\(\int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [1209]

Optimal result

Integrand size = 43, antiderivative size = 310 \[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {8 a^4 (185 A+208 B+247 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {8 a^4 (100 A+113 B+132 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {8 a^4 (100 A+113 B+132 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {4 a^4 (5255 A+6019 B+6721 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15015 d}+\frac {2 a (8 A+13 B) \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2 (13 A+17 B+11 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 d}+\frac {4 (1355 A+1612 B+1573 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{9009 d} \]

[Out]

8/195*a^4*(185*A+208*B+247*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(
1/2))/d+8/231*a^4*(100*A+113*B+132*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/
2*c),2^(1/2))/d+4/15015*a^4*(5255*A+6019*B+6721*C)*cos(d*x+c)^(3/2)*sin(d*x+c)/d+2/143*a*(8*A+13*B)*cos(d*x+c)
^(3/2)*(a+a*cos(d*x+c))^3*sin(d*x+c)/d+2/13*A*cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^4*sin(d*x+c)/d+2/99*(13*A+17*B
+11*C)*cos(d*x+c)^(3/2)*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/d+4/9009*(1355*A+1612*B+1573*C)*cos(d*x+c)^(3/2)*(a^
4+a^4*cos(d*x+c))*sin(d*x+c)/d+8/231*a^4*(100*A+113*B+132*C)*sin(d*x+c)*cos(d*x+c)^(1/2)/d

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {4197, 3124, 3055, 3047, 3102, 2827, 2719, 2715, 2720} \[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {8 a^4 (100 A+113 B+132 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {8 a^4 (185 A+208 B+247 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {4 a^4 (5255 A+6019 B+6721 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{15015 d}+\frac {4 (1355 A+1612 B+1573 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9009 d}+\frac {8 a^4 (100 A+113 B+132 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{231 d}+\frac {2 (13 A+17 B+11 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{99 d}+\frac {2 a (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{143 d}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d} \]

[In]

Int[Cos[c + d*x]^(13/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(8*a^4*(185*A + 208*B + 247*C)*EllipticE[(c + d*x)/2, 2])/(195*d) + (8*a^4*(100*A + 113*B + 132*C)*EllipticF[(
c + d*x)/2, 2])/(231*d) + (8*a^4*(100*A + 113*B + 132*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^4*(52
55*A + 6019*B + 6721*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15015*d) + (2*a*(8*A + 13*B)*Cos[c + d*x]^(3/2)*(a +
a*Cos[c + d*x])^3*Sin[c + d*x])/(143*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(13*d)
+ (2*(13*A + 17*B + 11*C)*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(99*d) + (4*(1355*A + 1
612*B + 1573*C)*Cos[c + d*x]^(3/2)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(9009*d)

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 2827

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3055

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x
])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*
x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))
*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &
& NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 3124

