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

Optimal. Leaf size=746 \[ -\frac{\left (15 a^2 C-16 b^2 (5 A+4 C)\right ) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{240 b d}+\frac{a \left (-15 a^2 C+240 A b^2+172 b^2 C\right ) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{320 b d}-\frac{\left (-12 a^2 b^2 (220 A+141 C)+45 a^4 C-256 b^4 (5 A+4 C)\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left (-12 a^2 b^2 (220 A+141 C)-30 a^3 b C+45 a^4 C-8 a b^3 (260 A+193 C)-256 b^4 (5 A+4 C)\right ) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{1920 b^2 d}+\frac{(a-b) \sqrt{a+b} \left (-12 a^2 b^2 (220 A+141 C)+45 a^4 C-256 b^4 (5 A+4 C)\right ) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{1920 a b^2 d}-\frac{a \sqrt{a+b} \left (40 a^2 b^2 (2 A+C)+3 a^4 C+80 b^4 (4 A+3 C)\right ) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{128 b^3 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}-\frac{3 a C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{40 b d} \]

[Out]

((a - b)*Sqrt[a + b]*(45*a^4*C - 256*b^4*(5*A + 4*C) - 12*a^2*b^2*(220*A + 141*C))*Cot[c + d*x]*EllipticE[ArcS
in[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[a + b]*(45*a^4*C - 30*a^3*b*C - 256*b^4
*(5*A + 4*C) - 12*a^2*b^2*(220*A + 141*C) - 8*a*b^3*(260*A + 193*C))*Cot[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)])/(1920*b^2*d) - (a*Sqrt[a + b]*(3*a^4*C + 40*a^2*b^2*(2*A + C) + 80*b^4*(4*A
+ 3*C))*Cot[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)])/(128*b^3*d) - (
(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]]*Sin[c + d*x])/(1920*b^2
*d*Sqrt[Cos[c + d*x]]) + (a*(240*A*b^2 - 15*a^2*C + 172*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin
[c + d*x])/(320*b*d) - ((15*a^2*C - 16*b^2*(5*A + 4*C))*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c +
d*x])/(240*b*d) - (3*a*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(40*b*d) + (C*Sqrt[Cos[c
+ d*x]]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(5*b*d)

________________________________________________________________________________________

Rubi [A]  time = 2.75778, antiderivative size = 746, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.216, Rules used = {3050, 3049, 3061, 3053, 2809, 2998, 2816, 2994} \[ -\frac{\left (15 a^2 C-16 b^2 (5 A+4 C)\right ) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{240 b d}+\frac{a \left (-15 a^2 C+240 A b^2+172 b^2 C\right ) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{320 b d}-\frac{\left (-12 a^2 b^2 (220 A+141 C)+45 a^4 C-256 b^4 (5 A+4 C)\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left (-12 a^2 b^2 (220 A+141 C)-30 a^3 b C+45 a^4 C-8 a b^3 (260 A+193 C)-256 b^4 (5 A+4 C)\right ) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{1920 b^2 d}+\frac{(a-b) \sqrt{a+b} \left (-12 a^2 b^2 (220 A+141 C)+45 a^4 C-256 b^4 (5 A+4 C)\right ) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{1920 a b^2 d}-\frac{a \sqrt{a+b} \left (40 a^2 b^2 (2 A+C)+3 a^4 C+80 b^4 (4 A+3 C)\right ) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{128 b^3 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}-\frac{3 a C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{40 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a - b)*Sqrt[a + b]*(45*a^4*C - 256*b^4*(5*A + 4*C) - 12*a^2*b^2*(220*A + 141*C))*Cot[c + d*x]*EllipticE[ArcS
in[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[a + b]*(45*a^4*C - 30*a^3*b*C - 256*b^4
*(5*A + 4*C) - 12*a^2*b^2*(220*A + 141*C) - 8*a*b^3*(260*A + 193*C))*Cot[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)])/(1920*b^2*d) - (a*Sqrt[a + b]*(3*a^4*C + 40*a^2*b^2*(2*A + C) + 80*b^4*(4*A
+ 3*C))*Cot[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)])/(128*b^3*d) - (
(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]]*Sin[c + d*x])/(1920*b^2
*d*Sqrt[Cos[c + d*x]]) + (a*(240*A*b^2 - 15*a^2*C + 172*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin
[c + d*x])/(320*b*d) - ((15*a^2*C - 16*b^2*(5*A + 4*C))*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c +
d*x])/(240*b*d) - (3*a*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(40*b*d) + (C*Sqrt[Cos[c
+ d*x]]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(5*b*d)

Rule 3050

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)
*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n
+ 1))/(d*f*(m + n + 2)), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n
*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*
x] + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0
] && NeQ[c, 0])))

