\(\int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} (A+C \sec ^2(c+d x)) \, dx\) [724]

Optimal result

Integrand size = 35, antiderivative size = 583 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {(a-b) \sqrt {a+b} \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{64 a^2 d}+\frac {\sqrt {a+b} \left (2 a A b^2-3 A b^3+8 a^3 (3 A+4 C)+a^2 (52 A b+80 b C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{64 a^2 d}-\frac {\sqrt {a+b} \left (3 A b^4+24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{64 a^3 d}-\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{64 a^2 d}+\frac {\left (A b^2+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 a d}+\frac {A b \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{8 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d} \]

[Out]

1/4*A*cos(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d-1/64*(a-b)*(3*A*b^2-4*a^2*(13*A+20*C))*cot(d*x+c)*Ellip
ticE((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+1/64*(2*a*A*b^2-3*A*b^3+8*a^3*(3*A+4*C)+a^2*(52*A*b+80*C*b))*cot(d*x+c)*Ellipt
icF((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-1/64*(3*A*b^4+24*a^2*b^2*(A+2*C)+16*a^4*(3*A+4*C))*cot(d*x+c)*EllipticPi((a+b*s
ec(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+se
c(d*x+c))/(a-b))^(1/2)/a^3/d-1/64*b*(3*A*b^2-4*a^2*(13*A+20*C))*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a^2/d+1/32*(
A*b^2+4*a^2*(3*A+4*C))*cos(d*x+c)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a/d+1/8*A*b*cos(d*x+c)^2*sin(d*x+c)*(a+b*s
ec(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 1.78 (sec) , antiderivative size = 583, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4180, 4179, 4189, 4143, 4006, 3869, 3917, 4089} \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {(a-b) \sqrt {a+b} \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{64 a^2 d}-\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{64 a^2 d}+\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{32 a d}+\frac {\sqrt {a+b} \left (8 a^3 (3 A+4 C)+a^2 (52 A b+80 b C)+2 a A b^2-3 A b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{64 a^2 d}-\frac {\sqrt {a+b} \left (16 a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+3 A b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{64 a^3 d}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}+\frac {A b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{8 d} \]

[In]

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

[Out]

-1/64*((a - b)*Sqrt[a + b]*(3*A*b^2 - 4*a^2*(13*A + 20*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))
])/(a^2*d) + (Sqrt[a + b]*(2*a*A*b^2 - 3*A*b^3 + 8*a^3*(3*A + 4*C) + a^2*(52*A*b + 80*b*C))*Cot[c + d*x]*Ellip
ticF[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))])/(64*a^2*d) - (Sqrt[a + b]*(3*A*b^4 + 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)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^3*d) - (b*(3*A*b^2 - 4*a^2*(13*A
+ 20*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(64*a^2*d) + ((A*b^2 + 4*a^2*(3*A + 4*C))*Cos[c + d*x]*Sqrt[a
+ b*Sec[c + d*x]]*Sin[c + d*x])/(32*a*d) + (A*b*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(8*d) +
(A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*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 4180

