3.1517 \(\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^{\frac{13}{2}}(c+d x) \, dx\)

Optimal. Leaf size=705 \[ \frac{2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt{a+b \cos (c+d x)}}{231 d}+\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (5 a^2 b (229 A+297 C)+539 a^3 B+825 a b^2 B+15 A b^3\right ) \sqrt{a+b \cos (c+d x)}}{3465 a d}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (-5 a^2 b^2 (205 A+297 C)-75 a^4 (9 A+11 C)-1793 a^3 b B-55 a b^3 B+20 A b^4\right ) \sqrt{a+b \cos (c+d x)}}{3465 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (15 a^2 b^2 (19 A-121 B+33 C)-6 a^3 b (505 A-209 B+660 C)+3 a^4 (225 A-539 B+275 C)+10 a b^3 (3 A-11 B)+40 A b^4\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{3465 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+1617 a^5 B-110 a b^4 B+40 A b^5\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{3465 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 (11 a B+5 A b) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{99 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{11 d} \]

[Out]

(2*(a - b)*Sqrt[a + b]*(40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a
^4*b*(247*A + 319*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*S
qrt[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)])/(3465*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(40*A*b^4 + 10*a*b^3*(3*A - 11*B) + 15*a^2*b^2*
(19*A - 121*B + 33*C) + 3*a^4*(225*A - 539*B + 275*C) - 6*a^3*b*(505*A - 209*B + 660*C))*Sqrt[Cos[c + d*x]]*Cs
c[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sq
rt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3465*a^3*d*Sqrt[Sec[c + d*x]]) - (2*
(20*A*b^4 - 1793*a^3*b*B - 55*a*b^3*B - 75*a^4*(9*A + 11*C) - 5*a^2*b^2*(205*A + 297*C))*Sqrt[a + b*Cos[c + d*
x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3465*a^2*d) + (2*(15*A*b^3 + 539*a^3*B + 825*a*b^2*B + 5*a^2*b*(229*A +
297*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3465*a*d) + (2*(5*A*b^2 + 44*a*b*B + 3*a^2*
(9*A + 11*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d) + (2*(5*A*b + 11*a*B)*(a + b*C
os[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/
2)*Sin[c + d*x])/(11*d)

________________________________________________________________________________________

Rubi [A]  time = 3.37183, antiderivative size = 705, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4221, 3047, 3055, 2998, 2816, 2994} \[ \frac{2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt{a+b \cos (c+d x)}}{231 d}+\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (5 a^2 b (229 A+297 C)+539 a^3 B+825 a b^2 B+15 A b^3\right ) \sqrt{a+b \cos (c+d x)}}{3465 a d}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (-5 a^2 b^2 (205 A+297 C)-75 a^4 (9 A+11 C)-1793 a^3 b B-55 a b^3 B+20 A b^4\right ) \sqrt{a+b \cos (c+d x)}}{3465 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (15 a^2 b^2 (19 A-121 B+33 C)-6 a^3 b (505 A-209 B+660 C)+3 a^4 (225 A-539 B+275 C)+10 a b^3 (3 A-11 B)+40 A b^4\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{3465 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+1617 a^5 B-110 a b^4 B+40 A b^5\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{3465 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 (11 a B+5 A b) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{99 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{11 d} \]

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)*Sec[c + d*x]^(13/2),x]

[Out]

