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

Optimal. Leaf size=774 \[ -\frac{\sqrt{a+b} \cot (c+d x) \left (-4 a^2 b^2 (423 A+295 B+660 C)-8 a^3 b (193 A+355 B+260 C)-16 a^4 (64 A+45 B+80 C)-30 a b^3 (A+5 B)+45 A b^4\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{1920 a^2 d}-\frac{\sin (c+d x) \left (-12 a^2 b^2 (141 A+220 C)-256 a^4 (4 A+5 C)-2840 a^3 b B-150 a b^3 B+45 A b^4\right ) \sqrt{a+b \sec (c+d x)}}{1920 a^2 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{240 d}+\frac{\sin (c+d x) \cos (c+d x) \left (4 a^2 b (193 A+260 C)+360 a^3 B+590 a b^2 B+15 A b^3\right ) \sqrt{a+b \sec (c+d x)}}{960 a d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left (-12 a^2 b^2 (141 A+220 C)-256 a^4 (4 A+5 C)-2840 a^3 b B-150 a b^3 B+45 A b^4\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{1920 a^2 b d}-\frac{\sqrt{a+b} \cot (c+d x) \left (40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+96 a^5 B-10 a b^4 B+3 A b^5\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{128 a^3 d}+\frac{(2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{8 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d} \]

[Out]

-((a - b)*Sqrt[a + b]*(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
))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c +
d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(1920*a^2*b*d) - (Sqrt[a + b]*(45*A*b^4 - 30*a*b^3*(A
+ 5*B) - 16*a^4*(64*A + 45*B + 80*C) - 8*a^3*b*(193*A + 355*B + 260*C) - 4*a^2*b^2*(423*A + 295*B + 660*C))*C
ot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]
))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(1920*a^2*d) - (Sqrt[a + b]*(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))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[S
qrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec
[c + d*x]))/(a - b))])/(128*a^3*d) - ((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*Sec[c + d*x]]*Sin[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))*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(960*a*d) + ((15*A*b^2 + 110*a*
b*B + 16*a^2*(4*A + 5*C))*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(240*d) + ((A*b + 2*a*B)*Cos[c
+ d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*Sin[c
+ d*x])/(5*d)

________________________________________________________________________________________

Rubi [A]  time = 3.20325, antiderivative size = 774, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {4094, 4104, 4058, 3921, 3784, 3832, 4004} \[ -\frac{\sin (c+d x) \left (-12 a^2 b^2 (141 A+220 C)-256 a^4 (4 A+5 C)-2840 a^3 b B-150 a b^3 B+45 A b^4\right ) \sqrt{a+b \sec (c+d x)}}{1920 a^2 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{240 d}+\frac{\sin (c+d x) \cos (c+d x) \left (4 a^2 b (193 A+260 C)+360 a^3 B+590 a b^2 B+15 A b^3\right ) \sqrt{a+b \sec (c+d x)}}{960 a d}-\frac{\sqrt{a+b} \cot (c+d x) \left (-4 a^2 b^2 (423 A+295 B+660 C)-8 a^3 b (193 A+355 B+260 C)-16 a^4 (64 A+45 B+80 C)-30 a b^3 (A+5 B)+45 A b^4\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{1920 a^2 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left (-12 a^2 b^2 (141 A+220 C)-256 a^4 (4 A+5 C)-2840 a^3 b B-150 a b^3 B+45 A b^4\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{1920 a^2 b d}-\frac{\sqrt{a+b} \cot (c+d x) \left (40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+96 a^5 B-10 a b^4 B+3 A b^5\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{128 a^3 d}+\frac{(2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{8 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a - b)*Sqrt[a + b]*(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
))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c +
d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(1920*a^2*b*d) - (Sqrt[a + b]*(45*A*b^4 - 30*a*b^3*(A
+ 5*B) - 16*a^4*(64*A + 45*B + 80*C) - 8*a^3*b*(193*A + 355*B + 260*C) - 4*a^2*b^2*(423*A + 295*B + 660*C))*C
ot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]
))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(1920*a^2*d) - (Sqrt[a + b]*(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))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[S
qrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec
[c + d*x]))/(a - b))])/(128*a^3*d) - ((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*Sec[c + d*x]]*Sin[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))*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(960*a*d) + ((15*A*b^2 + 110*a*
b*B + 16*a^2*(4*A + 5*C))*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(240*d) + ((A*b + 2*a*B)*Cos[c
+ d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*Sin[c
+ d*x])/(5*d)

