3.1135 \(\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)+C \cos ^2(c+d x))}{\cos ^{\frac{11}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=552 \[ \frac{2 \sin (c+d x) \left (a^2 b (163 A+231 C)+75 a^3 B+135 a b^2 B+5 A b^3\right ) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt{a+b \cos (c+d x)}}{315 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left (-6 a^2 b (19 A-60 B+28 C)+3 a^3 (49 A-25 B+63 C)+15 a b^2 (11 A-3 B+21 C)+10 A b^3\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{315 a^2 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left (-3 a^2 b^2 (93 A+161 C)-21 a^4 (7 A+9 C)-435 a^3 b B-45 a b^3 B+10 A b^4\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{315 a^3 d}+\frac{2 (9 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)} \]

[Out]

(-2*(a - b)*Sqrt[a + b]*(10*A*b^4 - 435*a^3*b*B - 45*a*b^3*B - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93*A + 161*C))*
Cot[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)])/(315*a^3*d) - (2*(a - b)*Sqrt[a + b
]*(10*A*b^3 + 15*a*b^2*(11*A - 3*B + 21*C) - 6*a^2*b*(19*A - 60*B + 28*C) + 3*a^3*(49*A - 25*B + 63*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)])/(315*a^2*d) + (2*(15*A*b^2 + 90*a*b*B + 7
*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*(5*A*b^3 + 75*a^3*B +
135*a*b^2*B + a^2*b*(163*A + 231*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(3/2)) + (2
*(5*A*b + 9*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*A*(a + b*Cos[c + d*x]
)^(5/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 2.04865, antiderivative size = 552, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3047, 3055, 2998, 2816, 2994} \[ \frac{2 \sin (c+d x) \left (a^2 b (163 A+231 C)+75 a^3 B+135 a b^2 B+5 A b^3\right ) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt{a+b \cos (c+d x)}}{315 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left (-6 a^2 b (19 A-60 B+28 C)+3 a^3 (49 A-25 B+63 C)+15 a b^2 (11 A-3 B+21 C)+10 A b^3\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{315 a^2 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left (-3 a^2 b^2 (93 A+161 C)-21 a^4 (7 A+9 C)-435 a^3 b B-45 a b^3 B+10 A b^4\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{315 a^3 d}+\frac{2 (9 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]

[Out]

(-2*(a - b)*Sqrt[a + b]*(10*A*b^4 - 435*a^3*b*B - 45*a*b^3*B - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93*A + 161*C))*
Cot[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)])/(315*a^3*d) - (2*(a - b)*Sqrt[a + b
]*(10*A*b^3 + 15*a*b^2*(11*A - 3*B + 21*C) - 6*a^2*b*(19*A - 60*B + 28*C) + 3*a^3*(49*A - 25*B + 63*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)])/(315*a^2*d) + (2*(15*A*b^2 + 90*a*b*B + 7
*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*(5*A*b^3 + 75*a^3*B +
135*a*b^2*B + a^2*b*(163*A + 231*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(3/2)) + (2
*(5*A*b + 9*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*A*(a + b*Cos[c + d*x]
)^(5/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))

Rule 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C - B*c*d + A*d^2)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(
c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c
*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*
d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)
))*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] && GtQ[m, 0] && LtQ[n, -1]

Rule 3055

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*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] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

