\(\int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [1312]

Optimal result

Integrand size = 43, antiderivative size = 404 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{693 d}+\frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (8 A b+11 a B) \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 A \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^4 \sin (c+d x)}{11 d} \]

[Out]

2/15*(7*B*a^4+54*B*a^2*b^2+15*B*b^4+12*a*b^3*(3*A+5*C)+4*a^3*b*(7*A+9*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+2/231*(220*B*a^3*b+308*B*a*b^3+77*b^4*(A+3*C)+66*a^2*b^2*(
5*A+7*C)+5*a^4*(9*A+11*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+2/3465*a*(192*A*b^3+539*B*a^3+1353*B*a*b^2+2*a^2*b*(673*A+891*C))*cos(d*x+c)^(3/2)*sin(d*x+c)/d+2/693*(64
*A*b^4+660*B*a^3*b+682*B*a*b^3+15*a^4*(9*A+11*C)+9*a^2*b^2*(101*A+143*C))*sin(d*x+c)*cos(d*x+c)^(1/2)/d+2/231*
(16*A*b^2+55*B*a*b+3*a^2*(9*A+11*C))*(b+a*cos(d*x+c))^2*sin(d*x+c)*cos(d*x+c)^(1/2)/d+2/99*(8*A*b+11*B*a)*(b+a
*cos(d*x+c))^3*sin(d*x+c)*cos(d*x+c)^(1/2)/d+2/11*A*(b+a*cos(d*x+c))^4*sin(d*x+c)*cos(d*x+c)^(1/2)/d

Rubi [A] (verified)

Time = 1.50 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4197, 3128, 3112, 3102, 2827, 2720, 2719} \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a \cos (c+d x)+b)^2}{231 d}+\frac {2 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{3465 d}+\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right )}{231 d}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{15 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{693 d}+\frac {2 (11 a B+8 A b) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^3}{99 d}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^4}{11 d} \]

[In]

Int[Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(2*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])
/(15*d) + (2*(220*a^3*b*B + 308*a*b^3*B + 77*b^4*(A + 3*C) + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*Elli
pticF[(c + d*x)/2, 2])/(231*d) + (2*(64*A*b^4 + 660*a^3*b*B + 682*a*b^3*B + 15*a^4*(9*A + 11*C) + 9*a^2*b^2*(1
01*A + 143*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(693*d) + (2*a*(192*A*b^3 + 539*a^3*B + 1353*a*b^2*B + 2*a^2*b
*(673*A + 891*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d) + (2*(16*A*b^2 + 55*a*b*B + 3*a^2*(9*A + 11*C))*Sq
rt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(231*d) + (2*(8*A*b + 11*a*B)*Sqrt[Cos[c + d*x]]*(b + a*
Cos[c + d*x])^3*Sin[c + d*x])/(99*d) + (2*A*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^4*Sin[c + d*x])/(11*d)

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 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 3112

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a +
b*Sin[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*
c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e
+ f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&
!LtQ[m, -1]

