3.1481 \(\int \frac {(a+b \cos (c+d x))^4 (A+B \cos (c+d x)+C \cos ^2(c+d x))}{\sqrt {\sec (c+d x)}} \, dx\)

Optimal. Leaf size=517 \[ \frac {2 \sin (c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b \sin (c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )}{231 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right )}{195 d}+\frac {2 (8 a C+13 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^4}{13 d \sec ^{\frac {3}{2}}(c+d x)} \]

[Out]

2/9009*b*(2171*a^2*b*B+1053*b^3*B+192*a^3*C+2*a*b^2*(1573*A+1259*C))*sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/6435*(345
8*a^3*b*B+4004*a*b^3*B+192*a^4*C+77*b^4*(13*A+11*C)+11*a^2*b^2*(637*A+491*C))*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/
1287*(143*A*b^2+221*B*a*b+48*C*a^2+121*C*b^2)*(a+b*cos(d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/143*(13*B*b+8
*C*a)*(a+b*cos(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/13*C*(a+b*cos(d*x+c))^4*sin(d*x+c)/d/sec(d*x+c)^(3/2)
+2/231*(77*a^4*B+330*a^2*b^2*B+45*b^4*B+44*a^3*b*(7*A+5*C)+20*a*b^3*(11*A+9*C))*sin(d*x+c)/d/sec(d*x+c)^(1/2)+
2/195*(468*a^3*b*B+364*a*b^3*B+39*a^4*(5*A+3*C)+78*a^2*b^2*(9*A+7*C)+7*b^4*(13*A+11*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))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+2/231*(77*
a^4*B+330*a^2*b^2*B+45*b^4*B+44*a^3*b*(7*A+5*C)+20*a*b^3*(11*A+9*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))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d

________________________________________________________________________________________

Rubi [A]  time = 1.53, antiderivative size = 517, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4221, 3049, 3033, 3023, 2748, 2639, 2635, 2641} \[ \frac {2 \sin (c+d x) \left (11 a^2 b^2 (637 A+491 C)+3458 a^3 b B+192 a^4 C+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b \sin (c+d x) \left (2171 a^2 b B+192 a^3 C+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \sin (c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )}{231 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+468 a^3 b B+364 a b^3 B+7 b^4 (13 A+11 C)\right )}{195 d}+\frac {2 (8 a C+13 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^4}{13 d \sec ^{\frac {3}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(468*a^3*b*B + 364*a*b^3*B + 39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*Sqrt[Cos[c
+ d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (2*(77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 44*a
^3*b*(7*A + 5*C) + 20*a*b^3*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(23
1*d) + (2*b*(2171*a^2*b*B + 1053*b^3*B + 192*a^3*C + 2*a*b^2*(1573*A + 1259*C))*Sin[c + d*x])/(9009*d*Sec[c +
d*x]^(5/2)) + (2*(3458*a^3*b*B + 4004*a*b^3*B + 192*a^4*C + 77*b^4*(13*A + 11*C) + 11*a^2*b^2*(637*A + 491*C))
*Sin[c + d*x])/(6435*d*Sec[c + d*x]^(3/2)) + (2*(143*A*b^2 + 221*a*b*B + 48*a^2*C + 121*b^2*C)*(a + b*Cos[c +
d*x])^2*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(3/2)) + (2*(13*b*B + 8*a*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(
143*d*Sec[c + d*x]^(3/2)) + (2*C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(13*d*Sec[c + d*x]^(3/2)) + (2*(77*a^4*B
+ 330*a^2*b^2*B + 45*b^4*B + 44*a^3*b*(7*A + 5*C) + 20*a*b^3*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c +
d*x]])

Rule 2635

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] && Integer
Q[2*n]

