3.10.57 \(\int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [957]

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

[Out]

1/8*(A*b+2*B*a)*cos(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d+1/5*A*cos(d*x+c)^4*(a+b*sec(d*x+c))^(5/2)*sin
(d*x+c)/d-1/1920*(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(d*x+
c)*EllipticE((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2
)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a^2/b/d-1/1920*(45*A*b^4-30*a*b^3*(A+5*B)-16*a^4*(64*A+45*B+80*C)-8*a^3*b*(1
93*A+355*B+260*C)-4*a^2*b^2*(423*A+295*B+660*C))*cot(d*x+c)*EllipticF((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b
)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a^2/d-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))*cot(d*x+c)*EllipticPi((a+b*sec(d*x+
c))^(1/2)/(a+b)^(1/2),(a+b)/a,((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c
))/(a-b))^(1/2)/a^3/d-1/1920*(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))*si
n(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a^2/d+1/960*(15*A*b^3+360*a^3*B+590*a*b^2*B+4*a^2*b*(193*A+260*C))*cos(d*x+c)*
sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a/d+1/240*(15*A*b^2+110*a*b*B+16*a^2*(4*A+5*C))*cos(d*x+c)^2*sin(d*x+c)*(a+b
*sec(d*x+c))^(1/2)/d

________________________________________________________________________________________

Rubi [A]
time = 2.19, antiderivative size = 774, normalized size of antiderivative = 1.00, 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 = {4179, 4189, 4143, 4006, 3869, 3917, 4089} \begin {gather*} \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 (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 \sec (c+d x)}}{960 a d}-\frac {\sqrt {a+b} \cot (c+d x) \left (-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)-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 (\text {ArcSin}\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 (-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 {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\text {ArcSin}\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 {\sin (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 \sec (c+d x)}}{1920 a^2 d}-\frac {\sqrt {a+b} \cot (c+d x) \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 {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{a};\text {ArcSin}\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} \end {gather*}

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]

-1/1920*((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 - S
ec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(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))
*Cot[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
[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 + S
ec[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 3869

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

Rule 3917

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

Rule 4006

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 4089

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 + C
sc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a
*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]

Rule 4143

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 4179

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 4189

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]

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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 21.25, size = 800, normalized size = 1.03 \begin {gather*} \frac {\cos ^4(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 ) \left (\frac {1}{480} \left (88 a^2 A+93 A b^2+170 a b B+80 a^2 C\right ) \sin (c+d x)+\frac {\left (1024 a^2 A b+15 A b^3+480 a^3 B+590 a b^2 B+1040 a^2 b C\right ) \sin (2 (c+d x))}{960 a}+\frac {1}{480} \left (100 a^2 A+93 A b^2+170 a b B+80 a^2 C\right ) \sin (3 (c+d x))+\frac {1}{160} a (21 A b+10 a B) \sin (4 (c+d x))+\frac {1}{40} a^2 A \sin (5 (c+d x))\right )}{d (b+a \cos (c+d x))^2 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}-\frac {\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 ) \left (\frac {i \left ((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 ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {-a+b}{a+b}} \tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a+b}{a-b}\right )-2 (a-b) \left (-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)\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {-a+b}{a+b}} \tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a+b}{a-b}\right )+30 \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 ) \Pi \left (-\frac {a+b}{a-b};i \sinh ^{-1}\left (\sqrt {\frac {-a+b}{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 {-a+b}{a+b}} (b+a \cos (c+d x)) \sqrt {\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )}}-\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 ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{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))} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(7028\) vs. \(2(721)=1442\).
time = 0.58, size = 7029, normalized size = 9.08

method	result	size
default	\(\text {Expression too large to display}\)	\(7029\)

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,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

integral((C*b^2*cos(d*x + c)^5*sec(d*x + c)^4 + (2*C*a*b + B*b^2)*cos(d*x + c)^5*sec(d*x + c)^3 + A*a^2*cos(d*
x + c)^5 + (C*a^2 + 2*B*a*b + A*b^2)*cos(d*x + c)^5*sec(d*x + c)^2 + (B*a^2 + 2*A*a*b)*cos(d*x + c)^5*sec(d*x
+ c))*sqrt(b*sec(d*x + c) + a), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\cos \left (c+d\,x\right )}^5\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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