3.1426 \(\int (a+b \cos (c+d x))^{5/2} (A+C \cos ^2(c+d x)) \sec ^{\frac{13}{2}}(c+d x) \, dx\)

Optimal. Leaf size=627 \[ \frac{2 \left (3 a^2 (9 A+11 C)+5 A b^2\right ) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{231 d}+\frac{2 b \left (a^2 (229 A+297 C)+3 A b^2\right ) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{693 a d}-\frac{2 \left (-a^2 b^2 (205 A+297 C)-15 a^4 (9 A+11 C)+4 A b^4\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{693 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left (3 a^2 b^2 (19 A+33 C)-6 a^3 b (101 A+132 C)+15 a^4 (9 A+11 C)+6 a A b^3+8 A b^4\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{693 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 b (a-b) \sqrt{a+b} \left (3 a^2 b^2 (17 A+33 C)+a^4 (741 A+957 C)+8 A b^4\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{693 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{11 d}+\frac{10 A b \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{99 d} \]

[Out]

(2*(a - b)*b*Sqrt[a + b]*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Sqrt[Cos[c + d*x]]*Csc[c +
d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*
(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(693*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)
*Sqrt[a + b]*(6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11*C) + 3*a^2*b^2*(19*A + 33*C) - 6*a^3*b*(101*A + 132*C))*S
qrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -(
(a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(693*a^3*d*Sqrt[S
ec[c + d*x]]) - (2*(4*A*b^4 - 15*a^4*(9*A + 11*C) - 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])/(693*a^2*d) + (2*b*(3*A*b^2 + a^2*(229*A + 297*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d
*x]^(5/2)*Sin[c + d*x])/(693*a*d) + (2*(5*A*b^2 + 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) + (10*A*b*(a + b*Cos[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 = 2.61462, antiderivative size = 627, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.189, Rules used = {4221, 3048, 3047, 3055, 2998, 2816, 2994} \[ \frac{2 \left (3 a^2 (9 A+11 C)+5 A b^2\right ) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{231 d}+\frac{2 b \left (a^2 (229 A+297 C)+3 A b^2\right ) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{693 a d}-\frac{2 \left (-a^2 b^2 (205 A+297 C)-15 a^4 (9 A+11 C)+4 A b^4\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{693 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left (3 a^2 b^2 (19 A+33 C)-6 a^3 b (101 A+132 C)+15 a^4 (9 A+11 C)+6 a A b^3+8 A b^4\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{693 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 b (a-b) \sqrt{a+b} \left (3 a^2 b^2 (17 A+33 C)+a^4 (741 A+957 C)+8 A b^4\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{693 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{11 d}+\frac{10 A b \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{99 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]

[Out]

(2*(a - b)*b*Sqrt[a + b]*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Sqrt[Cos[c + d*x]]*Csc[c +
d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*
(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(693*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)
*Sqrt[a + b]*(6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11*C) + 3*a^2*b^2*(19*A + 33*C) - 6*a^3*b*(101*A + 132*C))*S
qrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -(
(a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(693*a^3*d*Sqrt[S
ec[c + d*x]]) - (2*(4*A*b^4 - 15*a^4*(9*A + 11*C) - 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])/(693*a^2*d) + (2*b*(3*A*b^2 + a^2*(229*A + 297*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d
*x]^(5/2)*Sin[c + d*x])/(693*a*d) + (2*(5*A*b^2 + 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) + (10*A*b*(a + b*Cos[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 3048

