3.703 \(\int \frac{(a+b \cos (c+d x))^4 (A+C \cos ^2(c+d x))}{\cos ^{\frac{13}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=377 \[ \frac{2 \left (66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+77 b^4 (A+3 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}-\frac{8 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \left (9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+64 A b^4\right ) \sin (c+d x)}{693 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a b \left (a^2 (673 A+891 C)+96 A b^2\right ) \sin (c+d x)}{3465 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{8 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \cos ^{\frac{9}{2}}(c+d x)} \]

[Out]

(-8*a*b*(3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(77*b^4*(A + 3*C) + 66*a^
2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a*b*(96*A*b^2 + a^2*(673*A + 8
91*C))*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(5/2)) + (2*(64*A*b^4 + 15*a^4*(9*A + 11*C) + 9*a^2*b^2*(101*A + 143
*C))*Sin[c + d*x])/(693*d*Cos[c + d*x]^(3/2)) + (8*a*b*(3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sin[c + d*x])/(15
*d*Sqrt[Cos[c + d*x]]) + (2*(16*A*b^2 + 3*a^2*(9*A + 11*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(231*d*Cos[c
+ d*x]^(7/2)) + (16*A*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + b*Cos[c + d
*x])^4*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))

________________________________________________________________________________________

Rubi [A]  time = 1.23302, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {3048, 3047, 3031, 3021, 2748, 2636, 2639, 2641} \[ \frac{2 \left (66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+77 b^4 (A+3 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}-\frac{8 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \left (9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+64 A b^4\right ) \sin (c+d x)}{693 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a b \left (a^2 (673 A+891 C)+96 A b^2\right ) \sin (c+d x)}{3465 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{8 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \cos ^{\frac{9}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]

[Out]

(-8*a*b*(3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(77*b^4*(A + 3*C) + 66*a^
2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a*b*(96*A*b^2 + a^2*(673*A + 8
91*C))*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(5/2)) + (2*(64*A*b^4 + 15*a^4*(9*A + 11*C) + 9*a^2*b^2*(101*A + 143
*C))*Sin[c + d*x])/(693*d*Cos[c + d*x]^(3/2)) + (8*a*b*(3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sin[c + d*x])/(15
*d*Sqrt[Cos[c + d*x]]) + (2*(16*A*b^2 + 3*a^2*(9*A + 11*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(231*d*Cos[c
+ d*x]^(7/2)) + (16*A*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + b*Cos[c + d
*x])^4*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))

Rule 3048

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C + A*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[
e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m
- 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n +
2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(m + n + 2) + C*(c^2*(
m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] &
& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C - B*c*d + A*d^2)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(
c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c
*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*
d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)
))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2,
0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3031

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[((b*c - a*d)*(A*b^2 - a*b*B + a^2*C)*
Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)),
Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b
^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f*x] - b*C*d*(m +
1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && Ne
Q[a^2 - b^2, 0] && LtQ[m, -1]

Rule 3021

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m +
1)*(a^2 - b^2)), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B + a*
C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e,
f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

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 2636

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1))/(b*d*(n +
1)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[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]

Rubi steps

\begin{align*} \int \frac{(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac{13}{2}}(c+d x)} \, dx &=\frac{2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{2}{11} \int \frac{(a+b \cos (c+d x))^3 \left (4 A b+\frac{1}{2} a (9 A+11 C) \cos (c+d x)+\frac{1}{2} b (A+11 C) \cos ^2(c+d x)\right )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx\\ &=\frac{16 A b (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{4}{99} \int \frac{(a+b \cos (c+d x))^2 \left (\frac{3}{4} \left (16 A b^2+3 a^2 (9 A+11 C)\right )+\frac{1}{2} a b (73 A+99 C) \cos (c+d x)+\frac{1}{4} b^2 (17 A+99 C) \cos ^2(c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 \left (16 A b^2+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{16 A b (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{8}{693} \int \frac{(a+b \cos (c+d x)) \left (\frac{1}{4} b \left (96 A b^2+a^2 (673 A+891 C)\right )+\frac{1}{8} a \left (45 a^2 (9 A+11 C)+b^2 (1381 A+2079 C)\right ) \cos (c+d x)+\frac{1}{8} b \left (9 a^2 (9 A+11 C)+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{4 a b \left (96 A b^2+a^2 (673 A+891 C)\right ) \sin (c+d x)}{3465 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (16 A b^2+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{16 A b (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}-\frac{16 \int \frac{-\frac{15}{16} \left (64 A b^4+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right )-\frac{231}{4} a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \cos (c+d x)-\frac{5}{16} b^2 \left (9 a^2 (9 A+11 C)+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx}{3465}\\ &=\frac{4 a b \left (96 A b^2+a^2 (673 A+891 C)\right ) \sin (c+d x)}{3465 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (64 A b^4+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (16 A b^2+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{16 A b (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}-\frac{32 \int \frac{-\frac{693}{8} a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right )-\frac{45}{32} \left (77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx}{10395}\\ &=\frac{4 a b \left (96 A b^2+a^2 (673 A+891 C)\right ) \sin (c+d x)}{3465 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (64 A b^4+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (16 A b^2+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{16 A b (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{1}{15} \left (4 a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right )\right ) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx-\frac{1}{231} \left (-77 b^4 (A+3 C)-66 a^2 b^2 (5 A+7 C)-5 a^4 (9 A+11 C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{4 a b \left (96 A b^2+a^2 (673 A+891 C)\right ) \sin (c+d x)}{3465 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (64 A b^4+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 \left (16 A b^2+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{16 A b (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}-\frac{1}{15} \left (4 a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right )\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{8 a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{4 a b \left (96 A b^2+a^2 (673 A+891 C)\right ) \sin (c+d x)}{3465 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (64 A b^4+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 \left (16 A b^2+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{16 A b (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}\\ \end{align*}