Rule 2639

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

Rule 2641

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

Rule 2748

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 3023

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 3033

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 3049

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 4221

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*} \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{13} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (\frac {1}{2} a (13 A+3 C)+\frac {1}{2} (13 A b+13 a B+11 b C) \cos (c+d x)+\frac {1}{2} (13 b B+8 a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 (13 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{143} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \left (\frac {1}{4} a (143 a A+39 b B+57 a C)+\frac {1}{4} \left (286 a A b+143 a^2 B+117 b^2 B+226 a b C\right ) \cos (c+d x)+\frac {1}{4} \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (13 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (\frac {3}{8} a \left (338 a b B+11 b^2 (13 A+11 C)+3 a^2 (143 A+73 C)\right )+\frac {1}{8} \left (1287 a^3 B+2951 a b^2 B+77 b^3 (13 A+11 C)+3 a^2 b (1287 A+961 C)\right ) \cos (c+d x)+\frac {1}{8} \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \cos ^2(c+d x)\right ) \, dx}{1287}\\ &=\frac {2 b \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (13 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \left (\frac {21}{16} a^2 \left (338 a b B+11 b^2 (13 A+11 C)+3 a^2 (143 A+73 C)\right )+\frac {117}{16} \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \cos (c+d x)+\frac {7}{16} \left (3458 a^3 b B+4004 a b^3 B+192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^2(c+d x)\right ) \, dx}{9009}\\ &=\frac {2 b \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (3458 a^3 b B+4004 a b^3 B+192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (13 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (32 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \left (\frac {231}{32} \left (468 a^3 b B+364 a b^3 B+39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right )+\frac {585}{32} \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \cos (c+d x)\right ) \, dx}{45045}\\ &=\frac {2 b \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (3458 a^3 b B+4004 a b^3 B+192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (13 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{77} \left (\left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{195} \left (\left (468 a^3 b B+364 a b^3 B+39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 \left (468 a^3 b B+364 a b^3 B+39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {2 b \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (3458 a^3 b B+4004 a b^3 B+192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (13 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {1}{231} \left (\left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 \left (468 a^3 b B+364 a b^3 B+39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {2 \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 b \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (3458 a^3 b B+4004 a b^3 B+192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (13 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}\\ \end {align*}

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Mathematica [A]  time = 4.36, size = 400, normalized size = 0.77 \[ \frac {\sqrt {\sec (c+d x)} \left (\sin (2 (c+d x)) \left (154 \cos (c+d x) \left (936 a^4 C+3744 a^3 b B+156 a^2 b^2 (36 A+43 C)+4472 a b^3 B+b^4 (1118 A+1171 C)\right )+5 \left (77 b^2 \cos (3 (c+d x)) \left (312 a^2 C+208 a b B+52 A b^2+89 b^2 C\right )+1872 b \cos (2 (c+d x)) \left (22 a^3 C+33 a^2 b B+2 a b^2 (11 A+16 C)+8 b^3 B\right )+78 \left (616 a^4 B+176 a^3 b (14 A+13 C)+3432 a^2 b^2 B+4 a b^3 (572 A+531 C)+531 b^4 B\right )+1638 b^3 (4 a C+b B) \cos (4 (c+d x))+693 b^4 C \cos (5 (c+d x))\right )\right )+6240 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )+7392 \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right )\right )}{720720 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Sec[c + d*x]]*(7392*(468*a^3*b*B + 364*a*b^3*B + 39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13
*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 6240*(77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 44*a^3*
b*(7*A + 5*C) + 20*a*b^3*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (154*(3744*a^3*b*B + 447
2*a*b^3*B + 936*a^4*C + 156*a^2*b^2*(36*A + 43*C) + b^4*(1118*A + 1171*C))*Cos[c + d*x] + 5*(78*(616*a^4*B + 3
432*a^2*b^2*B + 531*b^4*B + 176*a^3*b*(14*A + 13*C) + 4*a*b^3*(572*A + 531*C)) + 1872*b*(33*a^2*b*B + 8*b^3*B
+ 22*a^3*C + 2*a*b^2*(11*A + 16*C))*Cos[2*(c + d*x)] + 77*b^2*(52*A*b^2 + 208*a*b*B + 312*a^2*C + 89*b^2*C)*Co
s[3*(c + d*x)] + 1638*b^3*(b*B + 4*a*C)*Cos[4*(c + d*x)] + 693*b^4*C*Cos[5*(c + d*x)]))*Sin[2*(c + d*x)]))/(72
0720*d)