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

Rule 4197

Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sec[(e_.)
+ (f_.)*(x_)] + (C_.)*sec[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[d^(m + 2), Int[(b + a*Cos[e + f*x])^m*(d*
Cos[e + f*x])^(n - m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}
, x] &&  !IntegerQ[n] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4 \left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2 \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4 \left (\frac {1}{2} a (3 A+13 C)+\frac {1}{2} a (8 A+13 B) \cos (c+d x)\right ) \, dx}{13 a} \\ & = \frac {2 a (8 A+13 B) \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {4 \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \left (\frac {1}{4} a^2 (57 A+39 B+143 C)+\frac {13}{4} a^2 (13 A+17 B+11 C) \cos (c+d x)\right ) \, dx}{143 a} \\ & = \frac {2 a (8 A+13 B) \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2 (13 A+17 B+11 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 d}+\frac {8 \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^2 \left (\frac {3}{4} a^3 (170 A+169 B+286 C)+\frac {1}{4} a^3 (1355 A+1612 B+1573 C) \cos (c+d x)\right ) \, dx}{1287 a} \\ & = \frac {2 a (8 A+13 B) \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2 (13 A+17 B+11 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 d}+\frac {4 (1355 A+1612 B+1573 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{9009 d}+\frac {16 \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x)) \left (\frac {15}{8} a^4 (509 A+559 B+715 C)+\frac {3}{8} a^4 (5255 A+6019 B+6721 C) \cos (c+d x)\right ) \, dx}{9009 a} \\ & = \frac {2 a (8 A+13 B) \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2 (13 A+17 B+11 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 d}+\frac {4 (1355 A+1612 B+1573 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{9009 d}+\frac {16 \int \sqrt {\cos (c+d x)} \left (\frac {15}{8} a^5 (509 A+559 B+715 C)+\left (\frac {15}{8} a^5 (509 A+559 B+715 C)+\frac {3}{8} a^5 (5255 A+6019 B+6721 C)\right ) \cos (c+d x)+\frac {3}{8} a^5 (5255 A+6019 B+6721 C) \cos ^2(c+d x)\right ) \, dx}{9009 a} \\ & = \frac {4 a^4 (5255 A+6019 B+6721 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15015 d}+\frac {2 a (8 A+13 B) \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2 (13 A+17 B+11 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 d}+\frac {4 (1355 A+1612 B+1573 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{9009 d}+\frac {32 \int \sqrt {\cos (c+d x)} \left (\frac {231}{8} a^5 (185 A+208 B+247 C)+\frac {585}{8} a^5 (100 A+113 B+132 C) \cos (c+d x)\right ) \, dx}{45045 a} \\ & = \frac {4 a^4 (5255 A+6019 B+6721 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15015 d}+\frac {2 a (8 A+13 B) \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2 (13 A+17 B+11 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 d}+\frac {4 (1355 A+1612 B+1573 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{9009 d}+\frac {1}{77} \left (4 a^4 (100 A+113 B+132 C)\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{195} \left (4 a^4 (185 A+208 B+247 C)\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = \frac {8 a^4 (185 A+208 B+247 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {8 a^4 (100 A+113 B+132 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {4 a^4 (5255 A+6019 B+6721 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15015 d}+\frac {2 a (8 A+13 B) \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2 (13 A+17 B+11 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 d}+\frac {4 (1355 A+1612 B+1573 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{9009 d}+\frac {1}{231} \left (4 a^4 (100 A+113 B+132 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {8 a^4 (185 A+208 B+247 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {8 a^4 (100 A+113 B+132 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {8 a^4 (100 A+113 B+132 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {4 a^4 (5255 A+6019 B+6721 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15015 d}+\frac {2 a (8 A+13 B) \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2 (13 A+17 B+11 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 d}+\frac {4 (1355 A+1612 B+1573 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{9009 d} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.54 (sec) , antiderivative size = 1416, normalized size of antiderivative = 4.57 \[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^4 \left (\sqrt {\cos (c+d x)} (1+\cos (c+d x))^4 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-\frac {(185 A+208 B+247 C) \cot (c)}{390 d}+\frac {(3764 A+4087 B+4488 C) \cos (d x) \sin (c)}{14784 d}+\frac {(15625 A+15392 B+13208 C) \cos (2 d x) \sin (2 c)}{149760 d}+\frac {(404 A+321 B+176 C) \cos (3 d x) \sin (3 c)}{9856 d}+\frac {(98 A+52 B+13 C) \cos (4 d x) \sin (4 c)}{7488 d}+\frac {(4 A+B) \cos (5 d x) \sin (5 c)}{1408 d}+\frac {A \cos (6 d x) \sin (6 c)}{3328 d}+\frac {(3764 A+4087 B+4488 C) \cos (c) \sin (d x)}{14784 d}+\frac {(15625 A+15392 B+13208 C) \cos (2 c) \sin (2 d x)}{149760 d}+\frac {(404 A+321 B+176 C) \cos (3 c) \sin (3 d x)}{9856 d}+\frac {(98 A+52 B+13 C) \cos (4 c) \sin (4 d x)}{7488 d}+\frac {(4 A+B) \cos (5 c) \sin (5 d x)}{1408 d}+\frac {A \cos (6 c) \sin (6 d x)}{3328 d}\right )-\frac {50 A (1+\cos (c+d x))^4 \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (d x-\arctan (\cot (c))) \sqrt {1-\sin (d x-\arctan (\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \sin (c) \sin (d x-\arctan (\cot (c)))} \sqrt {1+\sin (d x-\arctan (\cot (c)))}}{231 d \sqrt {1+\cot ^2(c)}}-\frac {113 B (1+\cos (c+d x))^4 \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (d x-\arctan (\cot (c))) \sqrt {1-\sin (d x-\arctan (\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \sin (c) \sin (d x-\arctan (\cot (c)))} \sqrt {1+\sin (d x-\arctan (\cot (c)))}}{462 d \sqrt {1+\cot ^2(c)}}-\frac {2 C (1+\cos (c+d x))^4 \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (d x-\arctan (\cot (c))) \sqrt {1-\sin (d x-\arctan (\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \sin (c) \sin (d x-\arctan (\cot (c)))} \sqrt {1+\sin (d x-\arctan (\cot (c)))}}{7 d \sqrt {1+\cot ^2(c)}}-\frac {37 A (1+\cos (c+d x))^4 \csc (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\arctan (\tan (c)))} \sqrt {1+\cos (d x+\arctan (\tan (c)))} \sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{156 d}-\frac {4 B (1+\cos (c+d x))^4 \csc (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\arctan (\tan (c)))} \sqrt {1+\cos (d x+\arctan (\tan (c)))} \sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{15 d}-\frac {19 C (1+\cos (c+d x))^4 \csc (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\arctan (\tan (c)))} \sqrt {1+\cos (d x+\arctan (\tan (c)))} \sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{60 d}\right ) \]

