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

Optimal. Leaf size=779 \[ \frac {\left (-15 a^2 C+50 a b B+64 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{240 b d}+\frac {\left (-15 a^3 C+50 a^2 b B+172 a b^2 C+120 b^3 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}{320 b d}+\frac {\left (-45 a^4 C+150 a^3 b B+1692 a^2 b^2 C+2840 a b^3 B+1024 b^4 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{1920 b^2 d \sqrt {\cos (c+d x)}}-\frac {\sqrt {a+b} \left (45 a^4 C-30 a^3 b (5 B+C)-4 a^2 b^2 (295 B+423 C)-8 a b^3 (355 B+193 C)-16 b^4 (45 B+64 C)\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{1920 b^2 d}-\frac {(a-b) \sqrt {a+b} \left (-45 a^4 C+150 a^3 b B+1692 a^2 b^2 C+2840 a b^3 B+1024 b^4 C\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{1920 a b^2 d}+\frac {\sqrt {a+b} \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 C-240 a^2 b^3 B-240 a b^4 C-96 b^5 B\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{128 b^3 d}+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{40 b d}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d} \]

[Out]

1/240*(50*B*a*b-15*C*a^2+64*C*b^2)*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)*cos(d*x+c)^(1/2)/b/d+1/40*(10*B*b-3*C*a)*
(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)*cos(d*x+c)^(1/2)/b/d+1/5*C*(a+b*cos(d*x+c))^(7/2)*sin(d*x+c)*cos(d*x+c)^(1/2
)/b/d+1/1920*(150*B*a^3*b+2840*B*a*b^3-45*C*a^4+1692*C*a^2*b^2+1024*C*b^4)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b
^2/d/cos(d*x+c)^(1/2)+1/320*(50*B*a^2*b+120*B*b^3-15*C*a^3+172*C*a*b^2)*sin(d*x+c)*cos(d*x+c)^(1/2)*(a+b*cos(d
*x+c))^(1/2)/b/d-1/1920*(a-b)*(150*B*a^3*b+2840*B*a*b^3-45*C*a^4+1692*C*a^2*b^2+1024*C*b^4)*cot(d*x+c)*Ellipti
cE((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)*(a*(1-sec(d*x+c))/(a+
b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a/b^2/d-1/1920*(45*a^4*C-30*a^3*b*(5*B+C)-16*b^4*(45*B+64*C)-8*a*b^3*
(355*B+193*C)-4*a^2*b^2*(295*B+423*C))*cot(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)*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/b^2/d+1/128*
(10*B*a^4*b-240*B*a^2*b^3-96*B*b^5-3*C*a^5-40*C*a^3*b^2-240*C*a*b^4)*cot(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)*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(
1+sec(d*x+c))/(a-b))^(1/2)/b^3/d

________________________________________________________________________________________

Rubi [A]  time = 3.29, antiderivative size = 779, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.204, Rules used = {3029, 2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994} \[ \frac {\left (-15 a^2 C+50 a b B+64 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{240 b d}+\frac {\left (50 a^2 b B-15 a^3 C+172 a b^2 C+120 b^3 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}{320 b d}+\frac {\left (1692 a^2 b^2 C+150 a^3 b B-45 a^4 C+2840 a b^3 B+1024 b^4 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{1920 b^2 d \sqrt {\cos (c+d x)}}-\frac {\sqrt {a+b} \left (-4 a^2 b^2 (295 B+423 C)-30 a^3 b (5 B+C)+45 a^4 C-8 a b^3 (355 B+193 C)-16 b^4 (45 B+64 C)\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{1920 b^2 d}-\frac {(a-b) \sqrt {a+b} \left (1692 a^2 b^2 C+150 a^3 b B-45 a^4 C+2840 a b^3 B+1024 b^4 C\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{1920 a b^2 d}+\frac {\sqrt {a+b} \left (-240 a^2 b^3 B-40 a^3 b^2 C+10 a^4 b B-3 a^5 C-240 a b^4 C-96 b^5 B\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{128 b^3 d}+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{40 b d}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a - b)*Sqrt[a + b]*(150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 1692*a^2*b^2*C + 1024*b^4*C)*Cot[c + d*x]*Ellip
ticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c
+ d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*a*b^2*d) - (Sqrt[a + b]*(45*a^4*C - 30*a^3*b*(5
*B + C) - 16*b^4*(45*B + 64*C) - 8*a*b^3*(355*B + 193*C) - 4*a^2*b^2*(295*B + 423*C))*Cot[c + d*x]*EllipticF[A
rcSin[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) + (Sqrt[a + b]*(10*a^4*b*B - 240*a^2*b^3*B - 9
6*b^5*B - 3*a^5*C - 40*a^3*b^2*C - 240*a*b^4*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) + ((150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 1692*a^2*b^2*C + 1024*b^4*C)*
Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1920*b^2*d*Sqrt[Cos[c + d*x]]) + ((50*a^2*b*B + 120*b^3*B - 15*a^3*C +
172*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(320*b*d) + ((50*a*b*B - 15*a^2*C + 64
*b^2*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(240*b*d) + ((10*b*B - 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)*Si
n[c + d*x])/(5*b*d)

