+C)+16 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}}}{64 b^4 d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 b d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (16 A b^2-8 a b B+5 a^2 C+12 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 b^2 d \sqrt{\sec (c+d x)}}+\frac{(8 b B-5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b^2 d \sqrt{\sec (c+d x)}}-\frac{\left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{192 b^3 d}+\frac{\left (a \left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 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}{384 b^3}+\frac{\left (a \left (15 a^3 C-2 a^2 b (12 B+5 C)+4 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 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}{384 b^3}\\ &=\frac{(a-b) \sqrt{a+b} \left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 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}}}{192 a b^3 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left (15 a^3 C-2 a^2 b (12 B+5 C)+4 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 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}}}{192 b^3 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left (8 a^3 b B+32 a b^3 B-5 a^4 C-8 a^2 b^2 (2 A+C)+16 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}}}{64 b^4 d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 b d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (16 A b^2-8 a b B+5 a^2 C+12 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 b^2 d \sqrt{\sec (c+d x)}}+\frac{(8 b B-5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b^2 d \sqrt{\sec (c+d x)}}-\frac{\left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{192 b^3 d}\\ \end{align*}

Mathematica [A]  time = 15.3834, size = 854, normalized size = 1.11 \[ \frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left (\frac{(8 b B+a C) \sin (c+d x)}{96 b}+\frac{\left (-5 C a^2+8 b B a+48 A b^2+48 b^2 C\right ) \sin (2 (c+d x))}{192 b^2}+\frac{(8 b B+a C) \sin (3 (c+d x))}{96 b}+\frac{1}{32} C \sin (4 (c+d x))\right )}{d}-\frac{-b (b-a) (a+b) \left (15 C a^3-24 b B a^2+4 b^2 (12 A+7 C) a+128 b^3 B\right ) E\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{b-a}{a+b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{\frac{a \tan ^2\left (\frac{1}{2} (c+d x)\right )-b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}} \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )+a (a-b) (a+b) \left (15 C a^3-6 b (4 B+5 C) a^2+4 b^2 (12 A+12 B+11 C) a-8 b^3 (12 A+16 B+9 C)\right ) F\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{b-a}{a+b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{\frac{a \tan ^2\left (\frac{1}{2} (c+d x)\right )-b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}} \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )-3 (a-b) \left (5 C a^4-8 b B a^3+8 b^2 (2 A+C) a^2-32 b^3 B a-16 b^4 (4 A+3 C)\right ) \left ((a-b) F\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{b-a}{a+b}\right )-2 b \Pi \left (-1;-\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{b-a}{a+b}\right )\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{\frac{a \tan ^2\left (\frac{1}{2} (c+d x)\right )-b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}} \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )-(a-b) b \left (15 C a^3-24 b B a^2+4 b^2 (12 A+7 C) a+128 b^3 B\right ) \tan \left (\frac{1}{2} (c+d x)\right ) \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )-1\right ) \left (a \tan ^2\left (\frac{1}{2} (c+d x)\right )-b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b\right )}{192 (a-b) b^4 d \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )-1\right ) \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right ) \sqrt{\frac{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )}} \sqrt{\frac{a \tan ^2\left (\frac{1}{2} (c+d x)\right )-b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(((8*b*B + a*C)*Sin[c + d*x])/(96*b) + ((48*A*b^2 + 8*a*b*B - 5*a
^2*C + 48*b^2*C)*Sin[2*(c + d*x)])/(192*b^2) + ((8*b*B + a*C)*Sin[3*(c + d*x)])/(96*b) + (C*Sin[4*(c + d*x)])/
32))/d - (-((a - b)*b*(-24*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(12*A + 7*C))*Tan[(c + d*x)/2]*(-1 + Tan[(
c + d*x)/2]^2)*(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)) - b*(-a + b)*(a + b)*(-24*a^2*b*B + 128*
b^3*B + 15*a^3*C + 4*a*b^2*(12*A + 7*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c
+ d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + a
*(a - b)*(a + b)*(15*a^3*C - 6*a^2*b*(4*B + 5*C) - 8*b^3*(12*A + 16*B + 9*C) + 4*a*b^2*(12*A + 12*B + 11*C))*E
llipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqr
t[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 3*(a - b)*(-8*a^3*b*B - 32*a*b^3*B + 5*a^4*
C + 8*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] -
2*b*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d
*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)])/(192*(a - b)*b^4*d*(-1 + Tan[(c
+ d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*Sqrt[(a + b + a
*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])

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Maple [B]  time = 0.503, size = 5307, normalized size = 6.9 \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)+C*cos(d*x+c)^2)*(a+b*cos(d*x+c))^(1/2)/sec(d*x+c)^(3/2),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 )} \sqrt{b \cos \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*(a+b*cos(d*x+c))^(1/2)/sec(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [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)+C*cos(d*x+c)^2)*(a+b*cos(d*x+c))^(1/2)/sec(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

Timed out

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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)+C*cos(d*x+c)**2)*(a+b*cos(d*x+c))**(1/2)/sec(d*x+c)**(3/2),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 )} \sqrt{b \cos \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*(a+b*cos(d*x+c))^(1/2)/sec(d*x+c)^(3/2),x, algorithm="giac")

[Out]

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