\(\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^6(c+d x) \, dx\) [1037]

Optimal result

Integrand size = 43, antiderivative size = 624 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{1920 a^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (15 A b^4-3560 a^3 b B-1330 a b^3 B-256 a^4 (4 A+5 C)-4 a^2 b^2 (809 A+1180 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{1920 a d \sqrt {a+b \cos (c+d x)}}+\frac {\left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{128 a^2 d \sqrt {a+b \cos (c+d x)}}-\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{1920 a^2 d}+\frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {(A b+2 a B) (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d} \]

[Out]

1/1920*(45*A*b^4-2840*B*a^3*b-150*B*a*b^3-256*a^4*(4*A+5*C)-12*a^2*b^2*(141*A+220*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)*(b/(a+b))^(1/2))*(a+b*cos(d*x+c))^(1/2)/a^2/d/((a
+b*cos(d*x+c))/(a+b))^(1/2)-1/1920*(15*A*b^4-3560*B*a^3*b-1330*B*a*b^3-256*a^4*(4*A+5*C)-4*a^2*b^2*(809*A+1180
*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)*(b/(a+b))^(1/2))*((a
+b*cos(d*x+c))/(a+b))^(1/2)/a/d/(a+b*cos(d*x+c))^(1/2)+1/128*(3*A*b^5+96*a^5*B+240*a^3*b^2*B-10*a*b^4*B+40*a^2
*b^3*(A+2*C)+80*a^4*b*(3*A+4*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c)
,2,2^(1/2)*(b/(a+b))^(1/2))*((a+b*cos(d*x+c))/(a+b))^(1/2)/a^2/d/(a+b*cos(d*x+c))^(1/2)+1/8*(A*b+2*B*a)*(a+b*c
os(d*x+c))^(3/2)*sec(d*x+c)^3*tan(d*x+c)/d+1/5*A*(a+b*cos(d*x+c))^(5/2)*sec(d*x+c)^4*tan(d*x+c)/d-1/1920*(45*A
*b^4-2840*B*a^3*b-150*B*a*b^3-256*a^4*(4*A+5*C)-12*a^2*b^2*(141*A+220*C))*(a+b*cos(d*x+c))^(1/2)*tan(d*x+c)/a^
2/d+1/960*(15*A*b^3+360*B*a^3+590*B*a*b^2+4*a^2*b*(193*A+260*C))*sec(d*x+c)*(a+b*cos(d*x+c))^(1/2)*tan(d*x+c)/
a/d+1/240*(15*A*b^2+110*B*a*b+16*a^2*(4*A+5*C))*sec(d*x+c)^2*(a+b*cos(d*x+c))^(1/2)*tan(d*x+c)/d

Rubi [A] (verified)

Time = 3.13 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {3126, 3134, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{240 d}+\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{960 a d}-\frac {\tan (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{1920 a^2 d}-\frac {\left (-256 a^4 (4 A+5 C)-3560 a^3 b B-4 a^2 b^2 (809 A+1180 C)-1330 a b^3 B+15 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{1920 a d \sqrt {a+b \cos (c+d x)}}+\frac {\left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{1920 a^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (96 a^5 B+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+40 a^2 b^3 (A+2 C)-10 a b^4 B+3 A b^5\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{128 a^2 d \sqrt {a+b \cos (c+d x)}}+\frac {(2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{8 d}+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d} \]

[In]

Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]

[Out]

((45*A*b^4 - 2840*a^3*b*B - 150*a*b^3*B - 256*a^4*(4*A + 5*C) - 12*a^2*b^2*(141*A + 220*C))*Sqrt[a + b*Cos[c +
d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(1920*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((15*A*b^4 - 3
560*a^3*b*B - 1330*a*b^3*B - 256*a^4*(4*A + 5*C) - 4*a^2*b^2*(809*A + 1180*C))*Sqrt[(a + b*Cos[c + d*x])/(a +
b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(1920*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b^5 + 96*a^5*B + 240*a^
3*b^2*B - 10*a*b^4*B + 40*a^2*b^3*(A + 2*C) + 80*a^4*b*(3*A + 4*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*Ellipti
cPi[2, (c + d*x)/2, (2*b)/(a + b)])/(128*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((45*A*b^4 - 2840*a^3*b*B - 150*a*b
^3*B - 256*a^4*(4*A + 5*C) - 12*a^2*b^2*(141*A + 220*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(1920*a^2*d) +
((15*A*b^3 + 360*a^3*B + 590*a*b^2*B + 4*a^2*b*(193*A + 260*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c +
d*x])/(960*a*d) + ((15*A*b^2 + 110*a*b*B + 16*a^2*(4*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c
+ d*x])/(240*d) + ((A*b + 2*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (A*(a + b*Cos
[c + d*x])^(5/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2884

