3.1473 \(\int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x)+C \cos ^2(c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=463 \[ \frac {2 b \sin (c+d x) \left (24 a^2 C+195 a b B+143 A b^2+121 b^2 C\right )}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (117 a^3 B+39 a^2 b (9 A+7 C)+273 a b^2 B+7 b^3 (13 A+11 C)\right )}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (24 a^3 C+338 a^2 b B+39 a b^2 (11 A+9 C)+117 b^3 B\right )}{1001 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (11 a^3 (7 A+5 C)+165 a^2 b B+15 a b^2 (11 A+9 C)+45 b^3 B\right )}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (11 a^3 (7 A+5 C)+165 a^2 b B+15 a b^2 (11 A+9 C)+45 b^3 B\right )}{231 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (117 a^3 B+39 a^2 b (9 A+7 C)+273 a b^2 B+7 b^3 (13 A+11 C)\right )}{195 d}+\frac {2 (6 a C+13 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{143 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^3}{13 d \sec ^{\frac {5}{2}}(c+d x)} \]

[Out]

2/1287*b*(143*A*b^2+195*B*a*b+24*C*a^2+121*C*b^2)*sin(d*x+c)/d/sec(d*x+c)^(7/2)+2/1001*(338*a^2*b*B+117*b^3*B+
24*a^3*C+39*a*b^2*(11*A+9*C))*sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/143*(13*B*b+6*C*a)*(a+b*cos(d*x+c))^2*sin(d*x+c)
/d/sec(d*x+c)^(5/2)+2/13*C*(a+b*cos(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/585*(117*a^3*B+273*a*b^2*B+39*a^
2*b*(9*A+7*C)+7*b^3*(13*A+11*C))*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/231*(165*a^2*b*B+45*b^3*B+11*a^3*(7*A+5*C)+15
*a*b^2*(11*A+9*C))*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2/195*(117*a^3*B+273*a*b^2*B+39*a^2*b*(9*A+7*C)+7*b^3*(13*A+1
1*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)*s
ec(d*x+c)^(1/2)/d+2/231*(165*a^2*b*B+45*b^3*B+11*a^3*(7*A+5*C)+15*a*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]  time = 1.14, antiderivative size = 463, 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, 2635, 2641, 2639} \[ \frac {2 \sin (c+d x) \left (39 a^2 b (9 A+7 C)+117 a^3 B+273 a b^2 B+7 b^3 (13 A+11 C)\right )}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (338 a^2 b B+24 a^3 C+39 a b^2 (11 A+9 C)+117 b^3 B\right )}{1001 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 b \sin (c+d x) \left (24 a^2 C+195 a b B+143 A b^2+121 b^2 C\right )}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (11 a^3 (7 A+5 C)+165 a^2 b B+15 a b^2 (11 A+9 C)+45 b^3 B\right )}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (11 a^3 (7 A+5 C)+165 a^2 b B+15 a b^2 (11 A+9 C)+45 b^3 B\right )}{231 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (39 a^2 b (9 A+7 C)+117 a^3 B+273 a b^2 B+7 b^3 (13 A+11 C)\right )}{195 d}+\frac {2 (6 a C+13 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{143 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^3}{13 d \sec ^{\frac {5}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]

[Out]