(2*(a - b)*Sqrt[a + b]*(40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a
^4*b*(247*A + 319*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*S
qrt[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)])/(3465*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(40*A*b^4 + 10*a*b^3*(3*A - 11*B) + 15*a^2*b^2*
(19*A - 121*B + 33*C) + 3*a^4*(225*A - 539*B + 275*C) - 6*a^3*b*(505*A - 209*B + 660*C))*Sqrt[Cos[c + d*x]]*Cs
c[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sq
rt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3465*a^3*d*Sqrt[Sec[c + d*x]]) - (2*
(20*A*b^4 - 1793*a^3*b*B - 55*a*b^3*B - 75*a^4*(9*A + 11*C) - 5*a^2*b^2*(205*A + 297*C))*Sqrt[a + b*Cos[c + d*
x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3465*a^2*d) + (2*(15*A*b^3 + 539*a^3*B + 825*a*b^2*B + 5*a^2*b*(229*A +
297*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3465*a*d) + (2*(5*A*b^2 + 44*a*b*B + 3*a^2*
(9*A + 11*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d) + (2*(5*A*b + 11*a*B)*(a + b*C
os[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/
2)*Sin[c + d*x])/(11*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 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 (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac{13}{2}}(c+d x) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \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{13}{2}}(c+d x)} \, dx\\ &=\frac{2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{1}{11} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^{3/2} \left (\frac{1}{2} (5 A b+11 a B)+\frac{1}{2} (9 a A+11 b B+11 a C) \cos (c+d x)+\frac{1}{2} b (4 A+11 C) \cos ^2(c+d x)\right )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx\\ &=\frac{2 (5 A b+11 a B) (a+b \cos (c+d x))^{3/2} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{1}{99} \left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+b \cos (c+d x)} \left (\frac{3}{4} \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right )+\frac{1}{4} \left (152 a A b+77 a^2 B+99 b^2 B+198 a b C\right ) \cos (c+d x)+\frac{1}{4} b (56 A b+44 a B+99 b C) \cos ^2(c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{2 (5 A b+11 a B) (a+b \cos (c+d x))^{3/2} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{1}{693} \left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{8} \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right )+\frac{1}{8} \left (1507 a^2 b B+693 b^3 B+45 a^3 (9 A+11 C)+a b^2 (1531 A+2079 C)\right ) \cos (c+d x)+\frac{1}{8} b \left (836 a b B+36 a^2 (9 A+11 C)+b^2 (452 A+693 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx\\ &=\frac{2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3465 a d}+\frac{2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{2 (5 A b+11 a B) (a+b \cos (c+d x))^{3/2} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{\left (16 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{3}{16} \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right )+\frac{1}{16} a \left (1617 a^3 B+6655 a b^2 B+15 a^2 b (337 A+429 C)+5 b^3 (461 A+693 C)\right ) \cos (c+d x)+\frac{1}{8} b \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{3465 a}\\ &=-\frac{2 \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3465 a^2 d}+\frac{2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3465 a d}+\frac{2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{2 (5 A b+11 a B) (a+b \cos (c+d x))^{3/2} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{\left (32 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{3}{32} \left (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)\right )+\frac{3}{32} a \left (10 A b^4+2871 a^3 b B+1705 a b^3 B+75 a^4 (9 A+11 C)+15 a^2 b^2 (221 A+297 C)\right ) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{10395 a^2}\\ &=-\frac{2 \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3465 a^2 d}+\frac{2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3465 a d}+\frac{2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{2 (5 A b+11 a B) (a+b \cos (c+d x))^{3/2} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{\left (\left (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 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}{3465 a^2}+\frac{\left ((a-b) \left (40 A b^4+10 a b^3 (3 A-11 B)+15 a^2 b^2 (19 A-121 B+33 C)+3 a^4 (225 A-539 B+275 C)-6 a^3 b (505 A-209 B+660 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}{3465 a^2}\\ &=\frac{2 (a-b) \sqrt{a+b} \left (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 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}}}{3465 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left (40 A b^4+10 a b^3 (3 A-11 B)+15 a^2 b^2 (19 A-121 B+33 C)+3 a^4 (225 A-539 B+275 C)-6 a^3 b (505 A-209 B+660 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}}}{3465 a^3 d \sqrt{\sec (c+d x)}}-\frac{2 \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3465 a^2 d}+\frac{2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3465 a d}+\frac{2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{2 (5 A b+11 a B) (a+b \cos (c+d x))^{3/2} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}\\ \end{align*}

