3.1480 \(\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sqrt{\sec (c+d x)} \, dx\)

Optimal. Leaf size=444 \[ \frac{2 b \sin (c+d x) \left (1353 a^2 b B+192 a^3 C+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (9 a^2 b^2 (143 A+101 C)+682 a^3 b B+64 a^4 C+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+308 a^3 b B+220 a b^3 B+5 b^4 (11 A+9 C)\right )}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (12 a^3 b (5 A+3 C)+54 a^2 b^2 B+15 a^4 B+4 a b^3 (9 A+7 C)+7 b^4 B\right )}{15 d}+\frac{2 (8 a C+11 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \sqrt{\sec (c+d x)}} \]

[Out]

(2*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*Ellipti
cE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(308*a^3*b*B + 220*a*b^3*B + 77*a^4*(3*A + C) + 66*a^2*b^2*
(7*A + 5*C) + 5*b^4*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (
2*b*(1353*a^2*b*B + 539*b^3*B + 192*a^3*C + 2*a*b^2*(891*A + 673*C))*Sin[c + d*x])/(3465*d*Sec[c + d*x]^(3/2))
+ (2*(682*a^3*b*B + 660*a*b^3*B + 64*a^4*C + 15*b^4*(11*A + 9*C) + 9*a^2*b^2*(143*A + 101*C))*Sin[c + d*x])/(
693*d*Sqrt[Sec[c + d*x]]) + (2*(33*A*b^2 + 55*a*b*B + 16*a^2*C + 27*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x]
)/(231*d*Sqrt[Sec[c + d*x]]) + (2*(11*b*B + 8*a*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(99*d*Sqrt[Sec[c + d*x
]]) + (2*C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(11*d*Sqrt[Sec[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.46011, antiderivative size = 444, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {4221, 3049, 3033, 3023, 2748, 2641, 2639} \[ \frac{2 b \sin (c+d x) \left (1353 a^2 b B+192 a^3 C+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (9 a^2 b^2 (143 A+101 C)+682 a^3 b B+64 a^4 C+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+308 a^3 b B+220 a b^3 B+5 b^4 (11 A+9 C)\right )}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (12 a^3 b (5 A+3 C)+54 a^2 b^2 B+15 a^4 B+4 a b^3 (9 A+7 C)+7 b^4 B\right )}{15 d}+\frac{2 (8 a C+11 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \sqrt{\sec (c+d x)}} \]

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*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*Ellipti
cE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(308*a^3*b*B + 220*a*b^3*B + 77*a^4*(3*A + C) + 66*a^2*b^2*
(7*A + 5*C) + 5*b^4*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (
2*b*(1353*a^2*b*B + 539*b^3*B + 192*a^3*C + 2*a*b^2*(891*A + 673*C))*Sin[c + d*x])/(3465*d*Sec[c + d*x]^(3/2))
+ (2*(682*a^3*b*B + 660*a*b^3*B + 64*a^4*C + 15*b^4*(11*A + 9*C) + 9*a^2*b^2*(143*A + 101*C))*Sin[c + d*x])/(
693*d*Sqrt[Sec[c + d*x]]) + (2*(33*A*b^2 + 55*a*b*B + 16*a^2*C + 27*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x]
)/(231*d*Sqrt[Sec[c + d*x]]) + (2*(11*b*B + 8*a*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(99*d*Sqrt[Sec[c + d*x
]]) + (2*C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(11*d*Sqrt[Sec[c + d*x]])

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]

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 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 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 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 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 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]