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C + 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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n +
2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(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, 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 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+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+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{5 A b}{2}+\frac{1}{2} a (9 A+11 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{10 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+3 a^2 (9 A+11 C)\right )+\frac{1}{2} a b (76 A+99 C) \cos (c+d x)+\frac{1}{4} b^2 (56 A+99 C) \cos ^2(c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 \left (5 A b^2+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{10 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{5}{8} b \left (3 A b^2+a^2 (229 A+297 C)\right )+\frac{1}{8} a \left (45 a^2 (9 A+11 C)+b^2 (1531 A+2079 C)\right ) \cos (c+d x)+\frac{1}{8} b \left (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 b \left (3 A b^2+a^2 (229 A+297 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 a d}+\frac{2 \left (5 A b^2+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{10 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{15}{16} \left (4 A b^4-15 a^4 (9 A+11 C)-a^2 b^2 (205 A+297 C)\right )+\frac{5}{16} a b \left (3 a^2 (337 A+429 C)+b^2 (461 A+693 C)\right ) \cos (c+d x)+\frac{5}{8} b^2 \left (3 A b^2+a^2 (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 (4 A b^4-15 a^4 (9 A+11 C)-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)}{693 a^2 d}+\frac{2 b \left (3 A b^2+a^2 (229 A+297 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 a d}+\frac{2 \left (5 A b^2+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{10 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{15}{32} b \left (8 A b^4+3 a^2 b^2 (17 A+33 C)+a^4 (741 A+957 C)\right )+\frac{15}{32} a \left (2 A b^4+15 a^4 (9 A+11 C)+3 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 (4 A b^4-15 a^4 (9 A+11 C)-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)}{693 a^2 d}+\frac{2 b \left (3 A b^2+a^2 (229 A+297 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 a d}+\frac{2 \left (5 A b^2+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{10 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 ((a-b) \left (6 a A b^3+8 A b^4+15 a^4 (9 A+11 C)+3 a^2 b^2 (19 A+33 C)-6 a^3 b (101 A+132 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}{693 a^2}+\frac{\left (b \left (8 A b^4+3 a^2 b^2 (17 A+33 C)+a^4 (741 A+957 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}{693 a^2}\\ &=\frac{2 (a-b) b \sqrt{a+b} \left (8 A b^4+3 a^2 b^2 (17 A+33 C)+a^4 (741 A+957 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}}}{693 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left (6 a A b^3+8 A b^4+15 a^4 (9 A+11 C)+3 a^2 b^2 (19 A+33 C)-6 a^3 b (101 A+132 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}}}{693 a^3 d \sqrt{\sec (c+d x)}}-\frac{2 \left (4 A b^4-15 a^4 (9 A+11 C)-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)}{693 a^2 d}+\frac{2 b \left (3 A b^2+a^2 (229 A+297 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 a d}+\frac{2 \left (5 A b^2+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{10 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 [B]  time = 26.1459, size = 3885, normalized size = 6.2 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]

[Out]