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]

Rule 2886

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/
(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 3081

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

Rule 3126

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 3134

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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + D
ist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*
(b*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(
b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*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] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3138

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) +
(f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Dist[C/(b*d), Int[Sqrt[a + b*Sin[e + f*x]]
, x], x] - Dist[1/(b*d), Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e +
f*x]]*(c + d*Sin[e + f*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]

Rubi steps \begin{align*} \text {integral}& = \frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {1}{5} \int (a+b \cos (c+d x))^{3/2} \left (\frac {5}{2} (A b+2 a B)+(4 a A+5 b B+5 a C) \cos (c+d x)+\frac {1}{2} b (3 A+10 C) \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx \\ & = \frac {(A b+2 a B) (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {1}{20} \int \sqrt {a+b \cos (c+d x)} \left (\frac {1}{4} \left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right )+\frac {1}{2} \left (30 a^2 B+40 b^2 B+a b (59 A+80 C)\right ) \cos (c+d x)+\frac {1}{4} b (39 A b+30 a B+80 b C) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx \\ & = \frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {(A b+2 a B) (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {1}{60} \int \frac {\left (\frac {1}{8} \left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right )+\frac {1}{4} \left (490 a^2 b B+240 b^3 B+32 a^3 (4 A+5 C)+3 a b^2 (167 A+240 C)\right ) \cos (c+d x)+\frac {3}{8} b \left (170 a b B+16 a^2 (4 A+5 C)+b^2 (93 A+160 C)\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx \\ & = \frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {(A b+2 a B) (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {\int \frac {\left (\frac {1}{16} \left (1024 a^4 A+1692 a^2 A b^2-45 A b^4+2840 a^3 b B+150 a b^3 B+1280 a^4 C+2640 a^2 b^2 C\right )+\frac {1}{8} a \left (360 a^3 B+1610 a b^2 B+3 b^3 (191 A+320 C)+4 a^2 b (289 A+380 C)\right ) \cos (c+d x)+\frac {1}{16} b \left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{120 a} \\ & = -\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{1920 a^2 d}+\frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {(A b+2 a B) (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {\int \frac {\left (\frac {15}{32} \left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right )+\frac {1}{16} a b \left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \cos (c+d x)+\frac {1}{32} b \left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{120 a^2} \\ & = -\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{1920 a^2 d}+\frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {(A b+2 a B) (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac {\int \frac {\left (-\frac {15}{32} b \left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right )+\frac {1}{32} a b \left (15 A b^4-3560 a^3 b B-1330 a b^3 B-256 a^4 (4 A+5 C)-4 a^2 b^2 (809 A+1180 C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{120 a^2 b}+\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{3840 a^2} \\ & = -\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{1920 a^2 d}+\frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {(A b+2 a B) (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {\left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{256 a^2}-\frac {\left (15 A b^4-3560 a^3 b B-1330 a b^3 B-256 a^4 (4 A+5 C)-4 a^2 b^2 (809 A+1180 C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{3840 a}+\frac {\left (\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{3840 a^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}} \\ & = \frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{1920 a^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{1920 a^2 d}+\frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {(A b+2 a B) (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {\left (\left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{256 a^2 \sqrt {a+b \cos (c+d x)}}-\frac {\left (\left (15 A b^4-3560 a^3 b B-1330 a b^3 B-256 a^4 (4 A+5 C)-4 a^2 b^2 (809 A+1180 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{3840 a \sqrt {a+b \cos (c+d x)}} \\ & = \frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{1920 a^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (15 A b^4-3560 a^3 b B-1330 a b^3 B-256 a^4 (4 A+5 C)-4 a^2 b^2 (809 A+1180 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{1920 a d \sqrt {a+b \cos (c+d x)}}+\frac {\left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{128 a^2 d \sqrt {a+b \cos (c+d x)}}-\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{1920 a^2 d}+\frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {(A b+2 a B) (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 8.12 (sec) , antiderivative size = 930, normalized size of antiderivative = 1.49 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {\frac {2 \left (3088 a^3 A b^2+60 a A b^4+1440 a^4 b B+2360 a^2 b^3 B+4160 a^3 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (6176 a^4 A b-492 a^2 A b^3+135 A b^5+2880 a^5 B+4360 a^3 b^2 B-450 a b^4 B+8320 a^4 b C-240 a^2 b^3 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (-1024 a^4 A b-1692 a^2 A b^3+45 A b^5-2840 a^3 b^2 B-150 a b^4 B-1280 a^4 b C-2640 a^2 b^3 C\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {b+b \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-b^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-b^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2\right )}}{7680 a^2 d}+\frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{40} \sec ^4(c+d x) \left (21 a A b \sin (c+d x)+10 a^2 B \sin (c+d x)\right )+\frac {1}{240} \sec ^3(c+d x) \left (64 a^2 A \sin (c+d x)+93 A b^2 \sin (c+d x)+170 a b B \sin (c+d x)+80 a^2 C \sin (c+d x)\right )+\frac {\sec ^2(c+d x) \left (772 a^2 A b \sin (c+d x)+15 A b^3 \sin (c+d x)+360 a^3 B \sin (c+d x)+590 a b^2 B \sin (c+d x)+1040 a^2 b C \sin (c+d x)\right )}{960 a}+\frac {\sec (c+d x) \left (1024 a^4 A \sin (c+d x)+1692 a^2 A b^2 \sin (c+d x)-45 A b^4 \sin (c+d x)+2840 a^3 b B \sin (c+d x)+150 a b^3 B \sin (c+d x)+1280 a^4 C \sin (c+d x)+2640 a^2 b^2 C \sin (c+d x)\right )}{1920 a^2}+\frac {1}{5} a^2 A \sec ^4(c+d x) \tan (c+d x)\right )}{d} \]