(2*(117*a^3*B + 273*a*b^2*B + 39*a^2*b*(9*A + 7*C) + 7*b^3*(13*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*
x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (2*(165*a^2*b*B + 45*b^3*B + 11*a^3*(7*A + 5*C) + 15*a*b^2*(11*A + 9*C)
)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*b*(143*A*b^2 + 195*a*b*B + 24*
a^2*C + 121*b^2*C)*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(7/2)) + (2*(338*a^2*b*B + 117*b^3*B + 24*a^3*C + 39*a*b
^2*(11*A + 9*C))*Sin[c + d*x])/(1001*d*Sec[c + d*x]^(5/2)) + (2*(13*b*B + 6*a*C)*(a + b*Cos[c + d*x])^2*Sin[c
+ d*x])/(143*d*Sec[c + d*x]^(5/2)) + (2*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(13*d*Sec[c + d*x]^(5/2)) + (2*
(117*a^3*B + 273*a*b^2*B + 39*a^2*b*(9*A + 7*C) + 7*b^3*(13*A + 11*C))*Sin[c + d*x])/(585*d*Sec[c + d*x]^(3/2)
) + (2*(165*a^2*b*B + 45*b^3*B + 11*a^3*(7*A + 5*C) + 15*a*b^2*(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))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{13} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left (\frac {1}{2} a (13 A+5 C)+\frac {1}{2} (13 A b+13 a B+11 b C) \cos (c+d x)+\frac {1}{2} (13 b B+6 a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 (13 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{143 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{143} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x)) \left (\frac {1}{4} a (143 a A+65 b B+85 a C)+\frac {1}{4} \left (286 a A b+143 a^2 B+117 b^2 B+230 a b C\right ) \cos (c+d x)+\frac {1}{4} \left (143 A b^2+195 a b B+24 a^2 C+121 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 b \left (143 A b^2+195 a b B+24 a^2 C+121 b^2 C\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (13 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{143 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {9}{8} a^2 (143 a A+65 b B+85 a C)+\frac {11}{8} \left (117 a^3 B+273 a b^2 B+39 a^2 b (9 A+7 C)+7 b^3 (13 A+11 C)\right ) \cos (c+d x)+\frac {9}{8} \left (338 a^2 b B+117 b^3 B+24 a^3 C+39 a b^2 (11 A+9 C)\right ) \cos ^2(c+d x)\right ) \, dx}{1287}\\ &=\frac {2 b \left (143 A b^2+195 a b B+24 a^2 C+121 b^2 C\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (338 a^2 b B+117 b^3 B+24 a^3 C+39 a b^2 (11 A+9 C)\right ) \sin (c+d x)}{1001 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (13 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{143 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {117}{16} \left (165 a^2 b B+45 b^3 B+11 a^3 (7 A+5 C)+15 a b^2 (11 A+9 C)\right )+\frac {77}{16} \left (117 a^3 B+273 a b^2 B+39 a^2 b (9 A+7 C)+7 b^3 (13 A+11 C)\right ) \cos (c+d x)\right ) \, dx}{9009}\\ &=\frac {2 b \left (143 A b^2+195 a b B+24 a^2 C+121 b^2 C\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (338 a^2 b B+117 b^3 B+24 a^3 C+39 a b^2 (11 A+9 C)\right ) \sin (c+d x)}{1001 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (13 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{143 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{77} \left (\left (165 a^2 b B+45 b^3 B+11 a^3 (7 A+5 C)+15 a 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}{117} \left (\left (117 a^3 B+273 a b^2 B+39 a^2 b (9 A+7 C)+7 b^3 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {5}{2}}(c+d x) \, dx\\ &=\frac {2 b \left (143 A b^2+195 a b B+24 a^2 C+121 b^2 C\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (338 a^2 b B+117 b^3 B+24 a^3 C+39 a b^2 (11 A+9 C)\right ) \sin (c+d x)}{1001 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (13 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{143 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (117 a^3 B+273 a b^2 B+39 a^2 b (9 A+7 C)+7 b^3 (13 A+11 C)\right ) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (165 a^2 b B+45 b^3 B+11 a^3 (7 A+5 C)+15 a b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {1}{231} \left (\left (165 a^2 b B+45 b^3 B+11 a^3 (7 A+5 C)+15 a 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 {1}{195} \left (\left (117 a^3 B+273 a b^2 B+39 a^2 b (9 A+7 C)+7 b^3 (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 (117 a^3 B+273 a b^2 B+39 a^2 b (9 A+7 C)+7 b^3 (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 (165 a^2 b B+45 b^3 B+11 a^3 (7 A+5 C)+15 a b^2 (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 (143 A b^2+195 a b B+24 a^2 C+121 b^2 C\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (338 a^2 b B+117 b^3 B+24 a^3 C+39 a b^2 (11 A+9 C)\right ) \sin (c+d x)}{1001 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (13 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{143 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (117 a^3 B+273 a b^2 B+39 a^2 b (9 A+7 C)+7 b^3 (13 A+11 C)\right ) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (165 a^2 b B+45 b^3 B+11 a^3 (7 A+5 C)+15 a b^2 (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 = 3.89, size = 355, normalized size = 0.77 \[ \frac {\sqrt {\sec (c+d x)} \left (\sin (2 (c+d x)) \left (154 \cos (c+d x) \left (936 a^3 B+78 a^2 b (36 A+43 C)+3354 a b^2 B+b^3 (1118 A+1171 C)\right )+5 \left (77 b \cos (3 (c+d x)) \left (156 a^2 C+156 a b B+52 A b^2+89 b^2 C\right )+936 \cos (2 (c+d x)) \left (11 a^3 C+33 a^2 b B+3 a b^2 (11 A+16 C)+16 b^3 B\right )+78 \left (44 a^3 (14 A+13 C)+1716 a^2 b B+3 a b^2 (572 A+531 C)+531 b^3 B\right )+1638 b^2 (3 a C+b B) \cos (4 (c+d x))+693 b^3 C \cos (5 (c+d x))\right )\right )+6240 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (11 a^3 (7 A+5 C)+165 a^2 b B+15 a b^2 (11 A+9 C)+45 b^3 B\right )+7392 \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (117 a^3 B+39 a^2 b (9 A+7 C)+273 a b^2 B+7 b^3 (13 A+11 C)\right )\right )}{720720 d} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]

