3.949 \(\int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=650 \[ -\frac{\sqrt{a+b} \cot (c+d x) \left (-4 a^2 b (39 A+28 B+60 C)-8 a^3 (9 A+16 B+12 C)-6 a b^2 (A+4 B)+9 A b^3\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{192 a^2 d}-\frac{\sin (c+d x) \left (-12 a^2 b (13 A+20 C)-128 a^3 B-24 a b^2 B+9 A b^3\right ) \sqrt{a+b \sec (c+d x)}}{192 a^2 d}+\frac{\sin (c+d x) \cos (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{96 a d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left (-12 a^2 b (13 A+20 C)-128 a^3 B-24 a b^2 B+9 A b^3\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{192 a^2 b d}-\frac{\sqrt{a+b} \cot (c+d x) \left (24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)+96 a^3 b B-8 a b^3 B+3 A b^4\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{64 a^3 d}+\frac{(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d} \]

[Out]

-((a - b)*Sqrt[a + b]*(9*A*b^3 - 128*a^3*B - 24*a*b^2*B - 12*a^2*b*(13*A + 20*C))*Cot[c + d*x]*EllipticE[ArcSi
n[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))])/(192*a^2*b*d) - (Sqrt[a + b]*(9*A*b^3 - 6*a*b^2*(A + 4*B) - 8*a^3*(9*A + 16*B + 12*C
) - 4*a^2*b*(39*A + 28*B + 60*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))])/(192*a^2*d) - (Sqrt[a
+ b]*(3*A*b^4 + 96*a^3*b*B - 8*a*b^3*B + 24*a^2*b^2*(A + 2*C) + 16*a^4*(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)]*S
qrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^3*d) - ((9*A*b^3 - 128*a^3*B - 24*a*b^2*B - 12*a^2*b*(13*A + 20*
C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*a^2*d) + ((3*A*b^2 + 56*a*b*B + 12*a^2*(3*A + 4*C))*Cos[c + d*
x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*a*d) + ((3*A*b + 8*a*B)*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*
Sin[c + d*x])/(24*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)

________________________________________________________________________________________

Rubi [A]  time = 1.90629, antiderivative size = 650, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {4094, 4104, 4058, 3921, 3784, 3832, 4004} \[ -\frac{\sin (c+d x) \left (-12 a^2 b (13 A+20 C)-128 a^3 B-24 a b^2 B+9 A b^3\right ) \sqrt{a+b \sec (c+d x)}}{192 a^2 d}+\frac{\sin (c+d x) \cos (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{96 a d}-\frac{\sqrt{a+b} \cot (c+d x) \left (-4 a^2 b (39 A+28 B+60 C)-8 a^3 (9 A+16 B+12 C)-6 a b^2 (A+4 B)+9 A b^3\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{192 a^2 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left (-12 a^2 b (13 A+20 C)-128 a^3 B-24 a b^2 B+9 A b^3\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{192 a^2 b d}-\frac{\sqrt{a+b} \cot (c+d x) \left (24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)+96 a^3 b B-8 a b^3 B+3 A b^4\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{64 a^3 d}+\frac{(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

-((a - b)*Sqrt[a + b]*(9*A*b^3 - 128*a^3*B - 24*a*b^2*B - 12*a^2*b*(13*A + 20*C))*Cot[c + d*x]*EllipticE[ArcSi
n[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))])/(192*a^2*b*d) - (Sqrt[a + b]*(9*A*b^3 - 6*a*b^2*(A + 4*B) - 8*a^3*(9*A + 16*B + 12*C
) - 4*a^2*b*(39*A + 28*B + 60*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))])/(192*a^2*d) - (Sqrt[a
+ b]*(3*A*b^4 + 96*a^3*b*B - 8*a*b^3*B + 24*a^2*b^2*(A + 2*C) + 16*a^4*(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)]*S
qrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^3*d) - ((9*A*b^3 - 128*a^3*B - 24*a*b^2*B - 12*a^2*b*(13*A + 20*
C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*a^2*d) + ((3*A*b^2 + 56*a*b*B + 12*a^2*(3*A + 4*C))*Cos[c + d*
x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*a*d) + ((3*A*b + 8*a*B)*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*
Sin[c + d*x])/(24*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*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 ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{1}{4} \int \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)} \left (\frac{1}{2} (3 A b+8 a B)+(3 a A+4 b B+4 a C) \sec (c+d x)+\frac{1}{2} b (3 A+8 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{(3 A b+8 a B) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{1}{12} \int \frac{\cos ^2(c+d x) \left (\frac{1}{4} \left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right )+\frac{1}{2} \left (33 a A b+16 a^2 B+24 b^2 B+48 a b C\right ) \sec (c+d x)+\frac{3}{4} b (9 A b+8 a B+16 b C) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{\left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 a d}+\frac{(3 A b+8 a B) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}-\frac{\int \frac{\cos (c+d x) \left (\frac{1}{8} \left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right )-\frac{1}{4} a \left (104 a b B+12 a^2 (3 A+4 C)+3 b^2 (19 A+32 C)\right ) \sec (c+d x)-\frac{1}{8} b \left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{24 a}\\ &=-\frac{\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a^2 d}+\frac{\left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 a d}+\frac{(3 A b+8 a B) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{\int \frac{\frac{3}{16} \left (3 A b^4+96 a^3 b B-8 a b^3 B+24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)\right )+\frac{1}{8} a b \left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \sec (c+d x)+\frac{1}{16} b \left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{24 a^2}\\ &=-\frac{\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a^2 d}+\frac{\left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 a d}+\frac{(3 A b+8 a B) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{\int \frac{\frac{3}{16} \left (3 A b^4+96 a^3 b B-8 a b^3 B+24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)\right )+\left (\frac{1}{8} a b \left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right )-\frac{1}{16} b \left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right )\right ) \sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{24 a^2}+\frac{\left (b \left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right )\right ) \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx}{384 a^2}\\ &=-\frac{(a-b) \sqrt{a+b} \left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 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}}}{192 a^2 b d}-\frac{\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a^2 d}+\frac{\left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 a d}+\frac{(3 A b+8 a B) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{\left (3 A b^4+96 a^3 b B-8 a b^3 B+24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)\right ) \int \frac{1}{\sqrt{a+b \sec (c+d x)}} \, dx}{128 a^2}-\frac{\left (b \left (9 A b^3-6 a b^2 (A+4 B)-8 a^3 (9 A+16 B+12 C)-4 a^2 b (39 A+28 B+60 C)\right )\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{384 a^2}\\ &=-\frac{(a-b) \sqrt{a+b} \left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 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}}}{192 a^2 b d}-\frac{\sqrt{a+b} \left (9 A b^3-6 a b^2 (A+4 B)-8 a^3 (9 A+16 B+12 C)-4 a^2 b (39 A+28 B+60 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}}}{192 a^2 d}-\frac{\sqrt{a+b} \left (3 A b^4+96 a^3 b B-8 a b^3 B+24 a^2 b^2 (A+2 C)+16 a^4 (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}}}{64 a^3 d}-\frac{\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a^2 d}+\frac{\left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 a d}+\frac{(3 A b+8 a B) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}\\ \end{align*}