Int[((A_.) + 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] - Dis
t[1/(d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[A*b*m - 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, 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*} \text {integral}& = \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 {3 A b}{2}+a (3 A+4 C) \sec (c+d x)+\frac {1}{2} b (3 A+8 C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {A b \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{8 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 {3}{4} \left (A b^2+4 a^2 (3 A+4 C)\right )+\frac {3}{2} a b (11 A+16 C) \sec (c+d x)+\frac {3}{4} b^2 (9 A+16 C) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {\left (A b^2+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 a d}+\frac {A b \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{8 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 {3}{8} b \left (3 A b^2-a^2 (52 A+80 C)\right )-\frac {3}{4} a \left (4 a^2 (3 A+4 C)+b^2 (19 A+32 C)\right ) \sec (c+d x)-\frac {3}{8} b \left (A b^2+4 a^2 (3 A+4 C)\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{24 a} \\ & = -\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{64 a^2 d}+\frac {\left (A b^2+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 a d}+\frac {A b \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{8 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+24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)\right )+\frac {3}{8} a b \left (A b^2+4 a^2 (3 A+4 C)\right ) \sec (c+d x)+\frac {3}{16} b^2 \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sec ^2(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{24 a^2} \\ & = -\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{64 a^2 d}+\frac {\left (A b^2+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 a d}+\frac {A b \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{8 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+24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)\right )+\left (\frac {3}{8} a b \left (A b^2+4 a^2 (3 A+4 C)\right )-\frac {3}{16} b^2 \left (3 A b^2-4 a^2 (13 A+20 C)\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{24 a^2}-\frac {1}{128} \left (b^2 \left (A \left (52-\frac {3 b^2}{a^2}\right )+80 C\right )\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {(a-b) \sqrt {a+b} \left (A \left (52-\frac {3 b^2}{a^2}\right )+80 C\right ) \cot (c+d x) E\left (\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}}}{64 d}-\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{64 a^2 d}+\frac {\left (A b^2+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 a d}+\frac {A b \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{8 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+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 (2 a A b^2-3 A b^3+8 a^3 (3 A+4 C)+a^2 (52 A b+80 b C)\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{128 a^2} \\ & = \frac {(a-b) \sqrt {a+b} \left (A \left (52-\frac {3 b^2}{a^2}\right )+80 C\right ) \cot (c+d x) E\left (\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}}}{64 d}+\frac {\sqrt {a+b} \left (2 a A b^2-3 A b^3+8 a^3 (3 A+4 C)+a^2 (52 A b+80 b C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\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}}}{64 a^2 d}-\frac {\sqrt {a+b} \left (3 A b^4+24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\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}}}{64 a^3 d}-\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{64 a^2 d}+\frac {\left (A b^2+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 a d}+\frac {A b \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{8 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 [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1904\) vs. \(2(583)=1166\).

Time = 21.46 (sec) , antiderivative size = 1904, normalized size of antiderivative = 3.27 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \left (\frac {3}{16} A b \sin (c+d x)+\frac {\left (16 a^2 A+A b^2+16 a^2 C\right ) \sin (2 (c+d x))}{32 a}+\frac {3}{16} A b \sin (3 (c+d x))+\frac {1}{16} a A \sin (4 (c+d x))\right )}{d (b+a \cos (c+d x)) (A+2 C+A \cos (2 c+2 d x))}+\frac {(a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}} \left (52 a^3 A b \tan \left (\frac {1}{2} (c+d x)\right )+52 a^2 A b^2 \tan \left (\frac {1}{2} (c+d x)\right )-3 a A b^3 \tan \left (\frac {1}{2} (c+d x)\right )-3 A b^4 \tan \left (\frac {1}{2} (c+d x)\right )+80 a^3 b C \tan \left (\frac {1}{2} (c+d x)\right )+80 a^2 b^2 C \tan \left (\frac {1}{2} (c+d x)\right )-104 a^3 A b \tan ^3\left (\frac {1}{2} (c+d x)\right )+6 a A b^3 \tan ^3\left (\frac {1}{2} (c+d x)\right )-160 a^3 b C \tan ^3\left (\frac {1}{2} (c+d x)\right )+52 a^3 A b \tan ^5\left (\frac {1}{2} (c+d x)\right )-52 a^2 A b^2 \tan ^5\left (\frac {1}{2} (c+d x)\right )-3 a A b^3 \tan ^5\left (\frac {1}{2} (c+d x)\right )+3 A b^4 \tan ^5\left (\frac {1}{2} (c+d x)\right )+80 a^3 b C \tan ^5\left (\frac {1}{2} (c+d x)\right )-80 a^2 b^2 C \tan ^5\left (\frac {1}{2} (c+d x)\right )+96 a^4 A \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+48 a^2 A b^2 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+6 A b^4 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+128 a^4 C \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+96 a^2 b^2 C \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+96 a^4 A \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+48 a^2 A b^2 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+6 A b^4 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+128 a^4 C \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+96 a^2 b^2 C \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+b (a+b) \left (-3 A b^2+a^2 (52 A+80 C)\right ) E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}-2 a \left (-A b^3+8 a^3 (3 A+4 C)-4 a^2 b (3 A+4 C)+2 a b^2 (19 A+32 C)\right ) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}\right )}{32 a^2 d (b+a \cos (c+d x))^{3/2} (A+2 C+A \cos (2 c+2 d x)) \sec ^{\frac {7}{2}}(c+d x) \left (-1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\frac {1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}{1-\tan ^2\left (\frac {1}{2} (c+d x)\right )}} \left (a \left (-1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-b \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )} \]