Rule 3049

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

Rule 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 \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac{\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{3}{2} a C \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{5 b}\\ &=-\frac{3 a C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac{C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac{\int \frac{(a+b \cos (c+d x))^{3/2} \left (\frac{5 a^2 C}{4}+\frac{1}{2} a b (40 A+27 C) \cos (c+d x)-\frac{1}{4} \left (15 a^2 C-16 b^2 (5 A+4 C)\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{20 b}\\ &=-\frac{\left (15 a^2 C-16 b^2 (5 A+4 C)\right ) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}-\frac{3 a C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac{C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac{\int \frac{\sqrt{a+b \cos (c+d x)} \left (\frac{1}{8} a \left (15 a^2 C+16 b^2 (5 A+4 C)\right )+\frac{1}{4} b \left (32 b^2 (5 A+4 C)+3 a^2 (80 A+49 C)\right ) \cos (c+d x)+\frac{3}{8} a \left (240 A b^2-15 a^2 C+172 b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{60 b}\\ &=\frac{a \left (240 A b^2-15 a^2 C+172 b^2 C\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d}-\frac{\left (15 a^2 C-16 b^2 (5 A+4 C)\right ) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}-\frac{3 a C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac{C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac{\int \frac{\frac{1}{16} a^2 \left (1040 A b^2+15 a^2 C+772 b^2 C\right )+\frac{1}{8} a b \left (4 b^2 (380 A+289 C)+a^2 (960 A+573 C)\right ) \cos (c+d x)-\frac{1}{16} \left (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 (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)} \sin (c+d x)}{1920 b^2 d \sqrt{\cos (c+d x)}}+\frac{a \left (240 A b^2-15 a^2 C+172 b^2 C\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d}-\frac{\left (15 a^2 C-16 b^2 (5 A+4 C)\right ) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}-\frac{3 a C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac{C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac{\int \frac{\frac{1}{16} a \left (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^2 b \left (1040 A b^2+15 a^2 C+772 b^2 C\right ) \cos (c+d x)+\frac{15}{16} a \left (3 a^4 C+40 a^2 b^2 (2 A+C)+80 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 (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)} \sin (c+d x)}{1920 b^2 d \sqrt{\cos (c+d x)}}+\frac{a \left (240 A b^2-15 a^2 C+172 b^2 C\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d}-\frac{\left (15 a^2 C-16 b^2 (5 A+4 C)\right ) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}-\frac{3 a C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac{C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac{\int \frac{\frac{1}{16} a \left (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^2 b \left (1040 A b^2+15 a^2 C+772 b^2 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 (a \left (3 a^4 C+40 a^2 b^2 (2 A+C)+80 b^4 (4 A+3 C)\right )\right ) \int \frac{\sqrt{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx}{256 b^2}\\ &=-\frac{a \sqrt{a+b} \left (3 a^4 C+40 a^2 b^2 (2 A+C)+80 b^4 (4 A+3 C)\right ) \cot (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}-\frac{\left (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)} \sin (c+d x)}{1920 b^2 d \sqrt{\cos (c+d x)}}+\frac{a \left (240 A b^2-15 a^2 C+172 b^2 C\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d}-\frac{\left (15 a^2 C-16 b^2 (5 A+4 C)\right ) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}-\frac{3 a C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac{C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac{\left (a \left (45 a^4 C-256 b^4 (5 A+4 C)-12 a^2 b^2 (220 A+141 C)\right )\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 C-256 b^4 (5 A+4 C)-12 a^2 b^2 (220 A+141 C)-8 a b^3 (260 A+193 C)\right )\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 (45 a^4 C-256 b^4 (5 A+4 C)-12 a^2 b^2 (220 A+141 C)\right ) \cot (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}-\frac{\sqrt{a+b} \left (45 a^4 C-30 a^3 b C-256 b^4 (5 A+4 C)-12 a^2 b^2 (220 A+141 C)-8 a b^3 (260 A+193 C)\right ) \cot (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}-\frac{a \sqrt{a+b} \left (3 a^4 C+40 a^2 b^2 (2 A+C)+80 b^4 (4 A+3 C)\right ) \cot (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}-\frac{\left (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)} \sin (c+d x)}{1920 b^2 d \sqrt{\cos (c+d x)}}+\frac{a \left (240 A b^2-15 a^2 C+172 b^2 C\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d}-\frac{\left (15 a^2 C-16 b^2 (5 A+4 C)\right ) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}-\frac{3 a C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac{C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}\\ \end{align*}