[In]

Integrate[Cos[c + d*x]^(13/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

a^4*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(-1/390*((185*A + 208*B + 247*C)*Cot[c])/d +
((3764*A + 4087*B + 4488*C)*Cos[d*x]*Sin[c])/(14784*d) + ((15625*A + 15392*B + 13208*C)*Cos[2*d*x]*Sin[2*c])/
(149760*d) + ((404*A + 321*B + 176*C)*Cos[3*d*x]*Sin[3*c])/(9856*d) + ((98*A + 52*B + 13*C)*Cos[4*d*x]*Sin[4*c
])/(7488*d) + ((4*A + B)*Cos[5*d*x]*Sin[5*c])/(1408*d) + (A*Cos[6*d*x]*Sin[6*c])/(3328*d) + ((3764*A + 4087*B
+ 4488*C)*Cos[c]*Sin[d*x])/(14784*d) + ((15625*A + 15392*B + 13208*C)*Cos[2*c]*Sin[2*d*x])/(149760*d) + ((404*
A + 321*B + 176*C)*Cos[3*c]*Sin[3*d*x])/(9856*d) + ((98*A + 52*B + 13*C)*Cos[4*c]*Sin[4*d*x])/(7488*d) + ((4*A
+ B)*Cos[5*c]*Sin[5*d*x])/(1408*d) + (A*Cos[6*c]*Sin[6*d*x])/(3328*d)) - (50*A*(1 + Cos[c + d*x])^4*Csc[c]*Hy
pergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*Sec[d*x - ArcTan[Cot[c]]]
*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin
[d*x - ArcTan[Cot[c]]]])/(231*d*Sqrt[1 + Cot[c]^2]) - (113*B*(1 + Cos[c + d*x])^4*Csc[c]*HypergeometricPFQ[{1/
4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x -
ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c
]]]])/(462*d*Sqrt[1 + Cot[c]^2]) - (2*C*(1 + Cos[c + d*x])^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d
*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqr
t[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*Sqrt[1 + C
ot[c]^2]) - (37*A*(1 + Cos[c + d*x])^4*Csc[c]*Sec[c/2 + (d*x)/2]^8*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Co
s[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos
[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[
d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(
Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(156*d) - (4*B*(1 + Cos[c +
d*x])^4*Csc[c]*Sec[c/2 + (d*x)/2]^8*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[
d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Co
s[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/S
qrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[
c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d) - (19*C*(1 + Cos[c + d*x])^4*Csc[c]*Sec[c/2 + (d*x)/
2]^8*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(
Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*
Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2
*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sq
rt[1 + Tan[c]^2]]))/(60*d))

Maple [A] (verified)

Time = 434.55 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.86

method	result	size
default	\(-\frac {8 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, a^{4} \left (-110880 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (594720 A +65520 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1345120 A -323960 B -40040 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (1667840 A +659620 B +183040 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1237490 A -713518 B -336622 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (572110 A +448448 B +322322 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-117945 A -110097 B -97383 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+19500 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-42735 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+22035 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-48048 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+25740 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-57057 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{45045 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\)	\(576\)

[In]

int(cos(d*x+c)^(13/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

-8/45045*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^4*(-110880*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x
+1/2*c)^14+(594720*A+65520*B)*sin(1/2*d*x+1/2*c)^12*cos(1/2*d*x+1/2*c)+(-1345120*A-323960*B-40040*C)*sin(1/2*d
*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(1667840*A+659620*B+183040*C)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-123749
0*A-713518*B-336622*C)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(572110*A+448448*B+322322*C)*sin(1/2*d*x+1/2*c)
^4*cos(1/2*d*x+1/2*c)+(-117945*A-110097*B-97383*C)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+19500*A*(sin(1/2*d*
x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-42735*A*(sin(1/2*d*x+
1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+22035*B*(sin(1/2*d*x+1/
2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-48048*B*(sin(1/2*d*x+1/2*
c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+25740*C*(sin(1/2*d*x+1/2*c)
^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-57057*C*(sin(1/2*d*x+1/2*c)^2
)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))/(-2*sin(1/2*d*x+1/2*c)^4+sin(1
/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.13 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.93 \[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (390 i \, \sqrt {2} {\left (100 \, A + 113 \, B + 132 \, C\right )} a^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 390 i \, \sqrt {2} {\left (100 \, A + 113 \, B + 132 \, C\right )} a^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 462 i \, \sqrt {2} {\left (185 \, A + 208 \, B + 247 \, C\right )} a^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 462 i \, \sqrt {2} {\left (185 \, A + 208 \, B + 247 \, C\right )} a^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (3465 \, A a^{4} \cos \left (d x + c\right )^{5} + 4095 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{4} + 385 \, {\left (89 \, A + 52 \, B + 13 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 585 \, {\left (80 \, A + 75 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 77 \, {\left (740 \, A + 832 \, B + 793 \, C\right )} a^{4} \cos \left (d x + c\right ) + 780 \, {\left (100 \, A + 113 \, B + 132 \, C\right )} a^{4}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{45045 \, d} \]