Mathematica [A]  time = 4.45073, size = 284, normalized size = 0.75 \[ \frac{10 \left (66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+77 b^4 (A+3 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-616 \left (a^3 b (7 A+9 C)+3 a b^3 (3 A+5 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\frac{45 \left (\left (66 a^2 A b^2+a^4 (9 A+11 C)\right ) \sin (2 (c+d x))+14 a^4 A \tan (c+d x)\right )+2 \sin (c+d x) \left (924 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \cos ^4(c+d x)+15 \left (66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+77 A b^4\right ) \cos ^3(c+d x)+308 a b \left (a^2 (7 A+9 C)+9 A b^2\right ) \cos ^2(c+d x)+1540 a^3 A b\right )}{3 \cos ^{\frac{9}{2}}(c+d x)}}{1155 d} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]

[Out]

(-616*(3*a*b^3*(3*A + 5*C) + a^3*b*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2] + 10*(77*b^4*(A + 3*C) + 66*a^2*b^2*
(5*A + 7*C) + 5*a^4*(9*A + 11*C))*EllipticF[(c + d*x)/2, 2] + (2*(1540*a^3*A*b + 308*a*b*(9*A*b^2 + a^2*(7*A +
9*C))*Cos[c + d*x]^2 + 15*(77*A*b^4 + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*Cos[c + d*x]^3 + 924*a*b*(
3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Cos[c + d*x]^4)*Sin[c + d*x] + 45*((66*a^2*A*b^2 + a^4*(9*A + 11*C))*Sin[
2*(c + d*x)] + 14*a^4*A*Tan[c + d*x]))/(3*Cos[c + d*x]^(9/2)))/(1155*d)

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Maple [B]  time = 2.655, size = 1521, normalized size = 4. \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+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x)

[Out]

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*C*b^4*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d
*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)
)-8/5*a*b*(A*b^2+C*a^2)/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+
1/2*c)^2*(12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/
2)*sin(1/2*d*x+1/2*c)^4-24*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(s
in(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^2+24*sin(1/2*d*x+1/2*c)^4*cos(1
/2*d*x+1/2*c)+3*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^
(1/2)-8*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+2*A*a^4*
(-1/352*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^6-
9/616*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^4-15
/154*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^2+15/
77*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^
2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2*b^2*(A*b^2+6*C*a^2)*(-1/6*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x
+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^2+1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1
/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(
1/2)))+2*a^2*(6*A*b^2+C*a^2)*(-1/56*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-
1/2+cos(1/2*d*x+1/2*c)^2)^4-5/42*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2
+cos(1/2*d*x+1/2*c)^2)^2+5/21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1
/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+8*A*a^3*b*(-1/144*cos(1/2*d*x+1/2*c
)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^5-7/180*cos(1/2*d*x+1/2*c)*
(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^3-14/15*sin(1/2*d*x+1/2*c)^2*
cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)+7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)
*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+
1/2*c),2^(1/2))-7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+s
in(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))))+8*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)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2
*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+2*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*cos(
1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2)/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1))/sin(1/2*d*x+1/2*c)/(2*co
s(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} + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac{13}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^4/cos(d*x + c)^(13/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{C b^{4} \cos \left (d x + c\right )^{6} + 4 \, C a b^{3} \cos \left (d x + c\right )^{5} + 4 \, A a^{3} b \cos \left (d x + c\right ) + A a^{4} +{\left (6 \, C a^{2} b^{2} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (C a^{3} b + A a b^{3}\right )} \cos \left (d x + c\right )^{3} +{\left (C a^{4} + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}}{\cos \left (d x + c\right )^{\frac{13}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x, algorithm="fricas")

[Out]

integral((C*b^4*cos(d*x + c)^6 + 4*C*a*b^3*cos(d*x + c)^5 + 4*A*a^3*b*cos(d*x + c) + A*a^4 + (6*C*a^2*b^2 + A*
b^4)*cos(d*x + c)^4 + 4*(C*a^3*b + A*a*b^3)*cos(d*x + c)^3 + (C*a^4 + 6*A*a^2*b^2)*cos(d*x + c)^2)/cos(d*x + c
)^(13/2), 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))**4*(A+C*cos(d*x+c)**2)/cos(d*x+c)**(13/2),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} + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac{13}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^4/cos(d*x + c)^(13/2), x)