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

[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]])

________________________________________________________________________________________

Rubi [A]  time = 1.14125, antiderivative size = 463, normalized size of antiderivative = 1., 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 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 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 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 \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*}

Mathematica [A]  time = 3.87686, size = 355, normalized size = 0.77 \[ \frac{\sqrt{\sec (c+d x)} \left (\sin (2 (c+d x)) \left (154 \cos (c+d x) \left (78 a^2 b (36 A+43 C)+936 a^3 B+3354 a b^2 B+b^3 (1118 A+1171 C)\right )+5 \left (77 b \cos (3 (c+d x)) \left (156 a^2 C+156 a b B+52 A b^2+89 b^2 C\right )+936 \cos (2 (c+d x)) \left (33 a^2 b B+11 a^3 C+3 a b^2 (11 A+16 C)+16 b^3 B\right )+78 \left (44 a^3 (14 A+13 C)+1716 a^2 b B+3 a b^2 (572 A+531 C)+531 b^3 B\right )+1638 b^2 (3 a C+b B) \cos (4 (c+d x))+693 b^3 C \cos (5 (c+d x))\right )\right )+6240 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (11 a^3 (7 A+5 C)+165 a^2 b B+15 a b^2 (11 A+9 C)+45 b^3 B\right )+7392 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (39 a^2 b (9 A+7 C)+117 a^3 B+273 a b^2 B+7 b^3 (13 A+11 C)\right )\right )}{720720 d} \]

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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Maple [B]  time = 1.567, size = 1188, normalized size = 2.6 \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))^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*a^3*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))+26325*C*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))-63063*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-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^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 \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} \end{align*}

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

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

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)