\(\int \frac {(a+b \cos (c+d x))^4 (A+C \cos ^2(c+d x))}{\sqrt {\sec (c+d x)}} \, dx\) [1390]

Optimal result

Integrand size = 35, antiderivative size = 422 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {2 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {16 a C (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}} \]

[Out]

4/9009*a*b*(1573*A*b^2+96*C*a^2+1259*C*b^2)*sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/6435*(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*(48*a^2*C+11*b^2*(13*A+11*C))*(a+b*cos(d*x+c))
^2*sin(d*x+c)/d/sec(d*x+c)^(3/2)+16/143*a*C*(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)+8/231*a*b*(11*a^2*(7*A+5*C)+5*b^2*(11*A+9*C))*sin(d*x+c)/d/sec(d*x+c)^(
1/2)+2/195*(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+8/231*a*b*(11*a^2*(7*A+5*C)+5
*b^2*(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 = 1.44 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {4306, 3129, 3128, 3112, 3102, 2827, 2719, 2715, 2720} \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {2 \left (192 a^4 C+11 a^2 b^2 (637 A+491 C)+77 b^4 (13 A+11 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^4}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {16 a C \sin (c+d x) (a+b \cos (c+d x))^3}{143 d \sec ^{\frac {3}{2}}(c+d x)} \]

[In]

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

[Out]

(2*(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) + (8*a*b*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Ellipt
icF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a*b*(1573*A*b^2 + 96*a^2*C + 1259*b^2*C)*Sin[c + d*x])/(9
009*d*Sec[c + d*x]^(5/2)) + (2*(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*(48*a^2*C + 11*b^2*(13*A + 11*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(1287*d
*Sec[c + d*x]^(3/2)) + (16*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*C
os[c + d*x])^4*Sin[c + d*x])/(13*d*Sec[c + d*x]^(3/2)) + (8*a*b*(11*a^2*(7*A + 5*C) + 5*b^2*(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 3129

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (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)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e +
f*x] + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, 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+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} b (13 A+11 C) \cos (c+d x)+4 a C \cos ^2(c+d x)\right ) \, dx \\ & = \frac {16 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^2 (143 A+57 C)+\frac {1}{2} a b (143 A+113 C) \cos (c+d x)+\frac {1}{4} \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {16 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 (11 b^2 (13 A+11 C)+3 a^2 (143 A+73 C)\right )+\frac {1}{8} b \left (77 b^2 (13 A+11 C)+3 a^2 (1287 A+961 C)\right ) \cos (c+d x)+\frac {1}{4} a \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx}{1287} \\ & = \frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {16 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 (11 b^2 (13 A+11 C)+3 a^2 (143 A+73 C)\right )+\frac {117}{4} a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \cos (c+d x)+\frac {7}{16} \left (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 {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (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 (48 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {16 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 (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}{8} a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \cos (c+d x)\right ) \, dx}{45045} \\ & = \frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (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 (48 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {16 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 (4 a b \left (11 a^2 (7 A+5 C)+5 b^2 (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 (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 (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 {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (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 (48 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {16 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 {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {1}{231} \left (4 a b \left (11 a^2 (7 A+5 C)+5 b^2 (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 (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 {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (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 {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (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 (48 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {16 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 {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.14 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.72 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {\sqrt {\sec (c+d x)} \left (14784 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+49920 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 \left (154 \left (936 a^4 C+156 a^2 b^2 (36 A+43 C)+b^4 (1118 A+1171 C)\right ) \cos (c+d x)+5 b \left (312 a \left (44 a^2 (14 A+13 C)+b^2 (572 A+531 C)\right )+3744 a \left (11 A b^2+11 a^2 C+16 b^2 C\right ) \cos (2 (c+d x))+77 \left (52 A b^3+312 a^2 b C+89 b^3 C\right ) \cos (3 (c+d x))+6552 a b^2 C \cos (4 (c+d x))+693 b^3 C \cos (5 (c+d x))\right )\right ) \sin (2 (c+d x))\right )}{1441440 d} \]