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{(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx &=\frac{2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2}{9} \int \frac{(a+b \cos (c+d x))^{3/2} \left (\frac{1}{2} (5 A b+9 a B)+\frac{1}{2} (7 a A+9 b B+9 a C) \cos (c+d x)+\frac{1}{2} b (2 A+9 C) \cos ^2(c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 (5 A b+9 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{4}{63} \int \frac{\sqrt{a+b \cos (c+d x)} \left (\frac{1}{4} \left (15 A b^2+90 a b B+7 a^2 (7 A+9 C)\right )+\frac{1}{4} \left (88 a A b+45 a^2 B+63 b^2 B+126 a b C\right ) \cos (c+d x)+\frac{3}{4} b (8 A b+6 a B+21 b C) \cos ^2(c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 \left (15 A b^2+90 a b B+7 a^2 (7 A+9 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (5 A b+9 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{8}{315} \int \frac{\frac{3}{8} \left (5 A b^3+75 a^3 B+135 a b^2 B+a^2 b (163 A+231 C)\right )+\frac{1}{8} \left (585 a^2 b B+315 b^3 B+21 a^3 (7 A+9 C)+5 a b^2 (121 A+189 C)\right ) \cos (c+d x)+\frac{1}{8} b \left (270 a b B+14 a^2 (7 A+9 C)+15 b^2 (10 A+21 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx\\ &=\frac{2 \left (15 A b^2+90 a b B+7 a^2 (7 A+9 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (5 A b^3+75 a^3 B+135 a b^2 B+a^2 b (163 A+231 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (5 A b+9 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{16 \int \frac{-\frac{3}{16} \left (10 A b^4-435 a^3 b B-45 a b^3 B-21 a^4 (7 A+9 C)-3 a^2 b^2 (93 A+161 C)\right )+\frac{3}{16} a \left (75 a^3 B+405 a b^2 B+5 b^3 (31 A+63 C)+3 a^2 b (87 A+119 C)\right ) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{945 a}\\ &=\frac{2 \left (15 A b^2+90 a b B+7 a^2 (7 A+9 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (5 A b^3+75 a^3 B+135 a b^2 B+a^2 b (163 A+231 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (5 A b+9 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}-\frac{\left ((a-b) \left (10 A b^3+15 a b^2 (11 A-3 B+21 C)-6 a^2 b (19 A-60 B+28 C)+3 a^3 (49 A-25 B+63 C)\right )\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx}{315 a}-\frac{\left (10 A b^4-435 a^3 b B-45 a b^3 B-21 a^4 (7 A+9 C)-3 a^2 b^2 (93 A+161 C)\right ) \int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{315 a}\\ &=-\frac{2 (a-b) \sqrt{a+b} \left (10 A b^4-435 a^3 b B-45 a b^3 B-21 a^4 (7 A+9 C)-3 a^2 b^2 (93 A+161 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}}}{315 a^3 d}-\frac{2 (a-b) \sqrt{a+b} \left (10 A b^3+15 a b^2 (11 A-3 B+21 C)-6 a^2 b (19 A-60 B+28 C)+3 a^3 (49 A-25 B+63 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}}}{315 a^2 d}+\frac{2 \left (15 A b^2+90 a b B+7 a^2 (7 A+9 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (5 A b^3+75 a^3 B+135 a b^2 B+a^2 b (163 A+231 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (5 A b+9 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}\\ \end{align*}

Mathematica [C]  time = 7.101, size = 1616, normalized size = 2.93 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]

[Out]

-((-4*a*(-114*a^4*A*b + 124*a^2*A*b^3 - 10*A*b^5 - 75*a^5*B + 30*a^3*b^2*B + 45*a*b^4*B - 168*a^4*b*C + 168*a^
2*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]]) - 4*a*(147*a^5*A + 279*a^3*A*b^2 - 10*a*A*b^4 + 435*a^4*b*B + 45*a^2*b^3*B + 189*a^5*C + 483*a^3*b^2*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), 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*(147*a^4*A*b + 279*a^2*A*b^3 - 10*A*b^5 + 435*a^3*b^2*B + 45*a*b^4*B + 189*a^4*b*C + 483*a^2*b^3*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]])))/(315*a^2*d) + (Sq
rt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^4*(19*a*A*b*Sin[c + d*x] + 9*a^2*B*Sin[c + d*x]))/6
3 + (2*Sec[c + d*x]^3*(49*a^2*A*Sin[c + d*x] + 75*A*b^2*Sin[c + d*x] + 135*a*b*B*Sin[c + d*x] + 63*a^2*C*Sin[c
+ d*x]))/315 + (2*Sec[c + d*x]^2*(163*a^2*A*b*Sin[c + d*x] + 5*A*b^3*Sin[c + d*x] + 75*a^3*B*Sin[c + d*x] + 1
35*a*b^2*B*Sin[c + d*x] + 231*a^2*b*C*Sin[c + d*x]))/(315*a) + (2*Sec[c + d*x]*(147*a^4*A*Sin[c + d*x] + 279*a
^2*A*b^2*Sin[c + d*x] - 10*A*b^4*Sin[c + d*x] + 435*a^3*b*B*Sin[c + d*x] + 45*a*b^3*B*Sin[c + d*x] + 189*a^4*C
*Sin[c + d*x] + 483*a^2*b^2*C*Sin[c + d*x]))/(315*a^2) + (2*a^2*A*Sec[c + d*x]^4*Tan[c + d*x])/9))/d

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Maple [B]  time = 0.509, size = 6176, normalized size = 11.2 \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+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\cos \left (d x + c\right )^{\frac{11}{2}}}\,{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+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/2)/cos(d*x + c)^(11/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C b^{2} \cos \left (d x + c\right )^{4} +{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} + A a^{2} +{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )\right )} \sqrt{b \cos \left (d x + c\right ) + a}}{\cos \left (d x + c\right )^{\frac{11}{2}}}, 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+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x, algorithm="fricas")

[Out]

integral((C*b^2*cos(d*x + c)^4 + (2*C*a*b + B*b^2)*cos(d*x + c)^3 + A*a^2 + (C*a^2 + 2*B*a*b + A*b^2)*cos(d*x
+ c)^2 + (B*a^2 + 2*A*a*b)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)/cos(d*x + c)^(11/2), 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+b*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/cos(d*x+c)**(11/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\cos \left (d x + c\right )^{\frac{11}{2}}}\,{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+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/2)/cos(d*x + c)^(11/2), x)