[In]

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

[Out]

(Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*((3*A*b*Sin[c + d*x])/16 + ((16*a^2*A + A*b^
2 + 16*a^2*C)*Sin[2*(c + d*x)])/(32*a) + (3*A*b*Sin[3*(c + d*x)])/16 + (a*A*Sin[4*(c + d*x)])/16))/(d*(b + a*C
os[c + d*x])*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*Sqrt[(a + b
- a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(52*a^3*A*b*Tan[(c + d*x)/2] + 52*a^2
*A*b^2*Tan[(c + d*x)/2] - 3*a*A*b^3*Tan[(c + d*x)/2] - 3*A*b^4*Tan[(c + d*x)/2] + 80*a^3*b*C*Tan[(c + d*x)/2]
+ 80*a^2*b^2*C*Tan[(c + d*x)/2] - 104*a^3*A*b*Tan[(c + d*x)/2]^3 + 6*a*A*b^3*Tan[(c + d*x)/2]^3 - 160*a^3*b*C*
Tan[(c + d*x)/2]^3 + 52*a^3*A*b*Tan[(c + d*x)/2]^5 - 52*a^2*A*b^2*Tan[(c + d*x)/2]^5 - 3*a*A*b^3*Tan[(c + d*x)
/2]^5 + 3*A*b^4*Tan[(c + d*x)/2]^5 + 80*a^3*b*C*Tan[(c + d*x)/2]^5 - 80*a^2*b^2*C*Tan[(c + d*x)/2]^5 + 96*a^4*
A*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(
c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^2*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)
/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6
*A*b^4*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*
Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 128*a^4*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a -
b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] +
96*a^2*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a +
b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 96*a^4*A*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]],
(a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(
c + d*x)/2]^2)/(a + b)] + 48*a^2*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)
/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*A*b
^4*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*S
qrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 128*a^4*C*EllipticPi[-1, ArcSin[Tan[(c +
d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2
+ b*Tan[(c + d*x)/2]^2)/(a + b)] + 96*a^2*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[
(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)
] + b*(a + b)*(-3*A*b^2 + a^2*(52*A + 80*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan
[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]
- 2*a*(-(A*b^3) + 8*a^3*(3*A + 4*C) - 4*a^2*b*(3*A + 4*C) + 2*a*b^2*(19*A + 32*C))*EllipticF[ArcSin[Tan[(c + d
*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/
2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(32*a^2*d*(b + a*Cos[c + d*x])^(3/2)*(A + 2*C + A*Cos[2*c + 2*d*x])*Se
c[c + d*x]^(7/2)*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Ta
n[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(5198\) vs. \(2(534)=1068\).

Time = 6.37 (sec) , antiderivative size = 5199, normalized size of antiderivative = 8.92

method	result	size
default	\(\text {Expression too large to display}\)	\(5199\)

[In]

int(cos(d*x+c)^4*(a+b*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F]

\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \,d x } \]

[In]

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

Sympy [F(-1)]

Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^4\,\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]

[In]

int(cos(c + d*x)^4*(A + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^(3/2),x)

[Out]

int(cos(c + d*x)^4*(A + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^(3/2), x)