[In]

Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]

[Out]

((2*(3088*a^3*A*b^2 + 60*a*A*b^4 + 1440*a^4*b*B + 2360*a^2*b^3*B + 4160*a^3*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(
a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(6176*a^4*A*b - 492*a^2*A*b^3 + 1
35*A*b^5 + 2880*a^5*B + 4360*a^3*b^2*B - 450*a*b^4*B + 8320*a^4*b*C - 240*a^2*b^3*C)*Sqrt[(a + b*Cos[c + d*x])
/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-1024*a^4*A*b - 1692*a
^2*A*b^3 + 45*A*b^5 - 2840*a^3*b^2*B - 150*a*b^4*B - 1280*a^4*b*C - 2640*a^2*b^3*C)*Sqrt[(b - b*Cos[c + d*x])/
(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)
^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b
*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d
*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*
a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[
c + d*x])^2)))/(7680*a^2*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]^4*(21*a*A*b*Sin[c + d*x] + 10*a^2*B*Sin
[c + d*x]))/40 + (Sec[c + d*x]^3*(64*a^2*A*Sin[c + d*x] + 93*A*b^2*Sin[c + d*x] + 170*a*b*B*Sin[c + d*x] + 80*
a^2*C*Sin[c + d*x]))/240 + (Sec[c + d*x]^2*(772*a^2*A*b*Sin[c + d*x] + 15*A*b^3*Sin[c + d*x] + 360*a^3*B*Sin[c
+ d*x] + 590*a*b^2*B*Sin[c + d*x] + 1040*a^2*b*C*Sin[c + d*x]))/(960*a) + (Sec[c + d*x]*(1024*a^4*A*Sin[c + d
*x] + 1692*a^2*A*b^2*Sin[c + d*x] - 45*A*b^4*Sin[c + d*x] + 2840*a^3*b*B*Sin[c + d*x] + 150*a*b^3*B*Sin[c + d*
x] + 1280*a^4*C*Sin[c + d*x] + 2640*a^2*b^2*C*Sin[c + d*x]))/(1920*a^2) + (a^2*A*Sec[c + d*x]^4*Tan[c + d*x])/
5))/d

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(5170\) vs. \(2(673)=1346\).

Time = 1260.12 (sec) , antiderivative size = 5171, normalized size of antiderivative = 8.29

method	result	size
default	\(\text {Expression too large to display}\)	\(5171\)
parts	\(\text {Expression too large to display}\)	\(7471\)

[In]

int((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\text {Timed out} \]

[In]

integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\text {Timed out} \]

[In]

integrate((a+b*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**6,x)

[Out]

Timed out

Maxima [F]

\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{6} \,d x } \]

[In]

integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/2)*sec(d*x + c)^6, x)

Giac [F]

\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{6} \,d x } \]

[In]

integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/2)*sec(d*x + c)^6, x)

Mupad [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\cos \left (c+d\,x\right )}^6} \,d x \]

[In]

int(((a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^6,x)

[Out]

int(((a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^6, x)