3.1418 \(\int \frac{\sqrt{a+b \cos (c+d x)} (A+C \cos ^2(c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=698 \[ \frac{a \left (15 a^2 C+48 A b^2+28 b^2 C\right ) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{192 b^3 d}+\frac{\left (5 a^2 C+4 b^2 (4 A+3 C)\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{32 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left (-10 a^2 b C+15 a^3 C+4 a b^2 (12 A+7 C)+24 b^3 (4 A+3 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{192 b^3 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left (15 a^2 C+48 A b^2+28 b^2 C\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{192 b^3 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left (8 a^2 b^2 (2 A+C)+5 a^4 C-16 b^4 (4 A+3 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{64 b^4 d \sqrt{\sec (c+d x)}}-\frac{5 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d \sec ^{\frac{3}{2}}(c+d x)} \]

[Out]

-((a - b)*Sqrt[a + b]*(48*A*b^2 + 15*a^2*C + 28*b^2*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*b^3*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(15*a^3*C - 10*a^2*b*C +
24*b^3*(4*A + 3*C) + 4*a*b^2*(12*A + 7*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]*(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)) + ((5*a^2*C + 4*b^2*(4*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b^2*d*Sqrt[Sec[c + d*x]]) - (
5*a*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*b^2*d*Sqrt[Sec[c + d*x]]) + (a*(48*A*b^2 + 15*a^2*C + 28*b^
2*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*b^3*d)

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Rubi [A]  time = 2.17408, antiderivative size = 698, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.243, Rules used = {4221, 3050, 3049, 3061, 3053, 2809, 2998, 2816, 2994} \[ \frac{a \left (15 a^2 C+48 A b^2+28 b^2 C\right ) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{192 b^3 d}+\frac{\left (5 a^2 C+4 b^2 (4 A+3 C)\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{32 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left (-10 a^2 b C+15 a^3 C+4 a b^2 (12 A+7 C)+24 b^3 (4 A+3 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{192 b^3 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left (15 a^2 C+48 A b^2+28 b^2 C\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{192 b^3 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left (8 a^2 b^2 (2 A+C)+5 a^4 C-16 b^4 (4 A+3 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{64 b^4 d \sqrt{\sec (c+d x)}}-\frac{5 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d \sec ^{\frac{3}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a - b)*Sqrt[a + b]*(48*A*b^2 + 15*a^2*C + 28*b^2*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*b^3*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(15*a^3*C - 10*a^2*b*C +
24*b^3*(4*A + 3*C) + 4*a*b^2*(12*A + 7*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]*(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)) + ((5*a^2*C + 4*b^2*(4*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b^2*d*Sqrt[Sec[c + d*x]]) - (
5*a*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*b^2*d*Sqrt[Sec[c + d*x]]) + (a*(48*A*b^2 + 15*a^2*C + 28*b^
2*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 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 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+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+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{5}{2} 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{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{5 a^2 C}{4}-\frac{1}{2} a b C \cos (c+d x)+\frac{3}{4} \left (5 a^2 C+4 b^2 (4 A+3 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 (5 a^2 C+4 b^2 (4 A+3 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 b^2 d \sqrt{\sec (c+d x)}}-\frac{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 (5 a^2 C-12 b^2 (4 A+3 C)\right )+\frac{1}{4} b \left (48 A b^2+\left (a^2+36 b^2\right ) C\right ) \cos (c+d x)+\frac{1}{8} a \left (48 A b^2+15 a^2 C+28 b^2 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 (5 a^2 C+4 b^2 (4 A+3 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 b^2 d \sqrt{\sec (c+d x)}}-\frac{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{a \left (48 A b^2+15 a^2 C+28 b^2 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^2 \left (48 A b^2+15 a^2 C+28 b^2 C\right )-\frac{1}{4} a b \left (5 a^2 C-12 b^2 (4 A+3 C)\right ) \cos (c+d x)-\frac{3}{8} \left (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 (5 a^2 C+4 b^2 (4 A+3 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 b^2 d \sqrt{\sec (c+d x)}}-\frac{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{a \left (48 A b^2+15 a^2 C+28 b^2 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^2 \left (48 A b^2+15 a^2 C+28 b^2 C\right )-\frac{1}{4} a b \left (5 a^2 C-12 b^2 (4 A+3 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 (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 (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 (5 a^2 C+4 b^2 (4 A+3 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 b^2 d \sqrt{\sec (c+d x)}}-\frac{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{a \left (48 A b^2+15 a^2 C+28 b^2 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^2 \left (48 A b^2+15 a^2 C+28 b^2 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-10 a^2 b C+24 b^3 (4 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}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx}{384 b^3}\\ &=-\frac{(a-b) \sqrt{a+b} \left (48 A b^2+15 a^2 C+28 b^2 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 b^3 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left (15 a^3 C-10 a^2 b C+24 b^3 (4 A+3 C)+4 a b^2 (12 A+7 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 (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 (5 a^2 C+4 b^2 (4 A+3 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 b^2 d \sqrt{\sec (c+d x)}}-\frac{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{a \left (48 A b^2+15 a^2 C+28 b^2 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 [B]  time = 26.2278, size = 4845, normalized size = 6.94 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + 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]]*((a*C*Sin[c + d*x])/(96*b) + ((48*A*b^2 - 5*a^2*C + 48*b^2*C)*Sin
[2*(c + d*x)])/(192*b^2) + (a*C*Sin[3*(c + d*x)])/(96*b) + (C*Sin[4*(c + d*x)])/32))/d + (((A*b)/(2*Sqrt[a + b
*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*C)/(96*b*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (3*b*C)/(8*S
qrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (3*a*A*Sqrt[Sec[c + d*x]])/(8*Sqrt[a + b*Cos[c + d*x]]) + (25*a*
C*Sqrt[Sec[c + d*x]])/(96*Sqrt[a + b*Cos[c + d*x]]) + (5*a^3*C*Sqrt[Sec[c + d*x]])/(384*b^2*Sqrt[a + b*Cos[c +
d*x]]) + (a*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(8*Sqrt[a + b*Cos[c + d*x]]) + (7*a*C*Cos[2*(c + d*x)]*Sqr
t[Sec[c + d*x]])/(96*Sqrt[a + b*Cos[c + d*x]]) + (5*a^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(128*b^2*Sqrt[a
+ b*Cos[c + d*x]]))*(-2*a*b*(a + b)*(48*A*b^2 + 15*a^2*C + 28*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sq
rt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Se
c[(c + d*x)/2]^2 - 2*a*(a + b)*(15*a^3*C - 30*a^2*b*C - 24*b^3*(4*A + 3*C) + 4*a*b^2*(12*A + 11*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]))]*EllipticF[ArcSin[Tan[(c +
d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 + 3*(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*(48*A*b^2 + 15*a^2*C + 28*b^2*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(
c + d*x)/2]))/(192*b^4*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^4)*(-(Sec[(c + d*x
)/2]^2*Tan[(c + d*x)/2]^3*(-2*a*b*(a + b)*(48*A*b^2 + 15*a^2*C + 28*b^2*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 - 2*a*(a + b)*(15*a^3*C - 30*a^2*b*C - 24*b^3*(4*A + 3*C) + 4*a*b^2*(12*A + 11*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]))]*EllipticF[ArcSin[Tan[
(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 + 3*(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*(48*A*b^2 + 15*a^2*C + 28*b^2*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*
Tan[(c + d*x)/2]))/(96*b^4*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^4)^2) + (Sin[c +
d*x]*(-2*a*b*(a + b)*(48*A*b^2 + 15*a^2*C + 28*b^2*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 - 2*a*(a + b)*(15*a^3*C - 30*a^2*b*C - 24*b^3*(4*A + 3*C) + 4*a*b^2*(12*A + 11*C))*Sqrt[Cos[c + d*x]/(1 + Co
s[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a +
b)/(a + b)]*Sec[(c + d*x)/2]^2 + 3*(5*a^4*C + 8*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*((a - b)*EllipticF[Ar
cSin[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*(48*A*b^2 + 15*a^2*C + 28*b^2*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(
384*b^3*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^4)) - (Sqrt[Sec[c + d*x]]*Sin[c +
d*x]*(-2*a*b*(a + b)*(48*A*b^2 + 15*a^2*C + 28*b^2*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 - 2*a*(a + b)*(15*a^3*C - 30*a^2*b*C - 24*b^3*(4*A + 3*C) + 4*a*b^2*(12*A + 11*C))*Sqrt[Cos[c + d*x]/(1 + Co