(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*b*(741*a^4*A + 51*a^2*A*b^2 + 8*A*b^4 + 957*a^4*C + 99*a^2*b^
2*C)*Sin[c + d*x])/(693*a^3) + (2*Sec[c + d*x]^3*(81*a^2*A*Sin[c + d*x] + 113*A*b^2*Sin[c + d*x] + 99*a^2*C*Si
n[c + d*x]))/693 + (2*Sec[c + d*x]^2*(229*a^2*A*b*Sin[c + d*x] + 3*A*b^3*Sin[c + d*x] + 297*a^2*b*C*Sin[c + d*
x]))/(693*a) + (2*Sec[c + d*x]*(135*a^4*A*Sin[c + d*x] + 205*a^2*A*b^2*Sin[c + d*x] - 4*A*b^4*Sin[c + d*x] + 1
65*a^4*C*Sin[c + d*x] + 297*a^2*b^2*C*Sin[c + d*x]))/(693*a^2) + (46*a*A*b*Sec[c + d*x]^3*Tan[c + d*x])/99 + (
2*a^2*A*Sec[c + d*x]^4*Tan[c + d*x])/11))/d + (2*((-247*a^2*A*b)/(231*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*
x]]) - (17*A*b^3)/(231*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*A*b^5)/(693*a^2*Sqrt[a + b*Cos[c + d*
x]]*Sqrt[Sec[c + d*x]]) - (29*a^2*b*C)/(21*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (b^3*C)/(7*Sqrt[a +
b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (15*a^3*A*Sqrt[Sec[c + d*x]])/(77*Sqrt[a + b*Cos[c + d*x]]) - (26*a*A*b^
2*Sqrt[Sec[c + d*x]])/(231*Sqrt[a + b*Cos[c + d*x]]) - (7*A*b^4*Sqrt[Sec[c + d*x]])/(99*a*Sqrt[a + b*Cos[c + d
*x]]) - (8*A*b^6*Sqrt[Sec[c + d*x]])/(693*a^3*Sqrt[a + b*Cos[c + d*x]]) + (5*a^3*C*Sqrt[Sec[c + d*x]])/(21*Sqr
t[a + b*Cos[c + d*x]]) - (2*a*b^2*C*Sqrt[Sec[c + d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) - (b^4*C*Sqrt[Sec[c + d*
x]])/(7*a*Sqrt[a + b*Cos[c + d*x]]) - (247*a*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(231*Sqrt[a + b*Cos[c
+ d*x]]) - (17*A*b^4*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(231*a*Sqrt[a + b*Cos[c + d*x]]) - (8*A*b^6*Cos[2*(c
+ d*x)]*Sqrt[Sec[c + d*x]])/(693*a^3*Sqrt[a + b*Cos[c + d*x]]) - (29*a*b^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*
x]])/(21*Sqrt[a + b*Cos[c + d*x]]) - (b^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(7*a*Sqrt[a + b*Cos[c + d*x]]
))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*b*(a + b)*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C
))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[Arc
Sin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11*C) + 3*a^2*b^2
*(19*A + 33*C) + a^3*(606*A*b + 792*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a
+ b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - b*(8*A*b^4 + 3*a^2*b^2*(17*A
+ 33*C) + a^4*(741*A + 957*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(693*a
^3*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*((b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*
(-2*b*(a + b)*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*
Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]
+ 2*a*(a + b)*(-6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11*C) + 3*a^2*b^2*(19*A + 33*C) + a^3*(606*A*b + 792*b*C))
*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSi
n[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - b*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Cos[c + d
*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(693*a^3*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c
+ d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*b*(a + b)*(8*A*b^4 + 3*a^2*b^2*(1
7*A + 33*C) + a^4*(741*A + 957*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1
+ Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-6*a*A*b^3 + 8*A*b^4 +
15*a^4*(9*A + 11*C) + 3*a^2*b^2*(19*A + 33*C) + a^3*(606*A*b + 792*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])
]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)
] - b*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x
)/2]^2*Tan[(c + d*x)/2]))/(693*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/
2]^2*Sec[c + d*x]]*(-(b*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Cos[c + d*x]*(a + b*Cos[c +
d*x])*Sec[(c + d*x)/2]^4)/2 - (b*(a + b)*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Sqrt[(a + b
*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d
*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x]
)] + (a*(a + b)*(-6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11*C) + 3*a^2*b^2*(19*A + 33*C) + a^3*(606*A*b + 792*b*C
))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b
)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1
+ Cos[c + d*x])] - (b*(a + b)*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Sqrt[Cos[c + d*x]/(1 +
Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c
+ d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a
+ b)*(1 + Cos[c + d*x]))] + (a*(a + b)*(-6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11*C) + 3*a^2*b^2*(19*A + 33*C)
+ a^3*(606*A*b + 792*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/
(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 +
Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + b^2*(8*A*b^4 + 3*a^2*b^2*(17*A +
33*C) + a^4*(741*A + 957*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + b*(8*A*b^4 + 3*a
^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)
/2] - b*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d
*x)/2]^2*Tan[(c + d*x)/2]^2 + (a*(a + b)*(-6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11*C) + 3*a^2*b^2*(19*A + 33*C)
+ a^3*(606*A*b + 792*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[
c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])
- (b*(a + b)*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*
Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]
^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(693*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-
2*b*(a + b)*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sq
rt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] +
2*a*(a + b)*(-6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11*C) + 3*a^2*b^2*(19*A + 33*C) + a^3*(606*A*b + 792*b*C))*S
qrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[
Tan[(c + d*x)/2]], (-a + b)/(a + b)] - b*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Cos[c + d*x
]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2])
+ Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(693*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*S
qrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))

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Maple [B]  time = 0.62, size = 4702, normalized size = 7.5 \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)*sec(d*x+c)^(13/2),x)

[Out]

