\(\int \frac {\cos ^{\frac {3}{2}}(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx\) [1364]

Optimal result

Integrand size = 45, antiderivative size = 350 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx=\frac {2 \left (8 A b^2-6 a b B+a^2 (A+3 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{3 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (8 A b^3+3 a^3 B-6 a b^2 B-a^2 (5 A b-3 b C)\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{3 a^3 \left (a^2-b^2\right ) d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (4 A b^2-3 a b B-a^2 (A-3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d} \]

[Out]

2*(A*b^2-a*(B*b-C*a))*sin(d*x+c)*cos(d*x+c)^(1/2)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))^(1/2)+2/3*(8*A*b^2-6*B*a*b+a^
2*(A+3*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2
))*((b+a*cos(d*x+c))/(a+b))^(1/2)/a^3/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)-2/3*(4*A*b^2-3*B*a*b-a^2*(A-3*
C))*sin(d*x+c)*cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)/a^2/(a^2-b^2)/d+2/3*(8*A*b^3+3*B*a^3-6*B*a*b^2-a^2*(5*A
*b-3*C*b))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2
))*cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)/a^3/(a^2-b^2)/d/((b+a*cos(d*x+c))/(a+b))^(1/2)

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4350, 4185, 4189, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx=-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^2 (A-3 C)\right )-3 a b B+4 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 a^2 d \left (a^2-b^2\right )}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (a^2 (A+3 C)-6 a b B+8 A b^2\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{3 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \left (3 a^3 B-a^2 (5 A b-3 b C)-6 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{3 a^3 d \left (a^2-b^2\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}}} \]

[In]

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

[Out]

(2*(8*A*b^2 - 6*a*b*B + a^2*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)
])/(3*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B - a^2*(5*A*b - 3*
b*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*
Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^
2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(4*A*b^2 - 3*a*b*B - a^2*(A - 3*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c +
d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d)

Rule 2732

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

Rule 2734

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

Rule 2740

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

Rule 2742

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

Rule 3941

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)], x_Symbol] :> Dist[Sqrt[a +
b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*Sqrt[b + a*Sin[e + f*x]]), Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; Free
Q[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3943

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[Sqrt[d*C
sc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]), Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; Fr
eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4120

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

Rule 4185

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*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a +
b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^n/(a*f*(m + 1)*(a^2 - b^2))), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), I
nt[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*
(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] &&  !(ILtQ[m + 1/2, 0] &
& ILtQ[n, 0])

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]

Rule 4350

Int[(cos[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Cos[a + b*x])^m*(c*Sec[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Sec[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSecantIntegrandQ[
u, x]