s[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a +
b)/(a + b)]*Sec[(c + d*x)/2]^2 + 3*(5*a^4*C + 8*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*((a - b)*EllipticF[Ar
cSin[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*(48*A*b^2 + 15*a^2*C + 28*b^2*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(
384*b^4*Sqrt[a + b*Cos[c + d*x]]*(-1 + Tan[(c + d*x)/2]^4)) + (-(a*b*(48*A*b^2 + 15*a^2*C + 28*b^2*C)*Cos[c +
d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^6)/2 - (a*b*(a + b)*(48*A*b^2 + 15*a^2*C + 28*b^2*C)*Sqrt[(a + b*Co
s[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*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(
1 + Cos[c + d*x])] - (a*(a + b)*(15*a^3*C - 30*a^2*b*C - 24*b^3*(4*A + 3*C) + 4*a*b^2*(12*A + 11*C))*Sqrt[(a +
b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c +
d*x)/2]^2*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d
*x]/(1 + Cos[c + d*x])] - (a*b*(a + b)*(48*A*b^2 + 15*a^2*C + 28*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*
EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos
[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/
((a + b)*(1 + Cos[c + d*x]))] - (a*(a + b)*(15*a^3*C - 30*a^2*b*C - 24*b^3*(4*A + 3*C) + 4*a*b^2*(12*A + 11*C)
)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]
^2*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c
+ d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - 2*a*b*(a + b)*(48*A*b^2 + 15*a^2*C + 2
8*b^2*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]))]*Ellipti
cE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] - 2*a*(a + b)*(15*a^3*C - 3
0*a^2*b*C - 24*b^3*(4*A + 3*C) + 4*a*b^2*(12*A + 11*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]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]
^2*Tan[(c + d*x)/2] + a*b^2*(48*A*b^2 + 15*a^2*C + 28*b^2*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[
(c + d*x)/2] + a*b*(48*A*b^2 + 15*a^2*C + 28*b^2*C)*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(
c + d*x)/2] - 2*a*b*(48*A*b^2 + 15*a^2*C + 28*b^2*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[
(c + d*x)/2]^2 + (9*(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[Cos[c + d*x]
*Sec[(c + d*x)/2]^2]*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]*(-(Sec[(c + d*x)/2]^
2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/2 + (3*(5*a^4*C + 8*a^2*b^2*(2*A + C) - 1
6*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)*Sec[c + d*x]*(-((b*Sec[(c + d*x
)/2]^2*Sin[c + d*x])/(a + b)) + ((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[(
(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) - (a*(a + b)*(15*a^3*C - 30*a^2*b*C - 24*b^3*(4*A + 3*C) +
4*a*b^2*(12*A + 11*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]))]*Sec[(c + d*x)/2]^4)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - (a
*b*(a + b)*(48*A*b^2 + 15*a^2*C + 28*b^2*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]))]*Sec[(c + d*x)/2]^4*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(
c + d*x)/2]^2] + 3*(5*a^4*C + 8*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*(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)*Sec[(c + d*x)/2]^2)/(2*Sqrt[1
- Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (b*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan
[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])) + 3*(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*El
lipticPi[-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]*Tan[c + d*x])/(192*b^4*Sqrt[a + b*Cos[c + d*x]]*Sqrt
[Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^4))))

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Maple [B]  time = 0.36, size = 3615, normalized size = 5.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192/d/b^3*(96*A*sin(d*x+c)*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c
),(-(a-b)/(a+b))^(1/2))*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a*b^3-96*A*sin(d*x+c)*cos(d*x+c)*(cos(
d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,(-(a-b)/(a+b))^(1/2))*(1/(a+b)*(a+b*cos(
d*x+c))/(1+cos(d*x+c)))^(1/2)*a^2*b^2+48*A*sin(d*x+c)*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticE((
-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^2*b^2+48*A*s
in(d*x+c)*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/
2))*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a*b^3-10*C*sin(d*x+c)*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)