Rubi steps

\begin{align*} \int (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 \frac{(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \sqrt{\sec (c+d x)}}+\frac{1}{11} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^3 \left (\frac{1}{2} a (11 A+C)+\frac{1}{2} (11 A b+11 a B+9 b C) \cos (c+d x)+\frac{1}{2} (11 b B+8 a C) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 (11 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \sqrt{\sec (c+d x)}}+\frac{1}{99} \left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^2 \left (\frac{1}{4} a (99 a A+11 b B+17 a C)+\frac{1}{4} \left (198 a A b+99 a^2 B+77 b^2 B+146 a b C\right ) \cos (c+d x)+\frac{3}{4} \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 (11 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \sqrt{\sec (c+d x)}}+\frac{1}{693} \left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x)) \left (\frac{1}{8} a \left (242 a b B+9 b^2 (11 A+9 C)+a^2 (693 A+167 C)\right )+\frac{1}{8} \left (693 a^3 B+1441 a b^2 B+45 b^3 (11 A+9 C)+a^2 b (2079 A+1381 C)\right ) \cos (c+d x)+\frac{1}{8} \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \sin (c+d x)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 (11 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \sqrt{\sec (c+d x)}}+\frac{\left (16 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{5}{16} a^2 \left (242 a b B+9 b^2 (11 A+9 C)+a^2 (693 A+167 C)\right )+\frac{231}{16} \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \cos (c+d x)+\frac{15}{16} \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{3465}\\ &=\frac{2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \sin (c+d x)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sin (c+d x)}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 (11 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \sqrt{\sec (c+d x)}}+\frac{\left (32 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{45}{32} \left (308 a^3 b B+220 a b^3 B+77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right )+\frac{693}{32} \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{10395}\\ &=\frac{2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \sin (c+d x)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sin (c+d x)}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 (11 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \sqrt{\sec (c+d x)}}+\frac{1}{15} \left (\left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{231} \left (\left (308 a^3 b B+220 a b^3 B+77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (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 (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 \left (308 a^3 b B+220 a b^3 B+77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (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 (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \sin (c+d x)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sin (c+d x)}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 (11 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \sqrt{\sec (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 3.8864, size = 338, normalized size = 0.76 \[ \frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (240 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+308 a^3 b B+220 a b^3 B+5 b^4 (11 A+9 C)\right )+3696 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (12 a^3 b (5 A+3 C)+54 a^2 b^2 B+15 a^4 B+4 a b^3 (9 A+7 C)+7 b^4 B\right )+\frac{\sin (2 (c+d x)) \left (154 b \cos (c+d x) \left (216 a^2 b B+144 a^3 C+4 a b^2 (36 A+43 C)+43 b^3 B\right )+5 \left (36 b^2 \cos (2 (c+d x)) \left (66 a^2 C+44 a b B+11 A b^2+16 b^2 C\right )+792 a^2 b^2 (14 A+13 C)+7392 a^3 b B+1848 a^4 C+154 b^3 (4 a C+b B) \cos (3 (c+d x))+6864 a b^3 B+3 b^4 (572 A+531 C)+63 b^4 C \cos (4 (c+d x))\right )\right )}{\sqrt{\cos (c+d x)}}\right )}{27720 d} \]

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[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(3696*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b
^3*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 240*(308*a^3*b*B + 220*a*b^3*B + 77*a^4*(3*A + C) + 66*a^2*b^2*(7*
A + 5*C) + 5*b^4*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2] + ((154*b*(216*a^2*b*B + 43*b^3*B + 144*a^3*C + 4*a*b
^2*(36*A + 43*C))*Cos[c + d*x] + 5*(7392*a^3*b*B + 6864*a*b^3*B + 1848*a^4*C + 792*a^2*b^2*(14*A + 13*C) + 3*b
^4*(572*A + 531*C) + 36*b^2*(11*A*b^2 + 44*a*b*B + 66*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 154*b^3*(b*B + 4*a*
C)*Cos[3*(c + d*x)] + 63*b^4*C*Cos[4*(c + d*x)]))*Sin[2*(c + d*x)])/Sqrt[Cos[c + d*x]]))/(27720*d)

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Maple [B]  time = 1.618, size = 1273, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}

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]

-2/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(20160*C*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/
2*c)^12+(-12320*B*b^4-49280*C*a*b^3-50400*C*b^4)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(7920*A*b^4+31680*B*
a*b^3+24640*B*b^4+47520*C*a^2*b^2+98560*C*a*b^3+56880*C*b^4)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-22176*A
*a*b^3-11880*A*b^4-33264*B*a^2*b^2-47520*B*a*b^3-22792*B*b^4-22176*C*a^3*b-71280*C*a^2*b^2-91168*C*a*b^3-34920
*C*b^4)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(27720*A*a^2*b^2+22176*A*a*b^3+9240*A*b^4+18480*B*a^3*b+33264*
B*a^2*b^2+36960*B*a*b^3+10472*B*b^4+4620*C*a^4+22176*C*a^3*b+55440*C*a^2*b^2+41888*C*a*b^3+13860*C*b^4)*sin(1/
2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-13860*A*a^2*b^2-5544*A*a*b^3-2640*A*b^4-9240*B*a^3*b-8316*B*a^2*b^2-10560*
B*a*b^3-1848*B*b^4-2310*C*a^4-5544*C*a^3*b-15840*C*a^2*b^2-7392*C*a*b^3-2790*C*b^4)*sin(1/2*d*x+1/2*c)^2*cos(1
/2*d*x+1/2*c)+3465*A*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))+6930*a^2*A*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))+825*A*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))-13860*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^3*b-8316*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*b^3+4620*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))*a^3*b+3300*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(c
os(1/2*d*x+1/2*c),2^(1/2))*a*b^3-3465*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)*Elliptic
E(cos(1/2*d*x+1/2*c),2^(1/2))*a^4-12474*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)*Ellipt
icE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2*b^2-1617*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*E
llipticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^4+1155*a^4*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))+4950*a^2*b^2*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))+675*C*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))-8316*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^3*b-6468*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*b^3)/(-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., size = 0, normalized size = 0. \begin{align*} \int{\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} \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (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 )\right )} \sqrt{\sec \left (d x + c\right )}, x\right ) \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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} \end{align*}

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)