[In]

integrate(cos(d*x+c)^(13/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

-2/45045*(390*I*sqrt(2)*(100*A + 113*B + 132*C)*a^4*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))
- 390*I*sqrt(2)*(100*A + 113*B + 132*C)*a^4*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 462*I*
sqrt(2)*(185*A + 208*B + 247*C)*a^4*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x
+ c))) + 462*I*sqrt(2)*(185*A + 208*B + 247*C)*a^4*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x
+ c) - I*sin(d*x + c))) - (3465*A*a^4*cos(d*x + c)^5 + 4095*(4*A + B)*a^4*cos(d*x + c)^4 + 385*(89*A + 52*B +
13*C)*a^4*cos(d*x + c)^3 + 585*(80*A + 75*B + 44*C)*a^4*cos(d*x + c)^2 + 77*(740*A + 832*B + 793*C)*a^4*cos(d*
x + c) + 780*(100*A + 113*B + 132*C)*a^4)*sqrt(cos(d*x + c))*sin(d*x + c))/d

Sympy [F(-1)]

Timed out. \[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**(13/2)*(a+a*sec(d*x+c))**4*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)^(13/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{\frac {13}{2}} \,d x } \]

[In]

integrate(cos(d*x+c)^(13/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^4*cos(d*x + c)^(13/2), x)

Mupad [B] (verification not implemented)

Time = 21.05 (sec) , antiderivative size = 764, normalized size of antiderivative = 2.46 \[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]

[In]

int(cos(c + d*x)^(13/2)*(a + a/cos(c + d*x))^4*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)

[Out]

(2*(3*C*a^4*ellipticE(c/2 + (d*x)/2, 2) + 4*C*a^4*ellipticF(c/2 + (d*x)/2, 2) + 4*C*a^4*cos(c + d*x)^(1/2)*sin
(c + d*x)))/(3*d) - (8*((13*A*a^4*cos(c + d*x)^(9/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2) + (5*A*a^4*cos(c + d
*x)^(13/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2))*hypergeom([1/2, 13/4], 17/4, cos(c + d*x)^2))/(117*d) - (136*
((11*A*a^4*cos(c + d*x)^(11/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2) + (51*A*a^4*cos(c + d*x)^(15/2)*sin(c + d*
x))/(sin(c + d*x)^2)^(1/2))*hypergeom([1/2, 15/4], 23/4, cos(c + d*x)^2))/(21945*d) - (2*((66*C*a^4*cos(c + d*
x)^(7/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2) - (17*C*a^4*cos(c + d*x)^(11/2)*sin(c + d*x))/(sin(c + d*x)^2)^(
1/2))*hypergeom([1/2, 11/4], 15/4, cos(c + d*x)^2))/(77*d) - (2*hypergeom([1/2, 15/4], 19/4, cos(c + d*x)^2)*(
(165*A*a^4*cos(c + d*x)^(7/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2) + (578*A*a^4*cos(c + d*x)^(11/2)*sin(c + d*
x))/(sin(c + d*x)^2)^(1/2) - (127*A*a^4*cos(c + d*x)^(15/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2)))/(1155*d) +
(B*a^4*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d - (160*A*a^4*cos(c + d*x
)^(13/2)*sin(c + d*x)*hypergeom([1/2, 13/4], 21/4, cos(c + d*x)^2))/(663*d*(sin(c + d*x)^2)^(1/2)) - (8*B*a^4*
cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2)) - (4
*B*a^4*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(3*d*(sin(c + d*x)^2)^(1/2
)) - (8*B*a^4*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15/4, cos(c + d*x)^2))/(11*d*(sin(c + d*
x)^2)^(1/2)) - (2*B*a^4*cos(c + d*x)^(13/2)*sin(c + d*x)*hypergeom([1/2, 13/4], 17/4, cos(c + d*x)^2))/(13*d*(
sin(c + d*x)^2)^(1/2)) - (8*C*a^4*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))
/(9*d*(sin(c + d*x)^2)^(1/2)) - (208*C*a^4*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 19/4, cos(c
+ d*x)^2))/(385*d*(sin(c + d*x)^2)^(1/2))