Rule 4094

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

Rule 4104

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m +
1)*(d*Csc[e + f*x])^n)/(a*f*n), x] + Dist[1/(a*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[
a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ
[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]

Rule 4058

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_
.) + (a_)], x_Symbol] :> Int[(A + (B - C)*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Dist[C, Int[(Csc[e + f*
x]*(1 + Csc[e + f*x]))/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0
]

Rule 3921

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[c, In
t[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[d, Int[Csc[e + f*x]/Sqrt[a + b*Csc[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]

Rule 3784

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(2*Rt[a + b, 2]*Sqrt[(b*(1 - Csc[c + d*x])
)/(a + b)]*Sqrt[-((b*(1 + Csc[c + d*x]))/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[c + d*x]]/Rt[a
+ b, 2]], (a + b)/(a - b)])/(a*d*Cot[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]

Rule 3832

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(-2*Rt[a + b, 2]*Sqr
t[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Csc[e + f*x]))/(a - b))]*EllipticF[ArcSin[Sqrt[a + b*Csc[e +
f*x]]/Rt[a + b, 2]], (a + b)/(a - b)])/(b*f*Cot[e + f*x]), x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4004

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)
], x_Symbol] :> Simp[(-2*(A*b - a*B)*Rt[a + (b*B)/A, 2]*Sqrt[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Cs
c[e + f*x]))/(a - b))]*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + (b*B)/A, 2]], (a*A + b*B)/(a*A - b*B)]
)/(b^2*f*Cot[e + f*x]), x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]

Rubi steps

\begin{align*} \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac{1}{5} \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac{5}{2} (A b+2 a B)+(4 a A+5 b B+5 a C) \sec (c+d x)+\frac{1}{2} b (3 A+10 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{(A b+2 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac{A \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac{1}{20} \int \cos ^3(c+d x) \sqrt{a+b \sec (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 ) \sec (c+d x)+\frac{1}{4} b (39 A b+30 a B+80 b C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{240 d}+\frac{(A b+2 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac{A \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac{1}{60} \int \frac{\cos ^2(c+d x) \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 ) \sec (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 ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (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 ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{960 a d}+\frac{\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{240 d}+\frac{(A b+2 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac{A \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}-\frac{\int \frac{\cos (c+d x) \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 ) \sec (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 ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (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 \sec (c+d x)} \sin (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 ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{960 a d}+\frac{\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{240 d}+\frac{(A b+2 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac{A \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac{\int \frac{\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 ) \sec (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 ) \sec ^2(c+d x)}{\sqrt{a+b \sec (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 \sec (c+d x)} \sin (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 ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{960 a d}+\frac{\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{240 d}+\frac{(A b+2 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac{A \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac{\int \frac{\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 )+\left (-\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 )+\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 )\right ) \sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{120 a^2}+\frac{\left (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 )\right ) \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx}{3840 a^2}\\ &=-\frac{(a-b) \sqrt{a+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 ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{1920 a^2 b d}-\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 \sec (c+d x)} \sin (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 ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{960 a d}+\frac{\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{240 d}+\frac{(A b+2 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac{A \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \sin (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{1}{\sqrt{a+b \sec (c+d x)}} \, dx}{256 a^2}-\frac{\left (b \left (45 A b^4-30 a b^3 (A+5 B)-16 a^4 (64 A+45 B+80 C)-8 a^3 b (193 A+355 B+260 C)-4 a^2 b^2 (423 A+295 B+660 C)\right )\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{3840 a^2}\\ &=-\frac{(a-b) \sqrt{a+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 ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{1920 a^2 b d}-\frac{\sqrt{a+b} \left (45 A b^4-30 a b^3 (A+5 B)-16 a^4 (64 A+45 B+80 C)-8 a^3 b (193 A+355 B+260 C)-4 a^2 b^2 (423 A+295 B+660 C)\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{1920 a^2 d}-\frac{\sqrt{a+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 ) \cot (c+d x) \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{128 a^3 d}-\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 \sec (c+d x)} \sin (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 ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{960 a d}+\frac{\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{240 d}+\frac{(A b+2 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac{A \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}\\ \end{align*}