))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(
1/2))*a^3*b-4*C*sin(d*x+c)*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c
)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^2*b^2+72*C*sin(d*x+c)*cos(d*x+c)*(cos(d
*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+
c),(-(a-b)/(a+b))^(1/2))*a*b^3-48*C*sin(d*x+c)*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(
d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,(-(a-b)/(a+b))^(1/2))*a^2*b^2+15*C*sin(
d*x+c)*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(
(-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^3*b+28*C*sin(d*x+c)*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^
(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2
))*a^2*b^2+48*C*cos(d*x+c)^6*b^4+28*C*sin(d*x+c)*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*co
s(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a*b^3+15*C*sin(d*x+
c)*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+
cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^4+96*A*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticF((
-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a*b^3-96*A*sin
(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticPi((-1+cos(d
*x+c))/sin(d*x+c),-1,(-(a-b)/(a+b))^(1/2))*a^2*b^2+48*A*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*
(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^2*b^2+48*A
*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+co
s(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a*b^3-10*C*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a
+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^3*b-4*C*sin(
d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x
+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^2*b^2+72*C*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*
cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a*b^3-48*C*sin(d*
x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticPi((-1+cos(d*x+
c))/sin(d*x+c),-1,(-(a-b)/(a+b))^(1/2))*a^2*b^2+15*C*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+
b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^3*b+28*C*sin(
d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x
+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^2*b^2+28*C*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*
cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a*b^3+56*C*cos(d*
x+c)^5*a*b^3-2*C*cos(d*x+c)^4*a^2*b^2+5*C*cos(d*x+c)^3*a^3*b-192*A*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2
)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*b
^4+384*A*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,(-(a-b)/(a+b))^
(1/2))*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*b^4-144*C*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*
(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*b^4
-30*C*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticPi(
(-1+cos(d*x+c))/sin(d*x+c),-1,(-(a-b)/(a+b))^(1/2))*a^4+288*C*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/
(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,(-(a-b)/(a+b))^(1/2))*b^
4+15*C*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(
(-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^4+24*C*cos(d*x+c)^4*b^4-96*A*cos(d*x+c)^2*b^4-72*C*cos(d*x+
c)^2*b^4-15*C*cos(d*x+c)*a^4+144*A*cos(d*x+c)^3*a*b^3+48*A*cos(d*x+c)^2*a^2*b^2+15*C*cos(d*x+c)^2*a^4+96*A*cos
(d*x+c)^4*b^4-28*C*cos(d*x+c)^2*a*b^3+10*C*cos(d*x+c)*a^3*b-28*C*cos(d*x+c)*a^2*b^2-72*C*cos(d*x+c)*a*b^3+44*C
*cos(d*x+c)^3*a*b^3-48*A*cos(d*x+c)^2*a*b^3-48*A*cos(d*x+c)*a^2*b^2-96*A*cos(d*x+c)*a*b^3-192*A*sin(d*x+c)*cos
(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*
x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*b^4+384*A*sin(d*x+c)*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a
+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,(-(a-b)/(a+b))^(1/2))*b^4-
144*C*sin(d*x+c)*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*
EllipticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*b^4-30*C*sin(d*x+c)*cos(d*x+c)*(cos(d*x+c)/(1+cos(d
*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,(-(a-b)
/(a+b))^(1/2))*a^4+288*C*sin(d*x+c)*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+
cos(d*x+c)))^(1/2)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,(-(a-b)/(a+b))^(1/2))*b^4-15*C*cos(d*x+c)^2*a^3*b+
30*C*cos(d*x+c)^2*a^2*b^2)*cos(d*x+c)*(1/cos(d*x+c))^(3/2)/sin(d*x+c)/(a+b*cos(d*x+c))^(1/2)

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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