\(\int \frac {\sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)+C \cos ^2(c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\) [1509]

Optimal result

Integrand size = 45, antiderivative size = 766 \[ \int \frac {\sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {(a-b) \sqrt {a+b} \left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{192 a b^3 d \sqrt {\sec (c+d x)}}+\frac {\sqrt {a+b} \left (15 a^3 C-2 a^2 b (12 B+5 C)+4 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{192 b^3 d \sqrt {\sec (c+d x)}}-\frac {\sqrt {a+b} \left (8 a^3 b B+32 a b^3 B-5 a^4 C-8 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{64 b^4 d \sqrt {\sec (c+d x)}}+\frac {C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 b d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (16 A b^2-8 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b^2 d \sqrt {\sec (c+d x)}}+\frac {(8 b B-5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b^2 d \sqrt {\sec (c+d x)}}-\frac {\left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{192 b^3 d} \]

[Out]

1/4*C*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b/d/sec(d*x+c)^(3/2)+1/24*(8*B*b-5*C*a)*(a+b*cos(d*x+c))^(3/2)*sin(d*x
+c)/b^2/d/sec(d*x+c)^(1/2)+1/32*(16*A*b^2-8*B*a*b+5*C*a^2+12*C*b^2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b^2/d/se
c(d*x+c)^(1/2)-1/192*(24*B*a^2*b-128*B*b^3-15*a^3*C-4*a*b^2*(12*A+7*C))*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)*sec(
d*x+c)^(1/2)/b^3/d+1/192*(a-b)*(24*B*a^2*b-128*B*b^3-15*a^3*C-4*a*b^2*(12*A+7*C))*csc(d*x+c)*EllipticE((a+b*co
s(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*cos(d*x+c)^(1/2)*(a*(1-sec(d*x+
c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a/b^3/d/sec(d*x+c)^(1/2)+1/192*(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))*csc(d*x+c)*EllipticF((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)
^(1/2),((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*cos(d*x+c)^(1/2)*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-
b))^(1/2)/b^3/d/sec(d*x+c)^(1/2)-1/64*(8*B*a^3*b+32*B*a*b^3-5*a^4*C-8*a^2*b^2*(2*A+C)+16*b^4*(4*A+3*C))*csc(d*
x+c)*EllipticPi((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),(a+b)/b,((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*
cos(d*x+c)^(1/2)*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/b^4/d/sec(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 2.94 (sec) , antiderivative size = 766, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.178, Rules used = {4306, 3128, 3140, 3132, 2888, 3077, 2895, 3073} \[ \int \frac {\sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sin (c+d x) \left (5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{32 b^2 d \sqrt {\sec (c+d x)}}+\frac {\sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (15 a^3 C-2 a^2 b (12 B+5 C)+4 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{192 b^3 d \sqrt {\sec (c+d x)}}+\frac {(a-b) \sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (-15 a^3 C+24 a^2 b B-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{192 a b^3 d \sqrt {\sec (c+d x)}}-\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-15 a^3 C+24 a^2 b B-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{192 b^3 d}-\frac {\sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (-5 a^4 C+8 a^3 b B-8 a^2 b^2 (2 A+C)+32 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{64 b^4 d \sqrt {\sec (c+d x)}}+\frac {(8 b B-5 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b^2 d \sqrt {\sec (c+d x)}}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d \sec ^{\frac {3}{2}}(c+d x)} \]

[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 2888

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

Rule 2895

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

Rule 3073

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]/(f*b*c^2))*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)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ
[A, B] && PosQ[(c + d)/b]

Rule 3077

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 3128

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 3132

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 3140

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/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Si
n[e + f*x]]))*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], 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 4306

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]

Rubi steps \begin{align*} \text {integral}& = \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) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \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 (\arcsin \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) \operatorname {EllipticF}\left (\arcsin \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) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \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] (verified)

Time = 14.64 (sec) , antiderivative size = 704, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {(8 b B+a C) \sin (c+d x)}{96 b}+\frac {\left (48 A b^2+8 a b B-5 a^2 C+48 b^2 C\right ) \sin (2 (c+d x))}{192 b^2}+\frac {(8 b B+a C) \sin (3 (c+d x))}{96 b}+\frac {1}{32} C \sin (4 (c+d x))\right )}{d}+\frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {\sec (c+d x)} \left (-2 b (-a+b) (a+b) \left (-24 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (12 A+7 C)\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {a+b \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {-a+b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )+a (a-b) (a+b) \left (15 a^3 C-6 a^2 b (4 B+5 C)-8 b^3 (12 A+16 B+9 C)+4 a b^2 (12 A+12 B+11 C)\right ) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right ) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2} \sqrt {\frac {(a+b \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}} \sec (c+d x)-3 (a-b) \left (-8 a^3 b B-32 a b^3 B+5 a^4 C+8 a^2 b^2 (2 A+C)-16 b^4 (4 A+3 C)\right ) \left ((a-b) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )+2 b \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )\right ) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2} \sqrt {\frac {(a+b \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}} \sec (c+d x)+(a-b) b \left (-24 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (12 A+7 C)\right ) \cos (c+d x) (a+b \cos (c+d x)) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{192 (a-b) b^4 d \sqrt {a+b \cos (c+d x)}} \]

[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 + (Cos[(c + d*x)/2]^4*Sqrt[Sec[c + d*x]]*(-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))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x
]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 + 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))*EllipticF[ArcSin[Tan[(c + d*x)
/2]], (-a + b)/(a + b)]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)
/(a + b)]*Sec[c + d*x] - 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)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/
(a + b)]*Sec[c + d*x] + (a - b)*b*(-24*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(12*A + 7*C))*Cos[c + d*x]*(a
+ b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(192*(a - b)*b^4*d*Sqrt[a + b*Cos[c + d*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(7080\) vs. \(2(700)=1400\).

Time = 17.68 (sec) , antiderivative size = 7081, normalized size of antiderivative = 9.24

method	result	size
parts	\(\text {Expression too large to display}\)	\(7081\)
default	\(\text {Expression too large to display}\)	\(7166\)

[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,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F]

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

[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]

integral((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)

Sympy [F]

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

[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]

Integral(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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\sqrt {a+b\,\cos \left (c+d\,x\right )}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

[In]

int(((a + b*cos(c + d*x))^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(1/cos(c + d*x))^(3/2),x)

[Out]

int(((a + b*cos(c + d*x))^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(1/cos(c + d*x))^(3/2), x)