\(\int \frac {(a+b \cos (c+d x))^{5/2} (A+C \cos ^2(c+d x))}{\cos ^{\frac {13}{2}}(c+d x)} \, dx\) [747]

Optimal result

Integrand size = 37, antiderivative size = 587 \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {2 (a-b) b \sqrt {a+b} \left (8 A b^4+3 a^2 b^2 (17 A+33 C)+a^4 (741 A+957 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{693 a^4 d}+\frac {2 (a-b) \sqrt {a+b} \left (6 a A b^3+8 A b^4+15 a^4 (9 A+11 C)+3 a^2 b^2 (19 A+33 C)-6 a^3 b (101 A+132 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{693 a^3 d}+\frac {2 \left (5 A b^2+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b \left (3 A b^2+a^2 (229 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {2 \left (4 A b^4-15 a^4 (9 A+11 C)-a^2 b^2 (205 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {10 A b (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]

[Out]

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

Rubi [A] (verified)

Time = 2.56 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {3127, 3126, 3134, 3077, 2895, 3073} \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {2 b \left (a^2 (229 A+297 C)+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{693 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (3 a^2 (9 A+11 C)+5 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b (a-b) \sqrt {a+b} \left (a^4 (741 A+957 C)+3 a^2 b^2 (17 A+33 C)+8 A b^4\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{693 a^4 d}-\frac {2 \left (-15 a^4 (9 A+11 C)-a^2 b^2 (205 A+297 C)+4 A b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{693 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (a-b) \sqrt {a+b} \left (15 a^4 (9 A+11 C)-6 a^3 b (101 A+132 C)+3 a^2 b^2 (19 A+33 C)+6 a A b^3+8 A b^4\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{693 a^3 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{99 d \cos ^{\frac {9}{2}}(c+d x)} \]

[In]

Int[((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[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))*Cot[c + d*x]*EllipticE[ArcS
in[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) + (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))*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)]*Sq
rt[(a*(1 + Sec[c + d*x]))/(a - b)])/(693*a^3*d) + (2*(5*A*b^2 + 3*a^2*(9*A + 11*C))*Sqrt[a + b*Cos[c + d*x]]*S
in[c + d*x])/(231*d*Cos[c + d*x]^(7/2)) + (2*b*(3*A*b^2 + a^2*(229*A + 297*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c
+ d*x])/(693*a*d*Cos[c + d*x]^(5/2)) - (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]]*Sin[c + d*x])/(693*a^2*d*Cos[c + d*x]^(3/2)) + (10*A*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])
/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))

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])))

