\(\int \frac {(a+b \sec (c+d x)) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx\) [989]

Optimal result

Integrand size = 41, antiderivative size = 266 \[ \int \frac {(a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2 (7 A b+7 a B+9 b C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {10 (9 a A+11 b B+11 a C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 a A \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (A b+a B) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (9 a A+11 b B+11 a C) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (7 A b+7 a B+9 b C) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {10 (9 a A+11 b B+11 a C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}} \]

[Out]

2/11*a*A*sin(d*x+c)/d/sec(d*x+c)^(9/2)+2/9*(A*b+B*a)*sin(d*x+c)/d/sec(d*x+c)^(7/2)+2/77*(9*A*a+11*B*b+11*C*a)*
sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/45*(7*A*b+7*B*a+9*C*b)*sin(d*x+c)/d/sec(d*x+c)^(3/2)+10/231*(9*A*a+11*B*b+11*C
*a)*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2/15*(7*A*b+7*B*a+9*C*b)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ell
ipticE(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+10/231*(9*A*a+11*B*b+11*C*a)*(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))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/
d

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4159, 4132, 3854, 3856, 2720, 4130, 2719} \[ \int \frac {(a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2 \sin (c+d x) (7 a B+7 A b+9 b C)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) (9 a A+11 a C+11 b B)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {10 \sin (c+d x) (9 a A+11 a C+11 b B)}{231 d \sqrt {\sec (c+d x)}}+\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) (9 a A+11 a C+11 b B)}{231 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (7 a B+7 A b+9 b C)}{15 d}+\frac {2 (a B+A b) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a A \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)} \]

[In]

Int[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]

[Out]

(2*(7*A*b + 7*a*B + 9*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (10*(9*a*
A + 11*b*B + 11*a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a*A*Sin[c +
d*x])/(11*d*Sec[c + d*x]^(9/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(9*a*A + 11*b*B
+ 11*a*C)*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (2*(7*A*b + 7*a*B + 9*b*C)*Sin[c + d*x])/(45*d*Sec[c + d*x
]^(3/2)) + (10*(9*a*A + 11*b*B + 11*a*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])

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 3854

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Csc[c + d*x])^(n + 1)/(b*d*n)), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[2*n]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 4130

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[A*Cot[e
+ f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + Dist[(C*m + A*(m + 1))/(b^2*m), Int[(b*Csc[e + f*x])^(m + 2), x], x] /
; FreeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]

Rule 4132

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(
C_.)), x_Symbol] :> Dist[B/b, Int[(b*Csc[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x
]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, x]

Rule 4159

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x]
+ Dist[1/(d*n), Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (n*(a*C + B*b) + A*a*(n + 1))*Csc[e + f*x]
+ b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {2 a A \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {2}{11} \int \frac {-\frac {11}{2} (A b+a B)-\frac {1}{2} (9 a A+11 b B+11 a C) \sec (c+d x)-\frac {11}{2} b C \sec ^2(c+d x)}{\sec ^{\frac {9}{2}}(c+d x)} \, dx \\ & = \frac {2 a A \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {2}{11} \int \frac {-\frac {11}{2} (A b+a B)-\frac {11}{2} b C \sec ^2(c+d x)}{\sec ^{\frac {9}{2}}(c+d x)} \, dx-\frac {1}{11} (-9 a A-11 b B-11 a C) \int \frac {1}{\sec ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 a A \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (A b+a B) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (9 a A+11 b B+11 a C) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{77} (5 (9 a A+11 b B+11 a C)) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx-\frac {1}{9} (-7 A b-7 a B-9 b C) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {2 a A \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (A b+a B) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (9 a A+11 b B+11 a C) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (7 A b+7 a B+9 b C) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {10 (9 a A+11 b B+11 a C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {1}{231} (5 (9 a A+11 b B+11 a C)) \int \sqrt {\sec (c+d x)} \, dx-\frac {1}{15} (-7 A b-7 a B-9 b C) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx \\ & = \frac {2 a A \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (A b+a B) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (9 a A+11 b B+11 a C) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (7 A b+7 a B+9 b C) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {10 (9 a A+11 b B+11 a C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {1}{231} \left (5 (9 a A+11 b B+11 a C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx-\frac {1}{15} \left ((-7 A b-7 a B-9 b C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = \frac {2 (7 A b+7 a B+9 b C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {10 (9 a A+11 b B+11 a C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 a A \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (A b+a B) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (9 a A+11 b B+11 a C) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (7 A b+7 a B+9 b C) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {10 (9 a A+11 b B+11 a C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 6.42 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {e^{-i d x} \sqrt {\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (1200 (9 a A+11 b B+11 a C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-1232 i (7 A b+7 a B+9 b C) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+\cos (c+d x) (25872 i A b+25872 i a B+33264 i b C+30 (435 a A+506 b B+506 a C) \sin (c+d x)+308 (19 A b+19 a B+18 b C) \sin (2 (c+d x))+2565 a A \sin (3 (c+d x))+1980 b B \sin (3 (c+d x))+1980 a C \sin (3 (c+d x))+770 A b \sin (4 (c+d x))+770 a B \sin (4 (c+d x))+315 a A \sin (5 (c+d x)))\right )}{27720 d} \]

[In]

Integrate[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]

[Out]

(Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(1200*(9*a*A + 11*b*B + 11*a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c +
d*x)/2, 2] - (1232*I)*(7*A*b + 7*a*B + 9*b*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[
1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((25872*I)*A*b + (25872*I)*a*B + (33264*I)*b*C + 30*(435*a
*A + 506*b*B + 506*a*C)*Sin[c + d*x] + 308*(19*A*b + 19*a*B + 18*b*C)*Sin[2*(c + d*x)] + 2565*a*A*Sin[3*(c + d
*x)] + 1980*b*B*Sin[3*(c + d*x)] + 1980*a*C*Sin[3*(c + d*x)] + 770*A*b*Sin[4*(c + d*x)] + 770*a*B*Sin[4*(c + d
*x)] + 315*a*A*Sin[5*(c + d*x)])))/(27720*d*E^(I*d*x))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(610\) vs. \(2(286)=572\).

Time = 16.64 (sec) , antiderivative size = 611, normalized size of antiderivative = 2.30

method	result	size
default	\(-\frac {2 \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}}\, \left (20160 a A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (-50400 a A -12320 A b -12320 B a \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (56880 a A +24640 A b +24640 B a +7920 B b +7920 C a \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-34920 a A -22792 A b -22792 B a -11880 B b -11880 C a -5544 C b \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (13860 a A +10472 A b +10472 B a +9240 B b +9240 C a +5544 C b \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-2790 a A -1848 A b -1848 B a -2640 B b -2640 C a -1386 C b \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+675 a 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 )-1617 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 ) b +825 B 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 )-1617 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 ) a +825 C 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 )-2079 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 ) b \right )}{3465 \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}\)	\(611\)
parts	\(\text {Expression too large to display}\)	\(854\)

[In]

int((a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(11/2),x,method=_RETURNVERBOSE)

[Out]

-2/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(20160*a*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*
c)^12+(-50400*A*a-12320*A*b-12320*B*a)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(56880*A*a+24640*A*b+24640*B*a
+7920*B*b+7920*C*a)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-34920*A*a-22792*A*b-22792*B*a-11880*B*b-11880*C*
a-5544*C*b)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(13860*A*a+10472*A*b+10472*B*a+9240*B*b+9240*C*a+5544*C*b)
*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-2790*A*a-1848*A*b-1848*B*a-2640*B*b-2640*C*a-1386*C*b)*sin(1/2*d*x+
1/2*c)^2*cos(1/2*d*x+1/2*c)+675*a*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(co
s(1/2*d*x+1/2*c),2^(1/2))-1617*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))*b+825*B*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))-1617*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))*a+825*C*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))-2079*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))*b)/(-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 = 0.14 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.05 \[ \int \frac {(a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {75 \, \sqrt {2} {\left (i \, {\left (9 \, A + 11 \, C\right )} a + 11 i \, B b\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 75 \, \sqrt {2} {\left (-i \, {\left (9 \, A + 11 \, C\right )} a - 11 i \, B b\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 \, \sqrt {2} {\left (-7 i \, B a - i \, {\left (7 \, A + 9 \, C\right )} b\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 \, \sqrt {2} {\left (7 i \, B a + i \, {\left (7 \, A + 9 \, C\right )} b\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (315 \, A a \cos \left (d x + c\right )^{5} + 385 \, {\left (B a + A b\right )} \cos \left (d x + c\right )^{4} + 45 \, {\left ({\left (9 \, A + 11 \, C\right )} a + 11 \, B b\right )} \cos \left (d x + c\right )^{3} + 77 \, {\left (7 \, B a + {\left (7 \, A + 9 \, C\right )} b\right )} \cos \left (d x + c\right )^{2} + 75 \, {\left ({\left (9 \, A + 11 \, C\right )} a + 11 \, B b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{3465 \, d} \]

[In]

integrate((a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(11/2),x, algorithm="fricas")

[Out]

-1/3465*(75*sqrt(2)*(I*(9*A + 11*C)*a + 11*I*B*b)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) +
75*sqrt(2)*(-I*(9*A + 11*C)*a - 11*I*B*b)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 231*sqrt
(2)*(-7*I*B*a - I*(7*A + 9*C)*b)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x +
c))) + 231*sqrt(2)*(7*I*B*a + I*(7*A + 9*C)*b)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c)
- I*sin(d*x + c))) - 2*(315*A*a*cos(d*x + c)^5 + 385*(B*a + A*b)*cos(d*x + c)^4 + 45*((9*A + 11*C)*a + 11*B*b)
*cos(d*x + c)^3 + 77*(7*B*a + (7*A + 9*C)*b)*cos(d*x + c)^2 + 75*((9*A + 11*C)*a + 11*B*b)*cos(d*x + c))*sin(d
*x + c)/sqrt(cos(d*x + c)))/d

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/sec(d*x+c)**(11/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}}{\sec \left (d x + c\right )^{\frac {11}{2}}} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(11/2),x, algorithm="maxima")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)/sec(d*x + c)^(11/2), x)

Giac [F]

\[ \int \frac {(a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}}{\sec \left (d x + c\right )^{\frac {11}{2}}} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(11/2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)/sec(d*x + c)^(11/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int \frac {\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}} \,d x \]

[In]

int(((a + b/cos(c + d*x))*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/cos(c + d*x))^(11/2),x)

[Out]

int(((a + b/cos(c + d*x))*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/cos(c + d*x))^(11/2), x)