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx \\ & = \frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{2} \left (4 A b^2-3 a b B-a^2 (A-3 C)\right )+\frac {1}{2} a (A b-a B+b C) \sec (c+d x)-\left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{a \left (a^2-b^2\right )} \\ & = \frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (4 A b^2-3 a b B-a^2 (A-3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{4} \left (8 A b^3+3 a^3 B-6 a b^2 B-a^2 (5 A b-3 b C)\right )+\frac {1}{4} a \left (2 A b^2-3 a b B+a^2 (A+3 C)\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{3 a^2 \left (a^2-b^2\right )} \\ & = \frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (4 A b^2-3 a b B-a^2 (A-3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (\left (8 A b^2-6 a b B+a^2 (A+3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{3 a^3}+\frac {\left (\left (8 A b^3+3 a^3 B-6 a b^2 B-a^2 (5 A b-3 b C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{3 a^3 \left (a^2-b^2\right )} \\ & = \frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (4 A b^2-3 a b B-a^2 (A-3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (\left (8 A b^2-6 a b B+a^2 (A+3 C)\right ) \sqrt {b+a \cos (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{3 a^3 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (8 A b^3+3 a^3 B-6 a b^2 B-a^2 (5 A b-3 b C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{3 a^3 \left (a^2-b^2\right ) \sqrt {b+a \cos (c+d x)}} \\ & = \frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (4 A b^2-3 a b B-a^2 (A-3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (\left (8 A b^2-6 a b B+a^2 (A+3 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{3 a^3 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (8 A b^3+3 a^3 B-6 a b^2 B-a^2 (5 A b-3 b C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{3 a^3 \left (a^2-b^2\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}}} \\ & = \frac {2 \left (8 A b^2-6 a b B+a^2 (A+3 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{3 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (8 A b^3+3 a^3 B-6 a b^2 B-a^2 (5 A b-3 b C)\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{3 a^3 \left (a^2-b^2\right ) d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (4 A b^2-3 a b B-a^2 (A-3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 40.90 (sec) , antiderivative size = 4889, normalized size of antiderivative = 13.97 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*A*Sin[c + d*x])/(3*a^2)
- (4*(A*b^3*Sin[c + d*x] - a*b^2*B*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(a^2*(a^2 - b^2)*(b + a*Cos[c + d*x]
))))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(3/2)) - (4*(b + a*Cos[c + d*x]
)^(3/2)*((-10*A*b*Sqrt[Cos[c + d*x]])/(3*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*A*b^3*
Sqrt[Cos[c + d*x]])/(3*a^2*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a*B*Sqrt[Cos[c + d*x]
])/((a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*b^2*B*Sqrt[Cos[c + d*x]])/(a*(a^2 - b^2)*Sqr
t[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b*C*Sqrt[Cos[c + d*x]])/((a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*S
qrt[Sec[c + d*x]]) + (2*a*A*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]) +
(4*A*b^2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]) - (2*b*B*Sqrt[Cos[c
+ d*x]]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]) + (2*a*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d
*x]])/((a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]))*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/
2]^2)^(-1)]*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)/(1 + Tan[(c + d*x)/2]^2)]*((8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Tan[(c + d*
x)/2]*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2) + I*(a + b)*(8*A*b^3
+ 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(1 + Ta
n[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*a*(a + b)*(8*A*b^2 -
6*a*b*(A + B) + a^2*(A + 3*(B + C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(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)]))/(3*a^3*(a^2 - b^2)*d*(A + 2*C
+ 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(a*(-1 + Tan[(c + d*x)/
2]^2) - b*(1 + Tan[(c + d*x)/2]^2))*((4*(a*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] - b*Sec[(c + d*x)/2]^2*Tan[(c +
d*x)/2])*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*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)/(1 + Tan[(c + d*x)/2]^2)]*((8*A*b^3 + 3*a^3*B - 6*a*b^2*B +
a^2*(-5*A*b + 3*b*C))*Tan[(c + d*x)/2]*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c
+ d*x)/2]^2) + I*(a + b)*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*EllipticE[I*ArcSinh[Tan[(c + d
*x)/2]], (-a + b)/(a + b)]*(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)] - I*a*(a + b)*(8*A*b^2 - 6*a*b*(A + B) + a^2*(A + 3*(B + C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]],
