\(\int (a+b \cos (c+d x))^{5/2} (A+C \cos ^2(c+d x)) \sec ^{\frac {13}{2}}(c+d x) \, dx\) [1426]

Optimal result

Integrand size = 37, antiderivative size = 627 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {2 (a-b) b \sqrt {a+b} \left (8 A b^4+3 a^2 b^2 (17 A+33 C)+a^4 (741 A+957 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{693 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 (a-b) \sqrt {a+b} \left (6 a A b^3+8 A b^4+15 a^4 (9 A+11 C)+3 a^2 b^2 (19 A+33 C)-6 a^3 b (101 A+132 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{693 a^3 d \sqrt {\sec (c+d x)}}-\frac {2 \left (4 A b^4-15 a^4 (9 A+11 C)-a^2 b^2 (205 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{693 a^2 d}+\frac {2 b \left (3 A b^2+a^2 (229 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 a d}+\frac {2 \left (5 A b^2+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {10 A b (a+b \cos (c+d x))^{3/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d} \]

[Out]

10/99*A*b*(a+b*cos(d*x+c))^(3/2)*sec(d*x+c)^(9/2)*sin(d*x+c)/d+2/11*A*(a+b*cos(d*x+c))^(5/2)*sec(d*x+c)^(11/2)
*sin(d*x+c)/d-2/693*(4*A*b^4-15*a^4*(9*A+11*C)-a^2*b^2*(205*A+297*C))*sec(d*x+c)^(3/2)*sin(d*x+c)*(a+b*cos(d*x
+c))^(1/2)/a^2/d+2/693*b*(3*A*b^2+a^2*(229*A+297*C))*sec(d*x+c)^(5/2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/a/d+2/
231*(5*A*b^2+3*a^2*(9*A+11*C))*sec(d*x+c)^(7/2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/d+2/693*(a-b)*b*(8*A*b^4+3*a
^2*b^2*(17*A+33*C)+a^4*(741*A+957*C))*csc(d*x+c)*EllipticE((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)*cos(d*x+c)^(1/2)*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1
/2)/a^4/d/sec(d*x+c)^(1/2)+2/693*(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))*csc(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)*cos(d*x+c)^(1/2)*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a^3/d/sec(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 2.86 (sec) , antiderivative size = 627, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {4306, 3127, 3126, 3134, 3077, 2895, 3073} \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {2 \left (3 a^2 (9 A+11 C)+5 A b^2\right ) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}{231 d}+\frac {2 b \left (a^2 (229 A+297 C)+3 A b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}{693 a d}+\frac {2 b (a-b) \sqrt {a+b} \left (a^4 (741 A+957 C)+3 a^2 b^2 (17 A+33 C)+8 A b^4\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{693 a^4 d \sqrt {\sec (c+d x)}}-\frac {2 \left (-15 a^4 (9 A+11 C)-a^2 b^2 (205 A+297 C)+4 A b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}{693 a^2 d}+\frac {2 (a-b) \sqrt {a+b} \left (15 a^4 (9 A+11 C)-6 a^3 b (101 A+132 C)+3 a^2 b^2 (19 A+33 C)+6 a A b^3+8 A b^4\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{693 a^3 d \sqrt {\sec (c+d x)}}+\frac {2 A \sin (c+d x) \sec ^{\frac {11}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{11 d}+\frac {10 A b \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{99 d} \]

[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 2895

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(
Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqrt[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]
*EllipticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2]], -(a + b)/(a - b)], x] /; Fr
eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]

Rule 3073

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]/(f*b*c^2))*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)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ
[A, B] && PosQ[(c + d)/b]

Rule 3077

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 3126

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 3127

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*Si
n[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 3134

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] + D
ist[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] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 4306

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]

Rubi steps \begin{align*} \text {integral}& = \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 (\arcsin \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) \operatorname {EllipticF}\left (\arcsin \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] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(3885\) vs. \(2(627)=1254\).

Time = 24.23 (sec) , antiderivative size = 3885, normalized size of antiderivative = 6.20 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Result too large to show} \]

[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]]*(-1/2*(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]^4) - (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 + Co
s[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])]*
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])*(-(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]
*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))

Maple [F(-1)]

Timed out.

hanged

[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]

int((a+b*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(13/2),x)

Fricas [F]

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

[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)

Sympy [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Timed out} \]

[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

Maxima [F]

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

[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)

Giac [F]

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

[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]

integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^(5/2)*sec(d*x + c)^(13/2), x)

Mupad [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{13/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]

[In]

int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(13/2)*(a + b*cos(c + d*x))^(5/2),x)

[Out]

int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(13/2)*(a + b*cos(c + d*x))^(5/2), x)