\(\int \frac {(A+C \cos ^2(c+d x)) \sec ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx\) [1239]

Optimal result

Integrand size = 37, antiderivative size = 315 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\frac {(19 A+11 C) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{2 \sqrt {2} a^{3/2} d}-\frac {(1201 A+665 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}+\frac {(397 A+245 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}-\frac {(67 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A+7 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}} \]

[Out]

-1/2*(A+C)*sec(d*x+c)^(7/2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(3/2)+1/210*(397*A+245*C)*sec(d*x+c)^(3/2)*sin(d*x+c
)/a/d/(a+a*cos(d*x+c))^(1/2)-1/70*(67*A+35*C)*sec(d*x+c)^(5/2)*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^(1/2)+1/14*(11*
A+7*C)*sec(d*x+c)^(7/2)*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^(1/2)+1/4*(19*A+11*C)*arctan(1/2*sin(d*x+c)*a^(1/2)*2^
(1/2)/cos(d*x+c)^(1/2)/(a+a*cos(d*x+c))^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/a^(3/2)/d*2^(1/2)-1/210*(1201
*A+665*C)*sin(d*x+c)*sec(d*x+c)^(1/2)/a/d/(a+a*cos(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {4306, 3121, 3063, 12, 2861, 211} \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\frac {(19 A+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(11 A+7 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{14 a d \sqrt {a \cos (c+d x)+a}}-\frac {(A+C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac {(67 A+35 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{70 a d \sqrt {a \cos (c+d x)+a}}+\frac {(397 A+245 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{210 a d \sqrt {a \cos (c+d x)+a}}-\frac {(1201 A+665 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{210 a d \sqrt {a \cos (c+d x)+a}} \]

[In]

Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/(a + a*Cos[c + d*x])^(3/2),x]

[Out]

((19*A + 11*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c
+ d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((1201*A + 665*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(210*a
*d*Sqrt[a + a*Cos[c + d*x]]) + ((397*A + 245*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c + d
*x]]) - ((67*A + 35*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d
*x]^(7/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((11*A + 7*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(14*
a*d*Sqrt[a + a*Cos[c + d*x]])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2861

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> D
ist[-2*(a/f), Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c +
d*Sin[e + f*x]]))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 -
d^2, 0]

Rule 3063

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*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x]
)^(n + 1)/(f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(b*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin
[e + f*x])^(n + 1)*Simp[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n + 1)) + b*(B*c - A*d)*(m + n + 2)*Sin[e +
f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2
- d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || EqQ[m + 1/2, 0])

Rule 3121

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[a*(A + C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x
])^(n + 1)/(f*(b*c - a*d)*(2*m + 1))), x] + Dist[1/(b*(b*c - a*d)*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)
*(c + d*Sin[e + f*x])^n*Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) - C*(a*c*m + b*d*(n + 1)) + (a*A*d*(m + n + 2
) + C*(b*c*(2*m + 1) - a*d*(m - n - 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] &&
NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)]

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+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx \\ & = -\frac {(A+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{2} a (11 A+7 C)-2 a (2 A+C) \cos (c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{2 a^2} \\ & = -\frac {(A+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A+7 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{4} a^2 (67 A+35 C)+\frac {3}{2} a^2 (11 A+7 C) \cos (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{7 a^3} \\ & = -\frac {(67 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A+7 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{8} a^3 (397 A+245 C)-\frac {1}{2} a^3 (67 A+35 C) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{35 a^4} \\ & = \frac {(397 A+245 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}-\frac {(67 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A+7 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{16} a^4 (1201 A+665 C)+\frac {1}{8} a^4 (397 A+245 C) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{105 a^5} \\ & = -\frac {(1201 A+665 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}+\frac {(397 A+245 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}-\frac {(67 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A+7 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {105 a^5 (19 A+11 C)}{32 \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{105 a^6} \\ & = -\frac {(1201 A+665 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}+\frac {(397 A+245 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}-\frac {(67 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A+7 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {\left ((19 A+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{4 a} \\ & = -\frac {(1201 A+665 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}+\frac {(397 A+245 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}-\frac {(67 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A+7 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}-\frac {\left ((19 A+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{2 a^2+a x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{2 d} \\ & = \frac {(19 A+11 C) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{2 \sqrt {2} a^{3/2} d}-\frac {(1201 A+665 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}+\frac {(397 A+245 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}-\frac {(67 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A+7 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 8.31 (sec) , antiderivative size = 3122, normalized size of antiderivative = 9.91 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\text {Result too large to show} \]

[In]

Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/(a + a*Cos[c + d*x])^(3/2),x]

[Out]