(-a + b)/(a + b)]*(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)
]))/(3*a^3*(a^2 - b^2)*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2))^2) - (2*Sec[(c + d*x)/2]^2*T
an[(c + d*x)/2]*((1 - Tan[(c + d*x)/2]^2)^(-1))^(3/2)*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)/(1 + Tan[(c + d*x)/2]^2)]*((8*A*b^3 + 3*a^3*B - 6*a
*b^2*B + a^2*(-5*A*b + 3*b*C))*Tan[(c + d*x)/2]*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a + b - a*Tan[(c + d*x)/2]^2 + b
*Tan[(c + d*x)/2]^2) + I*(a + b)*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*EllipticE[I*ArcSinh[Ta
n[(c + d*x)/2]], (-a + b)/(a + b)]*(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)] - I*a*(a + b)*(8*A*b^2 - 6*a*b*(A + B) + a^2*(A + 3*(B + C)))*EllipticF[I*ArcSinh[Tan[(c + d
*x)/2]], (-a + b)/(a + b)]*(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)]))/(3*a^3*(a^2 - b^2)*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2))) - (2*Sqrt[(1 - Tan[
(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(-(
(Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]*(1 - Tan[(c + d*x)/2]^2))/(1 + Tan[(c + d*x)/2]^2)^2) - (Sec[(c + d*x)/2]
^2*Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]^2))*((8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Tan[(c
+ d*x)/2]*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2) + I*(a + b)*(8*A*
b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(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)] - I*a*(a + b)*(8*A*b
^2 - 6*a*b*(A + B) + a^2*(A + 3*(B + C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(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)]))/(3*a^3*(a^2 - b^2)*Sqrt[(1
- Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2))) -
(2*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(1 - Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*((-(a*Sec[(c +
d*x)/2]^2*Tan[(c + d*x)/2]) + b*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]^2) - (Sec[(c + d*x
)/2]^2*Tan[(c + d*x)/2]*(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2))/(1 + Tan[(c + d*x)/2]^2)^2)*((8
*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Tan[(c + d*x)/2]*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a + b - a*
Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2) + I*(a + b)*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*
EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(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)] - I*a*(a + b)*(8*A*b^2 - 6*a*b*(A + B) + a^2*(A + 3*(B + C)))*Elliptic
F[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(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)]))/(3*a^3*(a^2 - b^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)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2))) - (4*Sqrt[(1 - Tan[(c
+ d*x)/2]^2)^(-1)]*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)/(1 + Tan[(c + d*x)/2]^2)]*((8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Tan[
(c + d*x)/2]*(-(a*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + b*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*Sqrt[1 + Tan[(
c + d*x)/2]^2] + ((8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2
*(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2))/(2*Sqrt[1 + Tan[(c + d*x)/2]^2]) + ((8*A*b^3 + 3*a^3*B
- 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Sec[(c + d*x)/2]^2*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a + b - a*Tan[(c + d*x)/
2]^2 + b*Tan[(c + d*x)/2]^2))/2 + ((I/2)*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*EllipticE[I*Ar
cSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-(a*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + b*Sec[(c + d*x)/2]^2*Ta
n[(c + d*x)/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)]
- ((I/2)*a*(8*A*b^2 - 6*a*b*(A + B) + a^2*(A + 3*(B + C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a
+ b)]*(-(a*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + b*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/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)] + I*(a + b)*(8*A*b^3 + 3*a^3*B - 6*a*
b^2*B + a^2*(-5*A*b + 3*b*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Tan[
(c + d*x)/2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*a*(a + b)*(8*A*b^2 - 6*a*
b*(A + B) + a^2*(A + 3*(B + C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*T
an[(c + d*x)/2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a*(a + b)*(8*A*b^2 - 6*
a*b*(A + B) + a^2*(A + 3*(B + C)))*Sec[(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)])/(2*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(8*A*
b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Sec[(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)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/2))/(
3*a^3*(a^2 - b^2)*(a*(-1 + Tan[(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. \(2407\) vs. \(2(382)=764\).