-2/693/d/a^3*(8*A*cos(d*x+c)^7*b^6-63*A*a^6-363*C*cos(d*x+c)^6*a^4*b^2+99*C*cos(d*x+c)^6*a^3*b^3-99*C*cos(d*x+
c)^6*a^2*b^4-566*A*cos(d*x+c)^5*a^5*b-140*A*cos(d*x+c)^5*a^3*b^3-4*A*cos(d*x+c)^5*a*b^5-726*C*cos(d*x+c)^5*a^5
*b-8*A*cos(d*x+c)^6*b^6+741*A*sin(d*x+c)*cos(d*x+c)^5*(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^5*b-396*C*cos(d*x+c)^5*
a^3*b^3-160*A*cos(d*x+c)^4*a^4*b^2+A*cos(d*x+c)^4*a^2*b^4-594*C*cos(d*x+c)^4*a^4*b^2-86*A*cos(d*x+c)^3*a^5*b+1
35*A*cos(d*x+c)^7*a^5*b+741*A*cos(d*x+c)^7*a^4*b^2+205*A*cos(d*x+c)^7*a^3*b^3+51*A*cos(d*x+c)^7*a^2*b^4-4*A*co
s(d*x+c)^7*a*b^5+165*C*cos(d*x+c)^7*a^5*b+957*C*cos(d*x+c)^7*a^4*b^2+297*C*cos(d*x+c)^7*a^3*b^3+99*C*cos(d*x+c
)^7*a^2*b^4+741*A*cos(d*x+c)^6*a^5*b-307*A*cos(d*x+c)^6*a^4*b^2+51*A*cos(d*x+c)^6*a^3*b^3-52*A*cos(d*x+c)^6*a^
2*b^4+8*A*cos(d*x+c)^6*a*b^5+957*C*cos(d*x+c)^6*a^5*b+135*A*cos(d*x+c)^6*a^6+165*C*cos(d*x+c)^6*a^6-99*C*cos(d
*x+c)^2*a^6-54*A*cos(d*x+c)^4*a^6-66*C*cos(d*x+c)^4*a^6-18*A*cos(d*x+c)^2*a^6-116*A*cos(d*x+c)^3*a^3*b^3-274*A
*cos(d*x+c)^2*a^4*b^2-224*A*cos(d*x+c)*a^5*b-396*C*cos(d*x+c)^3*a^5*b+663*A*sin(d*x+c)*cos(d*x+c)^5*(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^2+51*A*sin(d*x+c)*cos(d*x+c)^5*(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^3+2*A*sin(d*x+c
)*cos(d*x+c)^5*(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^4+8*A*sin(d*x+c)*cos(d*x+c)^5*(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^5-741*A*sin(d*x+c)*cos(d*x+c)^5*(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*b-741*A*sin(d*x+c)*cos(d*x+c)^5*(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^2-51*A*sin(d*x+c)*cos(d*x+c)^5*(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^3-51*A*si
n(d*x+c)*cos(d*x+c)^5*(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)*Ellipt
icE((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^2*b^4-8*A*sin(d*x+c)*cos(d*x+c)^5*(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^5+957*C*sin(d*x+c)*cos(d*x+c)^5*(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^5*b+891*C*sin(d*x+c)*cos(d*x+c)
^5*(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^2+99*C*sin(d*x+c)*cos(d*x+c)^5*(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^3-
957*C*sin(d*x+c)*cos(d*x+c)^5*(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*b-957*C*sin(d*x+c)*cos(d*x+c)^5*(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^2-99*C*sin(d*x+c)*cos(d*x+c)^5*(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^3-99*C*sin(d*x+c)*
cos(d*x+c)^5*(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+c
os(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^2*b^4+741*A*sin(d*x+c)*cos(d*x+c)^6*(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^5*b+663*A*sin(d*x+c)*cos(d*x+c)^6*(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^2+51*A*sin(d*x+c)*cos(d*x+c)^6*(co
s(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^3+2*A*sin(d*x+c)*cos(d*x+c)^6*(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^4+8*A*sin
(d*x+c)*cos(d*x+c)^6*(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)*Ellipti
cF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a*b^5-741*A*sin(d*x+c)*cos(d*x+c)^6*(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*b-741*A*sin(d*x+c)*cos(d*x+c)^6*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+co
s(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^4*b^2-51*A*sin(d*x+c)*cos(d*x+c)
^6*(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^3-51*A*sin(d*x+c)*cos(d*x+c)^6*(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^4-
8*A*sin(d*x+c)*cos(d*x+c)^6*(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^5+957*C*sin(d*x+c)*cos(d*x+c)^6*(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^5*b+891*C*sin(d*x+c)*cos(d*x+c)^6*(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^2+99*C*sin(d*x+c)*cos
(d*x+c)^6*(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^3-957*C*sin(d*x+c)*cos(d*x+c)^6*(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*b-957*C*sin(d*x+c)*cos(d*x+c)^6*(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^2-99*C*sin(d*x+c)*cos(d*x+c)^6*(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^3-99*C*sin(d*x+c)*cos(d*x+c)^6*(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^4+135*A*sin
(d*x+c)*cos(d*x+c)^6*(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)*Ellipti
cF((-1+cos(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^6-8*A*sin(d*x+c)*cos(d*x+c)^6*(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^6+165*C*sin(d*x+c)*cos(d*x+c)^6*(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^6+135*A*sin(d*x+c)*cos(d*x+c)^5*(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^6-8*A*sin(d*x+c)*cos(d*x+c)^5*(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^6+165*C*sin(d*x+c)*c
os(d*x+c)^5*(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+co
s(d*x+c))/sin(d*x+c),(-(a-b)/(a+b))^(1/2))*a^6)*cos(d*x+c)*(1/cos(d*x+c))^(13/2)/(a+b*cos(d*x+c))^(1/2)/sin(d*
x+c)

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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}} \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+C*cos(d*x+c)^2)*sec(d*x+c)^(13/2),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + 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} + 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} \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+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*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)*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+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+C*cos(d*x+c)^2)*sec(d*x+c)^(13/2),x, algorithm="giac")

[Out]

Timed out