(2*Cos[c/2 + (d*x)/2]^3*Sqrt[(1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]*((4*C*Sin[c/2
+ (d*x)/2])/(7*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)) - ((A + C)*(1 - 2*Sin[c/2 + (d*x)/2]))/(28*(1 + Sin[c/2 +
(d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)) + ((A + C)*(1 + 2*Sin[c/2 + (d*x)/2]))/(28*(1 - Sin[c/2 + (d*x)/
2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)) + (24*C*Sin[c/2 + (d*x)/2])/(35*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) +
(32*C*Sin[c/2 + (d*x)/2])/(35*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + (64*C*Sin[c/2 + (d*x)/2])/(35*Sqrt[1 - 2*S
in[c/2 + (d*x)/2]^2]) - ((A + C)*(315*ArcTan[(1 - 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] + (5
+ 3*Sin[c/2 + (d*x)/2])/((1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) - (11 + 17*Sin[c/2 + (d
*x)/2])/((1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + (61 + 71*Sin[c/2 + (d*x)/2])/((1 - Sin
[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) + (193*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 - Sin[c/2 + (d*
x)/2])))/70 + ((A + C)*(315*ArcTan[(1 + 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] + (5 - 3*Sin[c
/2 + (d*x)/2])/((1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) - (11 - 17*Sin[c/2 + (d*x)/2])/((
1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + (61 - 71*Sin[c/2 + (d*x)/2])/((1 + Sin[c/2 + (d*
x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) + (193*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 + Sin[c/2 + (d*x)/2])))/7
0 - ((-A + 7*C)*Csc[c/2 + (d*x)/2]^9*(363825*Sin[c/2 + (d*x)/2]^2 - 4729725*Sin[c/2 + (d*x)/2]^4 + 26785605*Si
n[c/2 + (d*x)/2]^6 - 86790165*Sin[c/2 + (d*x)/2]^8 + 177677808*Sin[c/2 + (d*x)/2]^10 - 239283044*Sin[c/2 + (d*
x)/2]^12 + 52080*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2
+ (d*x)/2]^12 + 560*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/
2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 213120160*Sin[c/2 + (d*x)/2]^14 - 168280*Hypergeometric2F1[2, 11/2, 1
3/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 2240*HypergeometricPFQ[{2, 2,
2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 121
497024*Sin[c/2 + (d*x)/2]^16 + 212520*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 +
(d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 3360*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*
x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 40125184*Sin[c/2 + (d*x)/2]^18 - 124320*Hyperge
ometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^18 - 2240*Hyp
ergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/
2 + (d*x)/2]^18 - 5840384*Sin[c/2 + (d*x)/2]^20 + 28000*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/
(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^20 + 560*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/
2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^20 + 363825*ArcTanh[Sqrt[Sin[c/2 +
(d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 5336100*
ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^2*Sqrt[Sin[c/2 + (d*x)/2]
^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 34636140*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]
*Sin[c/2 + (d*x)/2]^4*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 131060160*ArcTanh[Sqrt[Sin[c/
2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^6*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2
+ (d*x)/2]^2)] + 320535600*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2
]^8*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 530671680*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1
+ 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^10*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] +
604296000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^12*Sqrt[Sin[c/2
+ (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 468948480*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (
d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^14*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 237726720*ArcTan
h[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^16*Sqrt[Sin[c/2 + (d*x)/2]^2/(-
1 + 2*Sin[c/2 + (d*x)/2]^2)] - 70963200*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[
c/2 + (d*x)/2]^18*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 9461760*ArcTanh[Sqrt[Sin[c/2 + (d
*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^20*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*
x)/2]^2)] - 1120*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 11/2}, {1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1
+ 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12*(-6 + 5*Sin[c/2 + (d*x)/2]^2) + 280*Cos[(c + d*x)/2]^4*Hyperg
eometricPFQ[{2, 2, 11/2}, {1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12
*(103 - 164*Sin[c/2 + (d*x)/2]^2 + 70*Sin[c/2 + (d*x)/2]^4)))/(80850*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(9/2)*(-1 +
2*Sin[c/2 + (d*x)/2]^2))))/(d*(a*(1 + Cos[c + d*x]))^(3/2))

Maple [A] (verified)

Time = 1.25 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.51

method	result	size
default	\(-\frac {\sqrt {2}\, \left (\sec ^{\frac {9}{2}}\left (d x +c \right )\right ) \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (1995 A \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1155 C \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+3990 A \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1201 A \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}+2310 C \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+665 C \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}+1995 A \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+804 A \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}+1155 C \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+420 C \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}-196 A \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}-140 C \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}+36 A \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}-60 A \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {2}\right )}{420 a^{2} d \left (1+\cos \left (d x +c \right )\right )^{2}}\)	\(477\)
parts	\(-\frac {A \sqrt {2}\, \left (\sec ^{\frac {9}{2}}\left (d x +c \right )\right ) \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (1995 \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )+1201 \sqrt {2}\, \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3990 \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )+804 \sqrt {2}\, \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )+1995 \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )-196 \sqrt {2}\, \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+36 \sqrt {2}\, \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-60 \sqrt {2}\, \cos \left (d x +c \right ) \sin \left (d x +c \right )\right )}{420 d \,a^{2} \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {C \sqrt {2}\, \left (\sec ^{\frac {9}{2}}\left (d x +c \right )\right ) \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (33 \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )+66 \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )+19 \sqrt {2}\, \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )+33 \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )+12 \sqrt {2}\, \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )-4 \sqrt {2}\, \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )\right )}{12 d \,a^{2} \left (1+\cos \left (d x +c \right )\right )^{2}}\)	\(508\)