Mathematica [A]  time = 16.9781, size = 761, normalized size = 1.17 \[ \frac{\cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac{\sin (2 (c+d x)) \left (48 a^2 A+48 a^2 C+56 a b B+3 A b^2\right )}{96 a}+\frac{1}{48} (8 a B+9 A b) \sin (c+d x)+\frac{1}{48} (8 a B+9 A b) \sin (3 (c+d x))+\frac{1}{16} a A \sin (4 (c+d x))\right )}{d (a \cos (c+d x)+b) (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}-\frac{\cos ^5(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (b (a+b) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \left (12 a^2 b (7 A+4 (B+3 C))+8 a^3 (9 A+16 B+12 C)-6 a b^2 (3 A+4 B)+9 A b^3\right ) \sqrt{\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{a+b}} \text{EllipticF}\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right ),\frac{a-b}{a+b}\right )+3 \sec ^2\left (\frac{1}{2} (c+d x)\right ) \left (24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)+96 a^3 b B-8 a b^3 B+3 A b^4\right ) \sqrt{\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{a+b}} \left ((a-b) \text{EllipticF}\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right ),\frac{a-b}{a+b}\right )+2 a \Pi \left (-1;-\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a-b}{a+b}\right )\right )-a \tan \left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \left (12 a^2 b (13 A+20 C)+128 a^3 B+24 a b^2 B-9 A b^3\right ) \left (\cos (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^{3/2} (a \cos (c+d x)+b)-a (a+b) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \left (12 a^2 b (13 A+20 C)+128 a^3 B+24 a b^2 B-9 A b^3\right ) \sqrt{\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{a+b}} E\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a-b}{a+b}\right )\right )}{96 a^3 d \left (\cos (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^{3/2} (a \cos (c+d x)+b)^2 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((9*A*b + 8*a*B)*Sin[c + d
*x])/48 + ((48*a^2*A + 3*A*b^2 + 56*a*b*B + 48*a^2*C)*Sin[2*(c + d*x)])/(96*a) + ((9*A*b + 8*a*B)*Sin[3*(c + d
*x)])/48 + (a*A*Sin[4*(c + d*x)])/16))/(d*(b + a*Cos[c + d*x])*(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])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-(a*(a + b)*(-9*A*b^
3 + 128*a^3*B + 24*a*b^2*B + 12*a^2*b*(13*A + 20*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[
(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + b*(a + b)*(9*A*b^3 - 6*a*b^2*(3*A +
4*B) + 8*a^3*(9*A + 16*B + 12*C) + 12*a^2*b*(7*A + 4*(B + 3*C)))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(
a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 3*(3*A*b^4 + 96*a^3*b*B -
8*a*b^3*B + 24*a^2*b^2*(A + 2*C) + 16*a^4*(3*A + 4*C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(
a + b)] + 2*a*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[
c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - a*(-9*A*b^3 + 128*a^3*B + 24*a*b^2*B + 12*a^2*b*(13*A + 20*C))*(b + a
*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(96*a^3*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]*Sec[(c + d*x)/2]^2)^(3/2))

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Maple [B]  time = 0.738, size = 5474, normalized size = 8.4 \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)^4*(a+b*sec(d*x+c))^(3/2)*(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{3}{2}} \cos \left (d x + c\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^(3/2)*(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)^(3/2)*cos(d*x + c)^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C b \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{3} +{\left (C a + B b\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{2} + A a \cos \left (d x + c\right )^{4} +{\left (B a + A b\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )\right )} \sqrt{b \sec \left (d x + c\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

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

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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)**4*(a+b*sec(d*x+c))**(3/2)*(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{3}{2}} \cos \left (d x + c\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^(3/2)*(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)^(3/2)*cos(d*x + c)^4, x)