[In]

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

[Out]

(Sqrt[Sec[c + d*x]]*(14784*(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] + 49920*a*b*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Ellipti
cF[(c + d*x)/2, 2] + 2*(154*(936*a^4*C + 156*a^2*b^2*(36*A + 43*C) + b^4*(1118*A + 1171*C))*Cos[c + d*x] + 5*b
*(312*a*(44*a^2*(14*A + 13*C) + b^2*(572*A + 531*C)) + 3744*a*(11*A*b^2 + 11*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)
] + 77*(52*A*b^3 + 312*a^2*b*C + 89*b^3*C)*Cos[3*(c + d*x)] + 6552*a*b^2*C*Cos[4*(c + d*x)] + 693*b^3*C*Cos[5*
(c + d*x)]))*Sin[2*(c + d*x)]))/(1441440*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1016\) vs. \(2(438)=876\).

Time = 18.21 (sec) , antiderivative size = 1017, normalized size of antiderivative = 2.41

method	result	size
default	\(\text {Expression too large to display}\)	\(1017\)
parts	\(\text {Expression too large to display}\)	\(1446\)

[In]

int((a+b*cos(d*x+c))^4*(A+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+(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^4-960960*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+41
1840*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^4-72072*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+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^4-18018*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*s
in(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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^4+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)/(-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 = 0.16 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=-\frac {780 \, \sqrt {2} {\left (11 i \, {\left (7 \, A + 5 \, C\right )} a^{3} b + 5 i \, {\left (11 \, A + 9 \, C\right )} a b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 780 \, \sqrt {2} {\left (-11 i \, {\left (7 \, A + 5 \, C\right )} a^{3} b - 5 i \, {\left (11 \, A + 9 \, C\right )} a b^{3}\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} - 78 i \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} - 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} + 78 i \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} + 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} + 16380 \, C a b^{3} \cos \left (d x + c\right )^{5} + 385 \, {\left (78 \, C a^{2} b^{2} + {\left (13 \, A + 11 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 2340 \, {\left (11 \, C a^{3} b + {\left (11 \, A + 9 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} + 77 \, {\left (117 \, C a^{4} + 78 \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} + 7 \, {\left (13 \, A + 11 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 780 \, {\left (11 \, {\left (7 \, A + 5 \, C\right )} a^{3} b + 5 \, {\left (11 \, A + 9 \, C\right )} a b^{3}\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+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

-1/45045*(780*sqrt(2)*(11*I*(7*A + 5*C)*a^3*b + 5*I*(11*A + 9*C)*a*b^3)*weierstrassPInverse(-4, 0, cos(d*x + c
) + I*sin(d*x + c)) + 780*sqrt(2)*(-11*I*(7*A + 5*C)*a^3*b - 5*I*(11*A + 9*C)*a*b^3)*weierstrassPInverse(-4, 0
, cos(d*x + c) - I*sin(d*x + c)) + 231*sqrt(2)*(-39*I*(5*A + 3*C)*a^4 - 78*I*(9*A + 7*C)*a^2*b^2 - 7*I*(13*A +
11*C)*b^4)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 231*sqrt(2)*(3
9*I*(5*A + 3*C)*a^4 + 78*I*(9*A + 7*C)*a^2*b^2 + 7*I*(13*A + 11*C)*b^4)*weierstrassZeta(-4, 0, weierstrassPInv
erse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(3465*C*b^4*cos(d*x + c)^6 + 16380*C*a*b^3*cos(d*x + c)^5 + 38
5*(78*C*a^2*b^2 + (13*A + 11*C)*b^4)*cos(d*x + c)^4 + 2340*(11*C*a^3*b + (11*A + 9*C)*a*b^3)*cos(d*x + c)^3 +
77*(117*C*a^4 + 78*(9*A + 7*C)*a^2*b^2 + 7*(13*A + 11*C)*b^4)*cos(d*x + c)^2 + 780*(11*(7*A + 5*C)*a^3*b + 5*(
11*A + 9*C)*a*b^3)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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