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\(\int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx\) [1062]

   Optimal result
   Rubi [A] (verified)
   Mathematica [C] (warning: unable to verify)
   Maple [B] (warning: unable to verify)
   Fricas [C] (verification not implemented)
   Sympy [F(-1)]
   Maxima [F(-1)]
   Giac [F]
   Mupad [F(-1)]

Optimal result

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

[Out]

2*(A*b^2-a*(B*b-C*a))*sin(d*x+c)/a/(a^2-b^2)/d/sec(d*x+c)^(1/2)/(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)*sec(d*x+c)^(1/2)/a^3/d/(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)*(a+b*sec(d*x+c))^(1/2)/a^2/(a^2-b^2)/d/sec(d*x+c)^(1/2)+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
))*(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)/sec(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4185, 4189, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=-\frac {2 \sin (c+d x) \left (-\left (a^2 (A-3 C)\right )-3 a b B+4 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 a^2 d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)}}+\frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \sqrt {\sec (c+d x)} \left (a^2 (A+3 C)-6 a b B+8 A b^2\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{3 a^3 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^3 B-a^2 (5 A b-3 b C)-6 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{3 a^3 d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}} \]

[In]

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

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]

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 10.97 (sec) , antiderivative size = 4557, normalized size of antiderivative = 13.02 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

((b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(-5*a^2*A*b + 11*A*b^3 + 3*a^3*B - 9*a*b^
2*B + 6*a^2*b*C - 5*a^2*A*b*Cos[2*c] + 5*A*b^3*Cos[2*c] + 3*a^3*B*Cos[2*c] - 3*a*b^2*B*Cos[2*c])*Csc[c]*Sec[c]
)/(3*a^3*(a^2 - b^2)*d) + (4*A*Cos[d*x]*Sin[c])/(3*a^2*d) + (4*A*Cos[c]*Sin[d*x])/(3*a^2*d) + (4*Sec[c]*(A*b^4
*Sin[c] - a*b^3*B*Sin[c] + a^2*b^2*C*Sin[c] - a*A*b^3*Sin[d*x] + a^2*b^2*B*Sin[d*x] - a^3*b*C*Sin[d*x]))/(a^3*
(a^2 - b^2)*d*(b + a*Cos[c + d*x]))))/((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)) - (4*A*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x
 - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + C
ot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*(
b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqr
t[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*S
qrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b
- a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(3*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos
[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (8*A*b^2*AppellF1[1/2, 1/2,
 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*
Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt
[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x
] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - Ar
cTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x -
ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[
c]]]])/(3*a^2*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sqrt[Sec[c +
d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (4*b*B*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[
c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*
Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c
]^2])))]*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*S
qrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]
*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c]
)]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(a*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*
x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (4*C*AppellF1[1/2
, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1
 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(
a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c
 + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*
x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[
d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTa
n[Cot[c]]]])/((a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sqrt[Sec[c +
d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (10*A*b*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c +
d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]
))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[T
an[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTa
n[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c
]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan
[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]])
 - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]
]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan
[c]^2]]))/(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)) - (16*A*b^3*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1
[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan
[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + T
an[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])
/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] +
a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c
]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + Arc
Tan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^
2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(3*a^2*(
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))
- (2*a*B*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2
, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Se
c[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqr
t[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]
^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]
^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + T
an[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[
c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^
2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/((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)) + (4*b^2*B*(b + a*Cos[c
 + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b +
 a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan
[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 -
 (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 +
Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1
+ Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a
*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2
] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sq
rt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(a*(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)) - (2*b*C*(b + a*Cos[c + d*x])^(3/2)*Csc[c]
*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + A
rcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]
*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1
 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x
 + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d
*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + Arc
Tan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b +
a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d
*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/((a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sq
rt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(3194\) vs. \(2(382)=764\).