Mathematica [C]  time = 6.48163, size = 1341, normalized size = 1.8 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((-4*a*(-4720*a^2*A*b^2 - 1280*A*b^4 + 15*a^4*C - 3236*a^2*b^2*C - 1024*b^4*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]
^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]
^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a +
b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-3840*a^3*A*b - 6080*a*A*
b^3 - 2292*a^3*b*C - 4624*a*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*
Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[(
(a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c
+ d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]
*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), Arc
Sin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[C
os[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-2640*a^2*A*b^2 - 1280*A*b^4 + 45*a^4*C - 1692*a^2*b^2*C - 1024*b
^4*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]],
(-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x
])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)
/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c
+ d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt
[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d
*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((
a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]
]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(3840*b*d)
+ (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((a*(1040*A*b^2 + 15*a^2*C + 898*b^2*C)*Sin[c + d*x])/(960*b) +
((80*A*b^2 + 93*a^2*C + 88*b^2*C)*Sin[2*(c + d*x)])/480 + (21*a*b*C*Sin[3*(c + d*x)])/160 + (b^2*C*Sin[4*(c +
d*x)])/40))/d

________________________________________________________________________________________

Maple [B]  time = 0.546, size = 4724, normalized size = 6.3 \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+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x)

[Out]

-1/1920/d/(a+b*cos(d*x+c))^(1/2)*(-15*C*cos(d*x+c)^3*a^4*b+1752*C*cos(d*x+c)^5*a^2*b^3+774*C*cos(d*x+c)^4*a^3*
b^2+4720*A*cos(d*x+c)^3*a^2*b^3+2640*A*cos(d*x+c)^2*a^3*b^2-3840*A*cos(d*x+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^2+2720*A*cos(d*x+c)^4*a*b^4+45*C*cos(d*x+c)*a^5+384*C*cos(d*x+c)^7*b^5+128*C*cos(d*x+c)^5*b^5+
640*A*cos(d*x+c)^5*b^5-45*C*cos(d*x+c)^2*a^5+30*C*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*c
os(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^4*b-2292*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^2+1544*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^3-4624*C*s
in(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^4-45*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*b+1692*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^2+1692*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^3+1024*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+co
s(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a*b^4+1200*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^3*b^2+
7200*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*b^4+2080*A*cos(d*x+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^3-6080*A*cos(d*x+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+c
os(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a*b^4+2640*A*cos(d*x+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^2+2640*A*cos(d*x+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^3+1
280*A*cos(d*x+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^4+2400*A*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(1+c
os(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^3*b^2+9600*A*cos(d*x+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*b^4+30*C*cos(d*x+
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^4*b-2292*C*cos(d*x+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^2+1544*C*cos(d*x+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^3-4624*C*cos(d*x+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^4-45*C*cos(d*x+c)*sin(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^4*b+1692*C*cos(d*
x+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^2+1692*C*cos(d*x+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^3+1024*C*cos(d*x+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^4+1200*C*cos(d*x+c)*sin(d*x+c)*(c
os(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^3*b^2+7200*C*cos(d*x+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*b^4
+640*A*cos(d*x+c)^3*b^5-1280*A*cos(d*x+c)^2*b^5+512*C*cos(d*x+c)^3*b^5-1024*C*cos(d*x+c)^2*b^5-45*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))/s
in(d*x+c),(-(a-b)/(a+b))^(1/2))*a^5+1024*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))*b^5+90*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^5+1280*A*cos(d*x+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))*b^5-45*C*cos(d*x
+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^5+1024*C*cos(d*x+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))*
b^5+90*C*cos(d*x+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^5-3840*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)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b)
)^(1/2))*a^3*b^2+2080*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)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^2*b^3-6080*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)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(
a+b))^(1/2))*a*b^4+2640*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^3*b^2+2640*A*sin(d*x+c)*(cos(d*x+c)/(1+c
os(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^3+1280*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*b^4+2400*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^3*b^2+9600*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*b^4+1280*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))*b^5-30*C*cos(d*x+c)*a^4*b-1692*C*cos(d*x+c)*a^3*b^2-1544*C*cos(d*x+c)*a^2*b^3-1024*
C*cos(d*x+c)*a*b^4+45*C*cos(d*x+c)^2*a^4*b+918*C*cos(d*x+c)^2*a^3*b^2-1692*C*cos(d*x+c)^2*a^2*b^3-1032*C*cos(d
*x+c)^2*a*b^4+664*C*cos(d*x+c)^4*a*b^4+1484*C*cos(d*x+c)^3*a^2*b^3+1392*C*cos(d*x+c)^6*a*b^4-2640*A*cos(d*x+c)
^2*a^2*b^3-1440*A*cos(d*x+c)^2*a*b^4-2640*A*cos(d*x+c)*a^3*b^2-2080*A*cos(d*x+c)*a^2*b^3-1280*A*cos(d*x+c)*a*b
^4)/sin(d*x+c)/b^2/cos(d*x+c)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sqrt{\cos \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+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

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

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

[Out]

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

[Out]

Timed out

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Giac [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+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="giac")

[Out]

Timed out