\(\int \frac {(a+b \cos (c+d x))^4 (A+B \cos (c+d x)+C \cos ^2(c+d x))}{\sqrt {\sec (c+d x)}} \, dx\) [1481]

Optimal result

Integrand size = 43, antiderivative size = 517 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {2 \left (468 a^3 b B+364 a b^3 B+39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {2 \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 b \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (3458 a^3 b B+4004 a b^3 B+192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (13 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}} \]

[Out]

2/9009*b*(2171*B*a^2*b+1053*B*b^3+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*B*a^3*b+4004*B*a*b^3+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*B*a^4+330*B*a^2*b^2+45*B*b^4+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*B*a^3*b+364*B*a*b^3+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*
B*a^4+330*B*a^2*b^2+45*B*b^4+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] (verified)

Time = 1.66 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4306, 3128, 3112, 3102, 2827, 2719, 2715, 2720} \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {2 \sin (c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b \sin (c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )}{231 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right )}{195 d}+\frac {2 (8 a C+13 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^4}{13 d \sec ^{\frac {3}{2}}(c+d x)} \]

[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 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 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 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 \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)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),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*}

Mathematica [A] (verified)

Time = 7.23 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.77 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {\sqrt {\sec (c+d x)} \left (7392 \left (468 a^3 b B+364 a b^3 B+39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+6240 \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\left (154 \left (3744 a^3 b B+4472 a b^3 B+936 a^4 C+156 a^2 b^2 (36 A+43 C)+b^4 (1118 A+1171 C)\right ) \cos (c+d x)+5 \left (78 \left (616 a^4 B+3432 a^2 b^2 B+531 b^4 B+176 a^3 b (14 A+13 C)+4 a b^3 (572 A+531 C)\right )+1872 b \left (33 a^2 b B+8 b^3 B+22 a^3 C+2 a b^2 (11 A+16 C)\right ) \cos (2 (c+d x))+77 b^2 \left (52 A b^2+208 a b B+312 a^2 C+89 b^2 C\right ) \cos (3 (c+d x))+1638 b^3 (b B+4 a C) \cos (4 (c+d x))+693 b^4 C \cos (5 (c+d x))\right )\right ) \sin (2 (c+d x))\right )}{720720 d} \]

[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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1406\) vs. \(2(533)=1066\).

Time = 21.78 (sec) , antiderivative size = 1407, normalized size of antiderivative = 2.72

method	result	size
default	\(\text {Expression too large to display}\)	\(1407\)
parts	\(\text {Expression too large to display}\)	\(1485\)

[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,method=_RETURNVERBOSE)

[Out]

-2/45045*((-1+2*cos(1/2*d*x+1/2*c)^2)*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-177777
6*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)+(-
120120*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(
cos(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(co
s(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)*Ell
ipticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^4+15015*B*a^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))+64350*B*a^2*b^2*(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*si
n(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)/(-1+2*cos(1/2*d*x+1/2*c)^2)^(1/2)/d

Fricas [C] (verification not implemented)

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

Time = 0.18 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.08 \[ \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=-\frac {195 \, \sqrt {2} {\left (77 i \, B a^{4} + 44 i \, {\left (7 \, A + 5 \, C\right )} a^{3} b + 330 i \, B a^{2} b^{2} + 20 i \, {\left (11 \, A + 9 \, C\right )} a b^{3} + 45 i \, B b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 195 \, \sqrt {2} {\left (-77 i \, B a^{4} - 44 i \, {\left (7 \, A + 5 \, C\right )} a^{3} b - 330 i \, B a^{2} b^{2} - 20 i \, {\left (11 \, A + 9 \, C\right )} a b^{3} - 45 i \, B 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 (-39 i \, {\left (5 \, A + 3 \, C\right )} a^{4} - 468 i \, B a^{3} b - 78 i \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} - 364 i \, B a b^{3} - 7 i \, {\left (13 \, A + 11 \, C\right )} 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 (39 i \, {\left (5 \, A + 3 \, C\right )} a^{4} + 468 i \, B a^{3} b + 78 i \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} + 364 i \, B a b^{3} + 7 i \, {\left (13 \, A + 11 \, C\right )} 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 ) - \frac {2 \, {\left (3465 \, C b^{4} \cos \left (d x + c\right )^{6} + 4095 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{5} + 385 \, {\left (78 \, C a^{2} b^{2} + 52 \, B a b^{3} + {\left (13 \, A + 11 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 585 \, {\left (44 \, C a^{3} b + 66 \, B a^{2} b^{2} + 4 \, {\left (11 \, A + 9 \, C\right )} a b^{3} + 9 \, B b^{4}\right )} \cos \left (d x + c\right )^{3} + 77 \, {\left (117 \, C a^{4} + 468 \, B a^{3} b + 78 \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} + 364 \, B a b^{3} + 7 \, {\left (13 \, A + 11 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 195 \, {\left (77 \, B a^{4} + 44 \, {\left (7 \, A + 5 \, C\right )} a^{3} b + 330 \, B a^{2} b^{2} + 20 \, {\left (11 \, A + 9 \, C\right )} a b^{3} + 45 \, B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{45045 \, d} \]

[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]

-1/45045*(195*sqrt(2)*(77*I*B*a^4 + 44*I*(7*A + 5*C)*a^3*b + 330*I*B*a^2*b^2 + 20*I*(11*A + 9*C)*a*b^3 + 45*I*
B*b^4)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 195*sqrt(2)*(-77*I*B*a^4 - 44*I*(7*A + 5*C)
*a^3*b - 330*I*B*a^2*b^2 - 20*I*(11*A + 9*C)*a*b^3 - 45*I*B*b^4)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*s
in(d*x + c)) + 231*sqrt(2)*(-39*I*(5*A + 3*C)*a^4 - 468*I*B*a^3*b - 78*I*(9*A + 7*C)*a^2*b^2 - 364*I*B*a*b^3 -
7*I*(13*A + 11*C)*b^4)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 23
1*sqrt(2)*(39*I*(5*A + 3*C)*a^4 + 468*I*B*a^3*b + 78*I*(9*A + 7*C)*a^2*b^2 + 364*I*B*a*b^3 + 7*I*(13*A + 11*C)
*b^4)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(3465*C*b^4*cos(d*
x + c)^6 + 4095*(4*C*a*b^3 + B*b^4)*cos(d*x + c)^5 + 385*(78*C*a^2*b^2 + 52*B*a*b^3 + (13*A + 11*C)*b^4)*cos(d
*x + c)^4 + 585*(44*C*a^3*b + 66*B*a^2*b^2 + 4*(11*A + 9*C)*a*b^3 + 9*B*b^4)*cos(d*x + c)^3 + 77*(117*C*a^4 +
468*B*a^3*b + 78*(9*A + 7*C)*a^2*b^2 + 364*B*a*b^3 + 7*(13*A + 11*C)*b^4)*cos(d*x + c)^2 + 195*(77*B*a^4 + 44*
(7*A + 5*C)*a^3*b + 330*B*a^2*b^2 + 20*(11*A + 9*C)*a*b^3 + 45*B*b^4)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x
+ c)))/d

Sympy [F(-1)]

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

[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

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]

[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)