3.1094 \(\int \frac {(a+b \cos (c+d x))^4 (A+B \cos (c+d x)+C \cos ^2(c+d x))}{\cos ^{\frac {13}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=475 \[ \frac {2 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right )}{231 d}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{15 d}+\frac {2 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{693 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{15 d \sqrt {\cos (c+d x)}}+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]

[Out]

-2/15*(7*a^4*B+54*a^2*b^2*B+15*b^4*B+12*a*b^3*(3*A+5*C)+4*a^3*b*(7*A+9*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))/d+2/231*(220*a^3*b*B+308*a*b^3*B+77*b^4*(A+3*C)+66*a^2*b^2*
(5*A+7*C)+5*a^4*(9*A+11*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))/d+2/3465*a*(192*A*b^3+539*a^3*B+1353*a*b^2*B+2*a^2*b*(673*A+891*C))*sin(d*x+c)/d/cos(d*x+c)^(5/2)+2/693*(6
4*A*b^4+660*a^3*b*B+682*a*b^3*B+15*a^4*(9*A+11*C)+9*a^2*b^2*(101*A+143*C))*sin(d*x+c)/d/cos(d*x+c)^(3/2)+2/231
*(16*A*b^2+55*a*b*B+3*a^2*(9*A+11*C))*(a+b*cos(d*x+c))^2*sin(d*x+c)/d/cos(d*x+c)^(7/2)+2/99*(8*A*b+11*B*a)*(a+
b*cos(d*x+c))^3*sin(d*x+c)/d/cos(d*x+c)^(9/2)+2/11*A*(a+b*cos(d*x+c))^4*sin(d*x+c)/d/cos(d*x+c)^(11/2)+2/15*(7
*a^4*B+54*a^2*b^2*B+15*b^4*B+12*a*b^3*(3*A+5*C)+4*a^3*b*(7*A+9*C))*sin(d*x+c)/d/cos(d*x+c)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.40, antiderivative size = 475, normalized size of antiderivative = 1.00, 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 = {3047, 3031, 3021, 2748, 2636, 2639, 2641} \[ \frac {2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)\right )}{231 d}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{15 d}+\frac {2 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+660 a^3 b B+682 a b^3 B+64 A b^4\right )}{693 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a \sin (c+d x) \left (2 a^2 b (673 A+891 C)+539 a^3 B+1353 a b^2 B+192 A b^3\right )}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{15 d \sqrt {\cos (c+d x)}}+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]

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))/Cos[c + d*x]^(13/2),x]

[Out]

(-2*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2]
)/(15*d) + (2*(220*a^3*b*B + 308*a*b^3*B + 77*b^4*(A + 3*C) + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*Ell
ipticF[(c + d*x)/2, 2])/(231*d) + (2*a*(192*A*b^3 + 539*a^3*B + 1353*a*b^2*B + 2*a^2*b*(673*A + 891*C))*Sin[c
+ d*x])/(3465*d*Cos[c + d*x]^(5/2)) + (2*(64*A*b^4 + 660*a^3*b*B + 682*a*b^3*B + 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)) + (2*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^
3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(16*A*b^2 + 55*a*b*B + 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)) + (2*(8*A*b + 11*a*B)*(a + b*Co
s[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 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]

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 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 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 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]

Rubi steps

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

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Mathematica [A]  time = 6.37, size = 381, normalized size = 0.80 \[ \frac {10 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right )-154 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )+\frac {45 \left (14 a^4 A \tan (c+d x)+a^2 \sin (2 (c+d x)) \left (a^2 (9 A+11 C)+44 a b B+66 A b^2\right )\right )+2 \sin (c+d x) \left (385 a^3 (a B+4 A b)+77 a \cos ^2(c+d x) \left (7 a^3 B+4 a^2 b (7 A+9 C)+54 a b^2 B+36 A b^3\right )+231 \cos ^4(c+d x) \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )+15 \cos ^3(c+d x) \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 A b^4\right )\right )}{3 \cos ^{\frac {9}{2}}(c+d x)}}{1155 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-154*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*EllipticE[(c + d*x)/2,
2] + 10*(220*a^3*b*B + 308*a*b^3*B + 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*(385*a^3*(4*A*b + a*B) + 77*a*(36*A*b^3 + 7*a^3*B + 54*a*b^2*B + 4*a^2*b*(7*A + 9*C))*Co
s[c + d*x]^2 + 15*(77*A*b^4 + 220*a^3*b*B + 308*a*b^3*B + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*Cos[c +
d*x]^3 + 231*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*Cos[c + d*x]^4)
*Sin[c + d*x] + 45*(a^2*(66*A*b^2 + 44*a*b*B + a^2*(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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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b^{4} \cos \left (d x + c\right )^{6} + {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{5} + A a^{4} + {\left (6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (2 \, C a^{3} b + 3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{\frac {13}{2}}}, x\right ) \]