Mathematica [A]  time = 21.6258, size = 959, normalized size = 1.36 \[ \frac{2 \sqrt{\frac{1}{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )}} \left (-(a+b) \left (1617 B a^5+15 b (247 A+319 C) a^4+3069 b^2 B a^3+15 b^3 (17 A+33 C) a^2-110 b^4 B a+40 A b^5\right ) E\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{b-a}{a+b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{\frac{a \tan ^2\left (\frac{1}{2} (c+d x)\right )-b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}} \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )+a (a+b) \left (3 (225 A+539 B+275 C) a^4+6 b (505 A+209 B+660 C) a^3+15 b^2 (19 A+121 B+33 C) a^2-10 b^3 (3 A+11 B) a+40 A b^4\right ) F\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{b-a}{a+b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{\frac{a \tan ^2\left (\frac{1}{2} (c+d x)\right )-b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}} \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )+\left (1617 B a^5+15 b (247 A+319 C) a^4+3069 b^2 B a^3+15 b^3 (17 A+33 C) a^2-110 b^4 B a+40 A b^5\right ) \tan \left (\frac{1}{2} (c+d x)\right ) \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )-1\right ) \left (a \tan ^2\left (\frac{1}{2} (c+d x)\right )-b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b\right )\right )}{3465 a^3 d \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )^{3/2} \sqrt{\frac{a \tan ^2\left (\frac{1}{2} (c+d x)\right )-b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}}}+\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left (\frac{2}{99} \left (11 B \sin (c+d x) a^2+23 A b \sin (c+d x) a\right ) \sec ^4(c+d x)+\frac{2}{11} a^2 A \tan (c+d x) \sec ^4(c+d x)+\frac{2}{693} \left (81 A \sin (c+d x) a^2+99 C \sin (c+d x) a^2+209 b B \sin (c+d x) a+113 A b^2 \sin (c+d x)\right ) \sec ^3(c+d x)+\frac{2 \left (539 B \sin (c+d x) a^3+1145 A b \sin (c+d x) a^2+1485 b C \sin (c+d x) a^2+825 b^2 B \sin (c+d x) a+15 A b^3 \sin (c+d x)\right ) \sec ^2(c+d x)}{3465 a}+\frac{2 \left (675 A \sin (c+d x) a^4+825 C \sin (c+d x) a^4+1793 b B \sin (c+d x) a^3+1025 A b^2 \sin (c+d x) a^2+1485 b^2 C \sin (c+d x) a^2+55 b^3 B \sin (c+d x) a-20 A b^4 \sin (c+d x)\right ) \sec (c+d x)}{3465 a^2}+\frac{2 \left (1617 B a^5+3705 A b a^4+4785 b C a^4+3069 b^2 B a^3+255 A b^3 a^2+495 b^3 C a^2-110 b^4 B a+40 A b^5\right ) \sin (c+d x)}{3465 a^3}\right )}{d} \]

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)*Sec[c + d*x]^(13/2),x]

[Out]

(2*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17
*A + 33*C) + 15*a^4*b*(247*A + 319*C))*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)*(a + b + a*Tan[(c + d*x)/2]^
2 - b*Tan[(c + d*x)/2]^2) - (a + b)*(40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A +
33*C) + 15*a^4*b*(247*A + 319*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x
)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + a*(a +
b)*(40*A*b^4 - 10*a*b^3*(3*A + 11*B) + 15*a^2*b^2*(19*A + 121*B + 33*C) + 3*a^4*(225*A + 539*B + 275*C) + 6*a^
3*b*(505*A + 209*B + 660*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2
]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(3465*a^3*d*(
1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]
^2)]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B
+ 3069*a^3*b^2*B - 110*a*b^4*B + 4785*a^4*b*C + 495*a^2*b^3*C)*Sin[c + d*x])/(3465*a^3) + (2*Sec[c + d*x]^4*(
23*a*A*b*Sin[c + d*x] + 11*a^2*B*Sin[c + d*x]))/99 + (2*Sec[c + d*x]^3*(81*a^2*A*Sin[c + d*x] + 113*A*b^2*Sin[
c + d*x] + 209*a*b*B*Sin[c + d*x] + 99*a^2*C*Sin[c + d*x]))/693 + (2*Sec[c + d*x]^2*(1145*a^2*A*b*Sin[c + d*x]
+ 15*A*b^3*Sin[c + d*x] + 539*a^3*B*Sin[c + d*x] + 825*a*b^2*B*Sin[c + d*x] + 1485*a^2*b*C*Sin[c + d*x]))/(34
65*a) + (2*Sec[c + d*x]*(675*a^4*A*Sin[c + d*x] + 1025*a^2*A*b^2*Sin[c + d*x] - 20*A*b^4*Sin[c + d*x] + 1793*a
^3*b*B*Sin[c + d*x] + 55*a*b^3*B*Sin[c + d*x] + 825*a^4*C*Sin[c + d*x] + 1485*a^2*b^2*C*Sin[c + d*x]))/(3465*a
^2) + (2*a^2*A*Sec[c + d*x]^4*Tan[c + d*x])/11))/d

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Maple [B]  time = 0.911, size = 7237, normalized size = 10.3 \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+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(13/2),x)

[Out]

result too large to display

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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} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sec \left (d x + c\right )^{\frac{13}{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)*sec(d*x+c)^(13/2),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)*sec(d*x + c)^(13/2), 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} +{\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} \sec \left (d x + c\right )^{\frac{13}{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)*sec(d*x+c)^(13/2),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)*sec(d*x + c)^(13/2), 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)*sec(d*x+c)**(13/2),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+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(13/2),x, algorithm="giac")

[Out]

Timed out