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fricas [F]  time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b^{4} \cos \left (d x + c\right )^{6} + {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{5} + A a^{4} + {\left (6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (2 \, C a^{3} b + 3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )}{\sqrt {\sec \left (d x + c\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

integral((C*b^4*cos(d*x + c)^6 + (4*C*a*b^3 + B*b^4)*cos(d*x + c)^5 + A*a^4 + (6*C*a^2*b^2 + 4*B*a*b^3 + A*b^4
)*cos(d*x + c)^4 + 2*(2*C*a^3*b + 3*B*a^2*b^2 + 2*A*a*b^3)*cos(d*x + c)^3 + (C*a^4 + 4*B*a^3*b + 6*A*a^2*b^2)*
cos(d*x + c)^2 + (B*a^4 + 4*A*a^3*b)*cos(d*x + c))/sqrt(sec(d*x + c)), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\sec \left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 3.30, size = 1407, normalized size = 2.72 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x)

[Out]

-2/45045*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-443520*C*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x
+1/2*c)^14+(262080*B*b^4+1048320*C*a*b^3+1330560*C*b^4)*sin(1/2*d*x+1/2*c)^12*cos(1/2*d*x+1/2*c)+(-160160*A*b^
4-640640*B*a*b^3-655200*B*b^4-960960*C*a^2*b^2-2620800*C*a*b^3-1798720*C*b^4)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*
x+1/2*c)+(411840*A*a*b^3+320320*A*b^4+617760*B*a^2*b^2+1281280*B*a*b^3+739440*B*b^4+411840*C*a^3*b+1921920*C*a
^2*b^2+2957760*C*a*b^3+1379840*C*b^4)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-432432*A*a^2*b^2-617760*A*a*b^
3-296296*A*b^4-288288*B*a^3*b-926640*B*a^2*b^2-1185184*B*a*b^3-453960*B*b^4-72072*C*a^4-617760*C*a^3*b-1777776
*C*a^2*b^2-1815840*C*a*b^3-666512*C*b^4)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(240240*A*a^3*b+432432*A*a^2*
b^2+480480*A*a*b^3+136136*A*b^4+60060*B*a^4+288288*B*a^3*b+720720*B*a^2*b^2+544544*B*a*b^3+180180*B*b^4+72072*
C*a^4+480480*C*a^3*b+816816*C*a^2*b^2+720720*C*a*b^3+198352*C*b^4)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-1
20120*A*a^3*b-108108*A*a^2*b^2-137280*A*a*b^3-24024*A*b^4-30030*B*a^4-72072*B*a^3*b-205920*B*a^2*b^2-96096*B*a
*b^3-36270*B*b^4-18018*C*a^4-137280*C*a^3*b-144144*C*a^2*b^2-145080*C*a*b^3-27258*C*b^4)*sin(1/2*d*x+1/2*c)^2*
cos(1/2*d*x+1/2*c)+60060*A*a^3*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))+42900*a*A*b^3*(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(c
os(1/2*d*x+1/2*c),2^(1/2))-45045*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))*a^4-162162*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))*a^2*b^2-21021*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)*Elli
pticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^4+15015*a^4*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))+64350*a^2*b^2*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))+8775*B*b^4*(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))-108108*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^3*b-84084*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*b^3+42900*a^3*b*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))+35100*C*a*b^3*(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))-27027*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))*a^4-126126*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))*a^2*b^2-17787*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^4)/(-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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\sec \left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(1/cos(c + d*x))^(1/2),x)

[Out]

int(((a + b*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(1/cos(c + d*x))^(1/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))**4*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/sec(d*x+c)**(1/2),x)

[Out]

Timed out

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