Rule 3128

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/(d*(m + n + 2)), Int[(a + b*Sin[e + f*
x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n +
2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*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] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 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 \frac {(b+a \cos (c+d x))^4 \left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 A \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^4 \sin (c+d x)}{11 d}+\frac {2}{11} \int \frac {(b+a \cos (c+d x))^3 \left (\frac {1}{2} b (A+11 C)+\frac {1}{2} (9 a A+11 b B+11 a C) \cos (c+d x)+\frac {1}{2} (8 A b+11 a B) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 (8 A b+11 a B) \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 A \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^4 \sin (c+d x)}{11 d}+\frac {4}{99} \int \frac {(b+a \cos (c+d x))^2 \left (\frac {1}{4} b (17 A b+11 a B+99 b C)+\frac {1}{4} \left (146 a A b+77 a^2 B+99 b^2 B+198 a b C\right ) \cos (c+d x)+\frac {3}{4} \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (8 A b+11 a B) \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 A \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^4 \sin (c+d x)}{11 d}+\frac {8}{693} \int \frac {(b+a \cos (c+d x)) \left (\frac {1}{8} b \left (242 a b B+9 a^2 (9 A+11 C)+b^2 (167 A+693 C)\right )+\frac {1}{8} \left (1441 a^2 b B+693 b^3 B+45 a^3 (9 A+11 C)+a b^2 (1381 A+2079 C)\right ) \cos (c+d x)+\frac {1}{8} \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (8 A b+11 a B) \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 A \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^4 \sin (c+d x)}{11 d}+\frac {16 \int \frac {\frac {5}{16} b^2 \left (242 a b B+9 a^2 (9 A+11 C)+b^2 (167 A+693 C)\right )+\frac {231}{16} \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \cos (c+d x)+\frac {15}{16} \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{3465} \\ & = \frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{693 d}+\frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (8 A b+11 a B) \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 A \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^4 \sin (c+d x)}{11 d}+\frac {32 \int \frac {\frac {45}{32} \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right )+\frac {693}{32} \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{10395} \\ & = \frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{693 d}+\frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (8 A b+11 a B) \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 A \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^4 \sin (c+d x)}{11 d}+\frac {1}{15} \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{231} \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{693 d}+\frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (8 A b+11 a B) \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 A \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^4 \sin (c+d x)}{11 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 12.84 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.79 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {154 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {1}{12} \sqrt {\cos (c+d x)} \left (154 a \left (144 A b^3+43 a^3 B+216 a b^2 B+4 a^2 b (43 A+36 C)\right ) \cos (c+d x)+5 \left (36 a^2 \left (66 A b^2+44 a b B+a^2 (16 A+11 C)\right ) \cos (2 (c+d x))+154 a^3 (4 A b+a B) \cos (3 (c+d x))+3 \left (616 A b^4+2288 a^3 b B+2464 a b^3 B+264 a^2 b^2 (13 A+14 C)+a^4 (531 A+572 C)+21 a^4 A \cos (4 (c+d x))\right )\right )\right ) \sin (c+d x)}{1155 d} \]

[In]

Integrate[Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(154*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2
] + 10*(220*a^3*b*B + 308*a*b^3*B + 77*b^4*(A + 3*C) + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*EllipticF[
(c + d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(154*a*(144*A*b^3 + 43*a^3*B + 216*a*b^2*B + 4*a^2*b*(43*A + 36*C))*Cos[
c + d*x] + 5*(36*a^2*(66*A*b^2 + 44*a*b*B + a^2*(16*A + 11*C))*Cos[2*(c + d*x)] + 154*a^3*(4*A*b + a*B)*Cos[3*
(c + d*x)] + 3*(616*A*b^4 + 2288*a^3*b*B + 2464*a*b^3*B + 264*a^2*b^2*(13*A + 14*C) + a^4*(531*A + 572*C) + 21
*a^4*A*Cos[4*(c + d*x)])))*Sin[c + d*x])/12)/(1155*d)

Maple [F(-1)]

Timed out.

hanged

[In]

int(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

int(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)

Fricas [C] (verification not implemented)

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

Time = 0.15 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.21 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (315 \, A a^{4} \cos \left (d x + c\right )^{4} + 75 \, {\left (9 \, A + 11 \, C\right )} a^{4} + 3300 \, B a^{3} b + 990 \, {\left (5 \, A + 7 \, C\right )} a^{2} b^{2} + 4620 \, B a b^{3} + 1155 \, A b^{4} + 385 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{3} + 45 \, {\left ({\left (9 \, A + 11 \, C\right )} a^{4} + 44 \, B a^{3} b + 66 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 77 \, {\left (7 \, B a^{4} + 4 \, {\left (7 \, A + 9 \, C\right )} a^{3} b + 54 \, B a^{2} b^{2} + 36 \, A a b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 \, \sqrt {2} {\left (5 i \, {\left (9 \, A + 11 \, C\right )} a^{4} + 220 i \, B a^{3} b + 66 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b^{2} + 308 i \, B a b^{3} + 77 i \, {\left (A + 3 \, C\right )} b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 \, \sqrt {2} {\left (-5 i \, {\left (9 \, A + 11 \, C\right )} a^{4} - 220 i \, B a^{3} b - 66 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b^{2} - 308 i \, B a b^{3} - 77 i \, {\left (A + 3 \, C\right )} b^{4}\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^{4} - 4 i \, {\left (7 \, A + 9 \, C\right )} a^{3} b - 54 i \, B a^{2} b^{2} - 12 i \, {\left (3 \, A + 5 \, C\right )} a b^{3} - 15 i \, B b^{4}\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^{4} + 4 i \, {\left (7 \, A + 9 \, C\right )} a^{3} b + 54 i \, B a^{2} b^{2} + 12 i \, {\left (3 \, A + 5 \, C\right )} a b^{3} + 15 i \, B b^{4}\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 )}{3465 \, d} \]