Mathematica [C]  time = 20.6554, size = 800, normalized size = 1.03 \[ \frac{\cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (\frac{1}{40} A \sin (5 (c+d x)) a^2+\frac{1}{160} (21 A b+10 a B) \sin (4 (c+d x)) a+\frac{1}{480} \left (88 A a^2+80 C a^2+170 b B a+93 A b^2\right ) \sin (c+d x)+\frac{1}{480} \left (100 A a^2+80 C a^2+170 b B a+93 A b^2\right ) \sin (3 (c+d x))+\frac{\left (480 B a^3+1024 A b a^2+1040 b C a^2+590 b^2 B a+15 A b^3\right ) \sin (2 (c+d x))}{960 a}\right )}{d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{\cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (\frac{i \left ((a-b) \left (256 (4 A+5 C) a^4+2840 b B a^3+12 b^2 (141 A+220 C) a^2+150 b^3 B a-45 A b^4\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b-a}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a+b}{a-b}\right )-2 (a-b) \left (720 B a^4+8 b (161 A+45 B+220 C) a^3+4 b^2 (129 A+185 B+180 C) a^2-30 b^3 (A-5 B) a-45 A b^4\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{b-a}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right ),\frac{a+b}{a-b}\right )+30 \left (96 B a^5+80 b (3 A+4 C) a^4+240 b^2 B a^3+40 b^3 (A+2 C) a^2-10 b^4 B a+3 A b^5\right ) \Pi \left (-\frac{a+b}{a-b};i \sinh ^{-1}\left (\sqrt{\frac{b-a}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a+b}{a-b}\right )\right ) \sqrt{\frac{(b+a \cos (c+d x)) \sec ^2\left (\frac{1}{2} (c+d x)\right )}{a+b}}}{\sqrt{\frac{b-a}{a+b}} (b+a \cos (c+d x)) \sqrt{\cos (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right )}}-\left (256 (4 A+5 C) a^4+2840 b B a^3+12 b^2 (141 A+220 C) a^2+150 b^3 B a-45 A b^4\right ) \tan \left (\frac{1}{2} (c+d x)\right )\right )}{960 a^2 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((88*a^2*A + 93*A*b^2 + 17
0*a*b*B + 80*a^2*C)*Sin[c + d*x])/480 + ((1024*a^2*A*b + 15*A*b^3 + 480*a^3*B + 590*a*b^2*B + 1040*a^2*b*C)*Si
n[2*(c + d*x)])/(960*a) + ((100*a^2*A + 93*A*b^2 + 170*a*b*B + 80*a^2*C)*Sin[3*(c + d*x)])/480 + (a*(21*A*b +
10*a*B)*Sin[4*(c + d*x)])/160 + (a^2*A*Sin[5*(c + d*x)])/40))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c +
d*x] + A*Cos[2*c + 2*d*x])) - (Cos[c + d*x]^5*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]
^2)*((I*((a - b)*(-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))*E
llipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - 2*(a - b)*(-45*A*b^4 - 30*a*b^
3*(A - 5*B) + 720*a^4*B + 4*a^2*b^2*(129*A + 185*B + 180*C) + 8*a^3*b*(161*A + 45*B + 220*C))*EllipticF[I*ArcS
inh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] + 30*(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))*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(
a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)])*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(Sqrt[(-
a + b)/(a + b)]*(b + a*Cos[c + d*x])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]) - (-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))*Tan[(c + d*x)/2]))/(960*a^2*d*(b + a*Cos[c + d*x])
^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))

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Maple [B]  time = 0.938, size = 7029, normalized size = 9.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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