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 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 2990

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

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]

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 3029

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] :> Dist[1/b^2, Int[(a + b*Sin[e + f*x])
^(m + 1)*(c + d*Sin[e + f*x])^n*(b*B - a*C + b*C*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m,
n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*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 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 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]

Rubi steps

\begin {align*} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2} (B+C \cos (c+d x)) \, 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}+4 b C \cos (c+d x)+\frac {1}{2} (10 b B-3 a C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{5 b}\\ &=\frac {(10 b B-3 a C) \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}{4} a (2 b B+a C)+\frac {3}{2} b (10 b B+9 a C) \cos (c+d x)+\frac {1}{4} \left (50 a b B-15 a^2 C+64 b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{20 b}\\ &=\frac {\left (50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-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 (110 a b B+15 a^2 C+64 b^2 C\right )+\frac {1}{4} b \left (310 a b B+147 a^2 C+128 b^2 C\right ) \cos (c+d x)+\frac {3}{8} \left (50 a^2 b B+120 b^3 B-15 a^3 C+172 a b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{60 b}\\ &=\frac {\left (50 a^2 b B+120 b^3 B-15 a^3 C+172 a 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 (50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-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 (590 a^2 b B+360 b^3 B+15 a^3 C+772 a b^2 C\right )+\frac {1}{8} b \left (1610 a^2 b B+360 b^3 B+573 a^3 C+1156 a b^2 C\right ) \cos (c+d x)+\frac {1}{16} \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 C\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{120 b}\\ &=\frac {\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (50 a^2 b B+120 b^3 B-15 a^3 C+172 a 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 (50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-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 (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 C\right )+\frac {1}{8} a b \left (590 a^2 b B+360 b^3 B+15 a^3 C+772 a b^2 C\right ) \cos (c+d x)-\frac {15}{16} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 C-240 a b^4 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 (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (50 a^2 b B+120 b^3 B-15 a^3 C+172 a 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 (50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-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 (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 C\right )+\frac {1}{8} a b \left (590 a^2 b B+360 b^3 B+15 a^3 C+772 a 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 (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 C-240 a b^4 C\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}} \, dx}{256 b^2}\\ &=\frac {\sqrt {a+b} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 C-240 a b^4 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 (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (50 a^2 b B+120 b^3 B-15 a^3 C+172 a 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 (50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-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 (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 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 (5 B+C)-16 b^4 (45 B+64 C)-8 a b^3 (355 B+193 C)-4 a^2 b^2 (295 B+423 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 (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 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 (5 B+C)-16 b^4 (45 B+64 C)-8 a b^3 (355 B+193 C)-4 a^2 b^2 (295 B+423 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 {\sqrt {a+b} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 C-240 a b^4 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 (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (50 a^2 b B+120 b^3 B-15 a^3 C+172 a 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 (50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-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*}

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Mathematica [C]  time = 6.54, size = 1353, normalized size = 1.74 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/3840*((-4*a*(-1330*a^3*b*B - 3560*a*b^3*B + 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*(-6440*a^2*b^2*B -
1440*b^4*B - 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)*Sqr
t[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), 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]])) + 2*(-150*a^3*b*B - 2840*a*b^3*B + 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[S
qrt[((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]])))/(b*d)
+ (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((590*a^2*b*B + 420*b^3*B + 15*a^3*C + 898*a*b^2*C)*Sin[c + d
*x])/(960*b) + ((170*a*b*B + 93*a^2*C + 88*b^2*C)*Sin[2*(c + d*x)])/480 + (b*(10*b*B + 21*a*C)*Sin[3*(c + d*x)
])/160 + (b^2*C*Sin[4*(c + d*x)])/40))/d

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fricas [F]  time = 4.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{4} + B a^{2} \cos \left (d x + c\right ) + {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{2} + 2 \, B a b\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^(5/2)*(B*cos(d*x+c)+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 + B*a^2*cos(d*x + c) + (2*C*a*b + B*b^2)*cos(d*x + c)^3 + (C*a^2 + 2*B*a*b)*cos
(d*x + c)^2)*sqrt(b*cos(d*x + c) + a)*sqrt(cos(d*x + c)), x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^(5/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.95, size = 5164, normalized size = 6.63 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\cos \left (d x + c\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^(5/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {\cos \left (c+d\,x\right )}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))**(5/2)*(B*cos(d*x+c)+C*cos(d*x+c)**2)*cos(d*x+c)**(1/2),x)

[Out]

Timed out

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