[In]

integrate(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/3465*(2*(315*A*a^4*cos(d*x + c)^4 + 75*(9*A + 11*C)*a^4 + 3300*B*a^3*b + 990*(5*A + 7*C)*a^2*b^2 + 4620*B*a*
b^3 + 1155*A*b^4 + 385*(B*a^4 + 4*A*a^3*b)*cos(d*x + c)^3 + 45*((9*A + 11*C)*a^4 + 44*B*a^3*b + 66*A*a^2*b^2)*
cos(d*x + c)^2 + 77*(7*B*a^4 + 4*(7*A + 9*C)*a^3*b + 54*B*a^2*b^2 + 36*A*a*b^3)*cos(d*x + c))*sqrt(cos(d*x + c
))*sin(d*x + c) - 15*sqrt(2)*(5*I*(9*A + 11*C)*a^4 + 220*I*B*a^3*b + 66*I*(5*A + 7*C)*a^2*b^2 + 308*I*B*a*b^3
+ 77*I*(A + 3*C)*b^4)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 15*sqrt(2)*(-5*I*(9*A + 11*C
)*a^4 - 220*I*B*a^3*b - 66*I*(5*A + 7*C)*a^2*b^2 - 308*I*B*a*b^3 - 77*I*(A + 3*C)*b^4)*weierstrassPInverse(-4,
0, cos(d*x + c) - I*sin(d*x + c)) - 231*sqrt(2)*(-7*I*B*a^4 - 4*I*(7*A + 9*C)*a^3*b - 54*I*B*a^2*b^2 - 12*I*(
3*A + 5*C)*a*b^3 - 15*I*B*b^4)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)
)) - 231*sqrt(2)*(7*I*B*a^4 + 4*I*(7*A + 9*C)*a^3*b + 54*I*B*a^2*b^2 + 12*I*(3*A + 5*C)*a*b^3 + 15*I*B*b^4)*we
ierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))))/d

Sympy [F(-1)]

Timed out. \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**(11/2)*(a+b*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. \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)^(11/2)*(a+b*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]

\[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{\frac {11}{2}} \,d x } \]

[In]

integrate(cos(d*x+c)^(11/2)*(a+b*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)*(b*sec(d*x + c) + a)^4*cos(d*x + c)^(11/2), x)

Mupad [B] (verification not implemented)

Time = 20.19 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.49 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2\,\left (C\,b^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+4\,C\,a\,b^3\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+2\,C\,a^2\,b^2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+2\,C\,a^2\,b^2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}+\frac {A\,b^4\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {2\,B\,b^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {4\,B\,a\,b^3\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}-\frac {2\,A\,a^4\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{13\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a^4\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {8\,A\,a\,b^3\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {8\,A\,a^3\,b\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {8\,B\,a^3\,b\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {8\,C\,a^3\,b\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,A\,a^2\,b^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {12\,B\,a^2\,b^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]

[In]

int(cos(c + d*x)^(11/2)*(a + b/cos(c + d*x))^4*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)

[Out]

(2*(C*b^4*ellipticF(c/2 + (d*x)/2, 2) + 4*C*a*b^3*ellipticE(c/2 + (d*x)/2, 2) + 2*C*a^2*b^2*ellipticF(c/2 + (d
*x)/2, 2) + 2*C*a^2*b^2*cos(c + d*x)^(1/2)*sin(c + d*x)))/d + (A*b^4*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 +
(2*ellipticF(c/2 + (d*x)/2, 2))/3))/d + (2*B*b^4*ellipticE(c/2 + (d*x)/2, 2))/d + (4*B*a*b^3*((2*cos(c + d*x)^
(1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d - (2*A*a^4*cos(c + d*x)^(13/2)*sin(c + d*x)*hype
rgeom([1/2, 13/4], 17/4, cos(c + d*x)^2))/(13*d*(sin(c + d*x)^2)^(1/2)) - (2*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*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)) - (8*A*a*b^3*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)) - (8*A*a^3*b
*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))
- (8*B*a^3*b*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)) - (8*C*a^3*b*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*A*a^2*b^2*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)) - (12*B*a^2*b^2*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))