Rubi steps \begin{align*} \text {integral}& = \frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2}{11} \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} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {4}{99} \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)} \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {10 A b (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {8}{693} \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 \left (5 A b^2+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b \left (3 A b^2+a^2 (229 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {10 A b (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {16 \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 (5 A b^2+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b \left (3 A b^2+a^2 (229 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a d \cos ^{\frac {5}{2}}(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)} \sin (c+d x)}{693 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {10 A b (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {32 \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 (5 A b^2+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b \left (3 A b^2+a^2 (229 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a d \cos ^{\frac {5}{2}}(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)} \sin (c+d x)}{693 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {10 A b (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\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 )\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 )\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 ) \cot (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}+\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 ) \cot (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}+\frac {2 \left (5 A b^2+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b \left (3 A b^2+a^2 (229 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a d \cos ^{\frac {5}{2}}(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)} \sin (c+d x)}{693 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {10 A b (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.98 (sec) , antiderivative size = 1591, normalized size of antiderivative = 2.71 \[ \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 {-\frac {4 a \left (135 a^6 A-78 a^4 A b^2-49 a^2 A b^4-8 A b^6+165 a^6 C-66 a^4 b^2 C-99 a^2 b^4 C\right ) \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-4 a \left (-741 a^5 A b-51 a^3 A b^3-8 a A b^5-957 a^5 b C-99 a^3 b^3 C\right ) \left (\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}\right )+2 \left (-741 a^4 A b^2-51 a^2 A b^4-8 A b^6-957 a^4 b^2 C-99 a^2 b^4 C\right ) \left (\frac {i \cos \left (\frac {1}{2} (c+d x)\right ) \sqrt {a+b \cos (c+d x)} E\left (i \text {arcsinh}\left (\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\cos (c+d x)}}\right )|-\frac {2 a}{-a-b}\right ) \sec (c+d x)}{b \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)} \sqrt {\frac {(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac {2 a \left (\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}\right )}{b}+\frac {\sqrt {a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt {\cos (c+d x)}}\right )}{693 a^3 d}+\frac {\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (\frac {2}{693} \sec ^4(c+d x) \left (81 a^2 A \sin (c+d x)+113 A b^2 \sin (c+d x)+99 a^2 C \sin (c+d x)\right )+\frac {2 \sec ^3(c+d x) \left (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)\right )}{693 a}+\frac {2 \sec ^2(c+d x) \left (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)+165 a^4 C \sin (c+d x)+297 a^2 b^2 C \sin (c+d x)\right )}{693 a^2}+\frac {2 \sec (c+d x) \left (741 a^4 A b \sin (c+d x)+51 a^2 A b^3 \sin (c+d x)+8 A b^5 \sin (c+d x)+957 a^4 b C \sin (c+d x)+99 a^2 b^3 C \sin (c+d x)\right )}{693 a^3}+\frac {46}{99} a A b \sec ^4(c+d x) \tan (c+d x)+\frac {2}{11} a^2 A \sec ^5(c+d x) \tan (c+d x)\right )}{d} \]

[In]

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

[Out]

((-4*a*(135*a^6*A - 78*a^4*A*b^2 - 49*a^2*A*b^4 - 8*A*b^6 + 165*a^6*C - 66*a^4*b^2*C - 99*a^2*b^4*C)*Sqrt[((a
+ b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d
*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sq
rt[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*(-741
*a^5*A*b - 51*a^3*A*b^3 - 8*a*A*b^5 - 957*a^5*b*C - 99*a^3*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]]) - (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*(-741*a^4*A*b^2 - 51*a^2*A*b^4 - 8*
A*b^6 - 957*a^4*b^2*C - 99*a^2*b^4*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c
+ d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[(
(a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a
+ b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*Elli
pticF[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]*E
llipticPi[-(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*S
qrt[Cos[c + d*x]])))/(693*a^3*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^4*(81*a^2*A*S
in[c + d*x] + 113*A*b^2*Sin[c + d*x] + 99*a^2*C*Sin[c + d*x]))/693 + (2*Sec[c + d*x]^3*(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]^2*(135*a^4*A*Sin[c + d*x] + 2
05*a^2*A*b^2*Sin[c + d*x] - 4*A*b^4*Sin[c + d*x] + 165*a^4*C*Sin[c + d*x] + 297*a^2*b^2*C*Sin[c + d*x]))/(693*
a^2) + (2*Sec[c + d*x]*(741*a^4*A*b*Sin[c + d*x] + 51*a^2*A*b^3*Sin[c + d*x] + 8*A*b^5*Sin[c + d*x] + 957*a^4*
b*C*Sin[c + d*x] + 99*a^2*b^3*C*Sin[c + d*x]))/(693*a^3) + (46*a*A*b*Sec[c + d*x]^4*Tan[c + d*x])/99 + (2*a^2*
A*Sec[c + d*x]^5*Tan[c + d*x])/11))/d

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(6391\) vs. \(2(537)=1074\).

Time = 45.42 (sec) , antiderivative size = 6392, normalized size of antiderivative = 10.89

method	result	size
parts	\(\text {Expression too large to display}\)	\(6392\)
default	\(\text {Expression too large to display}\)	\(6491\)

[In]

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

[Out]

result too large to display

Fricas [F]

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

[In]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

Giac [F]

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

[In]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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