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)/cos(d*x+c)^(13/2),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))/cos(d*x + c)^(13/2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {13}{2}}}\,{d x} \]

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)/cos(d*x+c)^(13/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 17.85, size = 1550, normalized size = 3.26 \[ \text {result too large to display} \]

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)/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)
)-4/5*a*b*(2*A*b^2+3*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))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1
/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4-24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-12*EllipticE(cos(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)^(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*(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))-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*b^2*(A*b^2+4*B*a*b+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*b^3*(B*b+4*C*a)*(-(-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)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Elli
pticE(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)*si
n(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)+2*a^3*(4*A*b+B*a)*(-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+
sin(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))))+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*a^2*(6*A*b^2+4*B*a*b+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))))/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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {13}{2}}}\,{d x} \]

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)/cos(d*x+c)^(13/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 10.60, size = 1161, normalized size = 2.44 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8*hypergeom([-5/4, 1/2], -1/4, cos(c + d*x)^2)*((9*A*a*b^3*sin(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x)^2)
^(1/2)) + (4*A*a^3*b*sin(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)) + (5*A*a^3*b*sin(c + d*x))/(cos
(c + d*x)^(9/2)*(sin(c + d*x)^2)^(1/2))))/(45*d) + (8*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2)*((7*B*a*b^3*
sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (4*B*a^3*b*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(
c + d*x)^2)^(1/2)) + (3*B*a^3*b*sin(c + d*x))/(cos(c + d*x)^(7/2)*(sin(c + d*x)^2)^(1/2))))/(21*d) + (8*hyperg
eom([-3/4, 1/2], 5/4, cos(c + d*x)^2)*((9*A*a^4*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (7
*A*a^4*sin(c + d*x))/(cos(c + d*x)^(7/2)*(sin(c + d*x)^2)^(1/2)) + (66*A*a^2*b^2*sin(c + d*x))/(cos(c + d*x)^(
3/2)*(sin(c + d*x)^2)^(1/2))))/(231*d) - (8*hypergeom([-1/4, 1/2], 7/4, cos(c + d*x)^2)*((7*B*a^4*sin(c + d*x)
)/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (5*B*a^4*sin(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1
/2)) + (54*B*a^2*b^2*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2))))/(135*d) + (2*hypergeom([-3/4,
1/2], 1/4, cos(c + d*x)^2)*((36*A*a^4*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (20*A*a^4*s
in(c + d*x))/(cos(c + d*x)^(7/2)*(sin(c + d*x)^2)^(1/2)) + (21*A*a^4*sin(c + d*x))/(cos(c + d*x)^(11/2)*(sin(c
+ d*x)^2)^(1/2)) + (77*A*b^4*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (264*A*a^2*b^2*sin(c
+ d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (198*A*a^2*b^2*sin(c + d*x))/(cos(c + d*x)^(7/2)*(sin(c
+ d*x)^2)^(1/2))))/(231*d) + (2*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2)*((28*B*a^4*sin(c + d*x))/(cos(c +
d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (12*B*a^4*sin(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)) + (5
*B*a^4*sin(c + d*x))/(cos(c + d*x)^(9/2)*(sin(c + d*x)^2)^(1/2)) + (45*B*b^4*sin(c + d*x))/(cos(c + d*x)^(1/2)
*(sin(c + d*x)^2)^(1/2)) + (216*B*a^2*b^2*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (54*B*a^
2*b^2*sin(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2))))/(45*d) + (2*C*b^4*ellipticF(c/2 + (d*x)/2, 2
))/d + (2*C*a^4*sin(c + d*x)*hypergeom([-7/4, 1/2], -3/4, cos(c + d*x)^2))/(7*d*cos(c + d*x)^(7/2)*(sin(c + d*
x)^2)^(1/2)) - (32*A*a^3*b*sin(c + d*x)*hypergeom([-5/4, 1/2], 3/4, cos(c + d*x)^2))/(15*d*cos(c + d*x)^(5/2)*
(sin(c + d*x)^2)^(1/2)) + (32*B*a^3*b*sin(c + d*x)*hypergeom([-3/4, 1/2], 5/4, cos(c + d*x)^2))/(21*d*cos(c +
d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (8*C*a*b^3*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2))/(d*c
os(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (8*C*a^3*b*sin(c + d*x)*hypergeom([-5/4, 1/2], -1/4, cos(c + d*x)^
2))/(5*d*cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)) + (4*C*a^2*b^2*sin(c + d*x)*hypergeom([-3/4, 1/2], 1/4, co
s(c + d*x)^2))/(d*cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2))

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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))**4*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/cos(d*x+c)**(13/2),x)

[Out]

Timed out

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