Time = 11.08 (sec) , antiderivative size = 2408, normalized size of antiderivative = 6.88

method	result	size
default	\(\text {Expression too large to display}\)	\(2408\)

[In]

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

[Out]

2/3/d*(-3*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+
c)),(-(a+b)/(a-b))^(1/2))*a^3*cos(d*x+c)-3*C*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/
(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b*cos(d*x+c)+A*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)
*sin(d*x+c)*a^3*(1/(1+cos(d*x+c)))^(1/2)+A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^3*(1/(1+cos(d*x+c)))^(1
/2)+A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^2*b*(1/(1+cos(d*x+c)))^(1/2)-4*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a*b^2*(
1/(1+cos(d*x+c)))^(1/2)+3*B*sin(d*x+c)*cos(d*x+c)*((a-b)/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*a^3-6*A*cos(d*x
+c)*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a
+b)/(a-b))^(1/2))*a^2*b-8*A*cos(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))
^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^2-3*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^2*b
*(1/(1+cos(d*x+c)))^(1/2)+5*A*cos(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b
))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b+3*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^2*b*(1/(1+co
s(d*x+c)))^(1/2)+6*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a*b^2*(1/(1+cos(d*x+c)))^(1/2)-3*C*((a-b)/(a+b))^(1/2)*a^2
*b*(1/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+6*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a
+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^2*cos(d*x+c)+6*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos
(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b*cos(d*x+c)-3
*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+
b)/(a-b))^(1/2))*a^3-8*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*b^3*(1/(1+cos(d*x+c)))^(1/2)-8*A*(1/(a+b)*(b+a*cos(d*x
+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^2+3
*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+
b)/(a-b))^(1/2))*a^3*cos(d*x+c)-8*A*cos(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b
)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*b^3-A*cos(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos
(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^3+5*A*(1/(a+b)*(
b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/
2))*a^2*b-6*A*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*
x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b+6*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^
(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^2+6*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*
EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b-3*C*(1/(a+b)*(b+a*cos(d*x+c)
)/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b-3*C*
(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/
(a-b))^(1/2))*a^3*cos(d*x+c)-4*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a*b^2*(1/(1+cos(d*x+c)))^(1/2)+3*B*
cos(d*x+c)*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^2*b*(1/(1+cos(d*x+c)))^(1/2)+A*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*si
n(d*x+c)*a^2*b*(1/(1+cos(d*x+c)))^(1/2)+3*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(
a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^3-3*C*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1
/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^3-8*A*(1/(a+b)*(b+a*cos(d*x+
c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*b^3-A*(1
/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a
-b))^(1/2))*a^3)*cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)/(1/(1+cos(d*x+c)))^(1/2)/(b+a*cos(d*x+c))/(a+b)/((a-b
)/(a+b))^(1/2)/a^3/(1+cos(d*x+c))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.17 (sec) , antiderivative size = 872, normalized size of antiderivative = 2.49 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/9*(6*((A - 3*C)*a^4*b + 3*B*a^3*b^2 - 4*A*a^2*b^3 + (A*a^5 - A*a^3*b^2)*cos(d*x + c))*sqrt((a*cos(d*x + c) +
b)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) - (sqrt(2)*(3*I*(A + 3*C)*a^5 - 15*I*B*a^4*b + 2*I*(8*A - 3*
C)*a^3*b^2 + 12*I*B*a^2*b^3 - 16*I*A*a*b^4)*cos(d*x + c) + sqrt(2)*(3*I*(A + 3*C)*a^4*b - 15*I*B*a^3*b^2 + 2*I
*(8*A - 3*C)*a^2*b^3 + 12*I*B*a*b^4 - 16*I*A*b^5))*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*
(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/a) - (sqrt(2)*(-3*I*(A + 3*C)*a^5 + 1
5*I*B*a^4*b - 2*I*(8*A - 3*C)*a^3*b^2 - 12*I*B*a^2*b^3 + 16*I*A*a*b^4)*cos(d*x + c) + sqrt(2)*(-3*I*(A + 3*C)*
a^4*b + 15*I*B*a^3*b^2 - 2*I*(8*A - 3*C)*a^2*b^3 - 12*I*B*a*b^4 + 16*I*A*b^5))*sqrt(a)*weierstrassPInverse(-4/
3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a) + 3*(s
qrt(2)*(3*I*B*a^5 - I*(5*A - 3*C)*a^4*b - 6*I*B*a^3*b^2 + 8*I*A*a^2*b^3)*cos(d*x + c) + sqrt(2)*(3*I*B*a^4*b -
I*(5*A - 3*C)*a^3*b^2 - 6*I*B*a^2*b^3 + 8*I*A*a*b^4))*sqrt(a)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*
(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(
d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/a)) + 3*(sqrt(2)*(-3*I*B*a^5 + I*(5*A - 3*C)*a^4*b + 6*I*B*a^3*b^2 - 8*I*
A*a^2*b^3)*cos(d*x + c) + sqrt(2)*(-3*I*B*a^4*b + I*(5*A - 3*C)*a^3*b^2 + 6*I*B*a^2*b^3 - 8*I*A*a*b^4))*sqrt(a
)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^
2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a)))/((a^7 - a^5*b^2)*d*
cos(d*x + c) + (a^6*b - a^4*b^3)*d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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