3.1037 \(\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^6(c+d x) \, dx\)

Optimal. Leaf size=624 \[ -\frac{\tan (c+d x) \left (-12 a^2 b^2 (141 A+220 C)-256 a^4 (4 A+5 C)-2840 a^3 b B-150 a b^3 B+45 A b^4\right ) \sqrt{a+b \cos (c+d x)}}{1920 a^2 d}-\frac{\left (-4 a^2 b^2 (809 A+1180 C)-256 a^4 (4 A+5 C)-3560 a^3 b B-1330 a b^3 B+15 A b^4\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{1920 a d \sqrt{a+b \cos (c+d x)}}+\frac{\left (-12 a^2 b^2 (141 A+220 C)-256 a^4 (4 A+5 C)-2840 a^3 b B-150 a b^3 B+45 A b^4\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{1920 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left (40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+96 a^5 B-10 a b^4 B+3 A b^5\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{128 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt{a+b \cos (c+d x)}}{240 d}+\frac{\tan (c+d x) \sec (c+d x) \left (4 a^2 b (193 A+260 C)+360 a^3 B+590 a b^2 B+15 A b^3\right ) \sqrt{a+b \cos (c+d x)}}{960 a d}+\frac{(2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{8 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 2.71027, antiderivative size = 624, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233, Rules used = {3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac{\tan (c+d x) \left (-12 a^2 b^2 (141 A+220 C)-256 a^4 (4 A+5 C)-2840 a^3 b B-150 a b^3 B+45 A b^4\right ) \sqrt{a+b \cos (c+d x)}}{1920 a^2 d}-\frac{\left (-4 a^2 b^2 (809 A+1180 C)-256 a^4 (4 A+5 C)-3560 a^3 b B-1330 a b^3 B+15 A b^4\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{1920 a d \sqrt{a+b \cos (c+d x)}}+\frac{\left (-12 a^2 b^2 (141 A+220 C)-256 a^4 (4 A+5 C)-2840 a^3 b B-150 a b^3 B+45 A b^4\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{1920 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left (40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+96 a^5 B-10 a b^4 B+3 A b^5\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{128 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt{a+b \cos (c+d x)}}{240 d}+\frac{\tan (c+d x) \sec (c+d x) \left (4 a^2 b (193 A+260 C)+360 a^3 B+590 a b^2 B+15 A b^3\right ) \sqrt{a+b \cos (c+d x)}}{960 a d}+\frac{(2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{8 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d} \]

Antiderivative was successfully verified.

[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 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 3055

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] + Dis
t[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] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

Rule 3059

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]

Rule 2655

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*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

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

Rule 3002

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 2663

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*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

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

Rule 2807

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*
Sin[e + f*x])/(c + d)]), 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 2805

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[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]

Rubi steps

\begin{align*} \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{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}} F\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}} \Pi \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]  time = 7.23387, size = 930, normalized size = 1.49 \[ \frac{\frac{2 \left (1440 b B a^4+3088 A b^2 a^3+4160 b^2 C a^3+2360 b^3 B a^2+60 A b^4 a\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{\sqrt{a+b \cos (c+d x)}}+\frac{2 \left (2880 B a^5+6176 A b a^4+8320 b C a^4+4360 b^2 B a^3-492 A b^3 a^2-240 b^3 C a^2-450 b^4 B a+135 A b^5\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \left (45 A b^5-150 a B b^4-1692 a^2 A b^3-2640 a^2 C b^3-2840 a^3 B b^2-1024 a^4 A b-1280 a^4 C b\right ) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{\cos (c+d x) b+b}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right )|\frac{a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right )|\frac{a+b}{a-b}\right )-b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 (a+b \cos (c+d x)) a-b^2+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-4 (a+b \cos (c+d x)) a-b^2+2 (a+b \cos (c+d x))^2\right )}}{7680 a^2 d}+\frac{\sqrt{a+b \cos (c+d x)} \left (\frac{1}{40} \left (10 B \sin (c+d x) a^2+21 A b \sin (c+d x) a\right ) \sec ^4(c+d x)+\frac{1}{5} a^2 A \tan (c+d x) \sec ^4(c+d x)+\frac{1}{240} \left (64 A \sin (c+d x) a^2+80 C \sin (c+d x) a^2+170 b B \sin (c+d x) a+93 A b^2 \sin (c+d x)\right ) \sec ^3(c+d x)+\frac{\left (360 B \sin (c+d x) a^3+772 A b \sin (c+d x) a^2+1040 b C \sin (c+d x) a^2+590 b^2 B \sin (c+d x) a+15 A b^3 \sin (c+d x)\right ) \sec ^2(c+d x)}{960 a}+\frac{\left (1024 A \sin (c+d x) a^4+1280 C \sin (c+d x) a^4+2840 b B \sin (c+d x) a^3+1692 A b^2 \sin (c+d x) a^2+2640 b^2 C \sin (c+d x) a^2+150 b^3 B \sin (c+d x) a-45 A b^4 \sin (c+d x)\right ) \sec (c+d x)}{1920 a^2}\right )}{d} \]

Warning: Unable to verify antiderivative.

[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

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Maple [B]  time = 6.298, size = 5171, normalized size = 8.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

[Out]

result too large to display

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Fricas [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))^(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)]  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))**(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**6,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[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)