[In]

int((A+C*cos(d*x+c)^2)*sec(d*x+c)^(9/2)/(a+a*cos(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/420/a^2/d*2^(1/2)*sec(d*x+c)^(9/2)*((1+cos(d*x+c))*a)^(1/2)/(1+cos(d*x+c))^2*(1995*A*arcsin(cot(d*x+c)-csc(
d*x+c))*cos(d*x+c)^6*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+1155*C*arcsin(cot(d*x+c)-csc(d*x+c))*cos(d*x+c)^6*(cos(
d*x+c)/(1+cos(d*x+c)))^(1/2)+3990*A*arcsin(cot(d*x+c)-csc(d*x+c))*cos(d*x+c)^5*(cos(d*x+c)/(1+cos(d*x+c)))^(1/
2)+1201*A*cos(d*x+c)^5*sin(d*x+c)*2^(1/2)+2310*C*arcsin(cot(d*x+c)-csc(d*x+c))*cos(d*x+c)^5*(cos(d*x+c)/(1+cos
(d*x+c)))^(1/2)+665*C*cos(d*x+c)^5*sin(d*x+c)*2^(1/2)+1995*A*cos(d*x+c)^4*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ar
csin(cot(d*x+c)-csc(d*x+c))+804*A*cos(d*x+c)^4*sin(d*x+c)*2^(1/2)+1155*C*cos(d*x+c)^4*(cos(d*x+c)/(1+cos(d*x+c
)))^(1/2)*arcsin(cot(d*x+c)-csc(d*x+c))+420*C*cos(d*x+c)^4*sin(d*x+c)*2^(1/2)-196*A*cos(d*x+c)^3*sin(d*x+c)*2^
(1/2)-140*C*cos(d*x+c)^3*sin(d*x+c)*2^(1/2)+36*A*cos(d*x+c)^2*sin(d*x+c)*2^(1/2)-60*A*cos(d*x+c)*sin(d*x+c)*2^
(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.73 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=-\frac {105 \, \sqrt {2} {\left ({\left (19 \, A + 11 \, C\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (19 \, A + 11 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (19 \, A + 11 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) + \frac {2 \, {\left ({\left (1201 \, A + 665 \, C\right )} \cos \left (d x + c\right )^{4} + 12 \, {\left (67 \, A + 35 \, C\right )} \cos \left (d x + c\right )^{3} - 28 \, {\left (7 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{2} + 36 \, A \cos \left (d x + c\right ) - 60 \, A\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{420 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} + 2 \, a^{2} d \cos \left (d x + c\right )^{4} + a^{2} d \cos \left (d x + c\right )^{3}\right )}} \]

[In]

integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^(9/2)/(a+a*cos(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

-1/420*(105*sqrt(2)*((19*A + 11*C)*cos(d*x + c)^5 + 2*(19*A + 11*C)*cos(d*x + c)^4 + (19*A + 11*C)*cos(d*x + c
)^3)*sqrt(a)*arctan(sqrt(2)*sqrt(a*cos(d*x + c) + a)*sqrt(cos(d*x + c))/(sqrt(a)*sin(d*x + c))) + 2*((1201*A +
665*C)*cos(d*x + c)^4 + 12*(67*A + 35*C)*cos(d*x + c)^3 - 28*(7*A + 5*C)*cos(d*x + c)^2 + 36*A*cos(d*x + c) -
60*A)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/sqrt(cos(d*x + c)))/(a^2*d*cos(d*x + c)^5 + 2*a^2*d*cos(d*x + c)^
4 + a^2*d*cos(d*x + c)^3)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate((A+C*cos(d*x+c)**2)*sec(d*x+c)**(9/2)/(a+a*cos(d*x+c))**(3/2),x)

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^(9/2)/(a+a*cos(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{\frac {9}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^(9/2)/(a+a*cos(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + A)*sec(d*x + c)^(9/2)/(a*cos(d*x + c) + a)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]

[In]

int(((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(9/2))/(a + a*cos(c + d*x))^(3/2),x)

[Out]

int(((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(9/2))/(a + a*cos(c + d*x))^(3/2), x)