[Out]

(Sqrt[Sec[c + d*x]]*(7392*(117*a^3*B + 273*a*b^2*B + 39*a^2*b*(9*A + 7*C) + 7*b^3*(13*A + 11*C))*Sqrt[Cos[c +
d*x]]*EllipticE[(c + d*x)/2, 2] + 6240*(165*a^2*b*B + 45*b^3*B + 11*a^3*(7*A + 5*C) + 15*a*b^2*(11*A + 9*C))*S
qrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (154*(936*a^3*B + 3354*a*b^2*B + 78*a^2*b*(36*A + 43*C) + b^3*(1
118*A + 1171*C))*Cos[c + d*x] + 5*(78*(1716*a^2*b*B + 531*b^3*B + 44*a^3*(14*A + 13*C) + 3*a*b^2*(572*A + 531*
C)) + 936*(33*a^2*b*B + 16*b^3*B + 11*a^3*C + 3*a*b^2*(11*A + 16*C))*Cos[2*(c + d*x)] + 77*b*(52*A*b^2 + 156*a
*b*B + 156*a^2*C + 89*b^2*C)*Cos[3*(c + d*x)] + 1638*b^2*(b*B + 3*a*C)*Cos[4*(c + d*x)] + 693*b^3*C*Cos[5*(c +
d*x)]))*Sin[2*(c + d*x)]))/(720720*d)

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fricas [F]  time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b^{3} \cos \left (d x + c\right )^{5} + {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{4} + A a^{3} + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac {3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^3/sec(d*x + c)^(3/2), x)

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maple [B]  time = 3.58, size = 1188, normalized size = 2.57 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),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^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x
+1/2*c)^14+(262080*B*b^3+786240*C*a*b^2+1330560*C*b^3)*sin(1/2*d*x+1/2*c)^12*cos(1/2*d*x+1/2*c)+(-160160*A*b^3
-480480*B*a*b^2-655200*B*b^3-480480*C*a^2*b-1965600*C*a*b^2-1798720*C*b^3)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1
/2*c)+(308880*A*a*b^2+320320*A*b^3+308880*B*a^2*b+960960*B*a*b^2+739440*B*b^3+102960*C*a^3+960960*C*a^2*b+2218
320*C*a*b^2+1379840*C*b^3)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-216216*A*a^2*b-463320*A*a*b^2-296296*A*b^
3-72072*B*a^3-463320*B*a^2*b-888888*B*a*b^2-453960*B*b^3-154440*C*a^3-888888*C*a^2*b-1361880*C*a*b^2-666512*C*
b^3)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(60060*A*a^3+216216*A*a^2*b+360360*A*a*b^2+136136*A*b^3+72072*B*a
^3+360360*B*a^2*b+408408*B*a*b^2+180180*B*b^3+120120*C*a^3+408408*C*a^2*b+540540*C*a*b^2+198352*C*b^3)*sin(1/2
*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-30030*A*a^3-54054*A*a^2*b-102960*A*a*b^2-24024*A*b^3-18018*B*a^3-102960*B*a
^2*b-72072*B*a*b^2-36270*B*b^3-34320*C*a^3-72072*C*a^2*b-108810*C*a*b^2-27258*C*b^3)*sin(1/2*d*x+1/2*c)^2*cos(
1/2*d*x+1/2*c)+15015*A*a^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))+32175*A*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))-81081*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-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^3+32175*a^2*b*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)*Ellipti
cF(cos(1/2*d*x+1/2*c),2^(1/2))+8775*b^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)*Ellipt
icF(cos(1/2*d*x+1/2*c),2^(1/2))-27027*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^3-63063*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*b^2+10725*C*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*El
lipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^3+26325*C*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*
EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^2-63063*C*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1
/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2*b-17787*C*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2
)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^3/sec(d*x + c)^(3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(1/cos(c + d*x))^(3/2),x)

[Out]

int(((a + b*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(1/cos(c + d*x))^(3/2), 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))**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/sec(d*x+c)**(3/2),x)

[Out]

Timed out

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