Time = 13.78 (sec) , antiderivative size = 3195, normalized size of antiderivative = 9.13

method result size
parts \(\text {Expression too large to display}\) \(3195\)
default \(\text {Expression too large to display}\) \(3532\)

[In]

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

[Out]

2/3*A/d/a^3/(a+b)/((a-b)/(a+b))^(1/2)*(a+b*sec(d*x+c))^(1/2)/(b+a*cos(d*x+c))/sec(d*x+c)^(3/2)/(1+cos(d*x+c))*
(EllipticF(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos
(d*x+c)))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*a^3*cos(d*x+c)+6*EllipticF(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c
)),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*a^2*b*cos(d*
x+c)+8*EllipticF(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/
(1+cos(d*x+c)))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*a*b^2*cos(d*x+c)-5*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+c
sc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*a^2*
b*cos(d*x+c)+8*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(
d*x+c))/(1+cos(d*x+c)))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*b^3*cos(d*x+c)+2*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c
)))^(1/2)*(1/(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+12*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1/(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+16*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1/(
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^2-10*(1/
(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1/(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+16*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1/(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-b)/(a+b))^(1/2)*
a^3*cos(d*x+c)*sin(d*x+c)+((a-b)/(a+b))^(1/2)*a^2*b*cos(d*x+c)*sin(d*x+c)+(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x
+c)))^(1/2)*(1/(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*sec(d*x+c)+6*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1/(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*sec(d*x+c)+8*(1/(a+b)*(b+a*cos(d*x+c))/(1+c
os(d*x+c)))^(1/2)*(1/(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*sec(d*x+c)-5*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*Elliptic
E(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b*sec(d*x+c)+8*(1/(a+b)*(b+a*cos(d*x+
c))/(1+cos(d*x+c)))^(1/2)*(1/(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*sec(d*x+c)+((a-b)/(a+b))^(1/2)*a^3*sin(d*x+c)-3*((a-b)/(a+b))^(1/2)*a^2*b*sin(d*x+c)-4*(
(a-b)/(a+b))^(1/2)*a*b^2*sin(d*x+c)+((a-b)/(a+b))^(1/2)*a^2*b*tan(d*x+c)-4*((a-b)/(a+b))^(1/2)*a*b^2*tan(d*x+c
)-8*((a-b)/(a+b))^(1/2)*b^3*tan(d*x+c))+2*B/d/((a-b)/(a+b))^(1/2)/(a+b)/a^2*((a*(1-cos(d*x+c))^2*csc(d*x+c)^2-
(1-cos(d*x+c))^2*b*csc(d*x+c)^2-a-b)/((1-cos(d*x+c))^2*csc(d*x+c)^2-1))^(1/2)*(((a-b)/(a+b))^(1/2)*(1-cos(d*x+
c))^3*a^2*csc(d*x+c)^3-2*((a-b)/(a+b))^(1/2)*(1-cos(d*x+c))^3*b^2*csc(d*x+c)^3+(-(a*(1-cos(d*x+c))^2*csc(d*x+c
)^2-(1-cos(d*x+c))^2*b*csc(d*x+c)^2-a-b)/(a+b))^(1/2)*((1-cos(d*x+c))^2*csc(d*x+c)^2+1)^(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+2*(-(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-(1-cos(d
*x+c))^2*b*csc(d*x+c)^2-a-b)/(a+b))^(1/2)*((1-cos(d*x+c))^2*csc(d*x+c)^2+1)^(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-(-(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-(1-cos(d*x+c))^2*b*csc
(d*x+c)^2-a-b)/(a+b))^(1/2)*((1-cos(d*x+c))^2*csc(d*x+c)^2+1)^(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+2*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))
^(1/2))*b^2*(-(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-(1-cos(d*x+c))^2*b*csc(d*x+c)^2-a-b)/(a+b))^(1/2)*((1-cos(d*x+c
))^2*csc(d*x+c)^2+1)^(1/2)-((a-b)/(a+b))^(1/2)*a^2*(-cot(d*x+c)+csc(d*x+c))-2*((a-b)/(a+b))^(1/2)*a*b*(-cot(d*
x+c)+csc(d*x+c))-2*((a-b)/(a+b))^(1/2)*b^2*(-cot(d*x+c)+csc(d*x+c)))/(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-(1-cos(d
*x+c))^2*b*csc(d*x+c)^2-a-b)/(-((1-cos(d*x+c))^2*csc(d*x+c)^2+1)/((1-cos(d*x+c))^2*csc(d*x+c)^2-1))^(1/2)-2*C/
d/a/(a+b)/((a-b)/(a+b))^(1/2)*(-((1-cos(d*x+c))^2*csc(d*x+c)^2+1)/((1-cos(d*x+c))^2*csc(d*x+c)^2-1))^(1/2)*((1
-cos(d*x+c))^2*csc(d*x+c)^2-1)*((a*(1-cos(d*x+c))^2*csc(d*x+c)^2-(1-cos(d*x+c))^2*b*csc(d*x+c)^2-a-b)/((1-cos(
d*x+c))^2*csc(d*x+c)^2-1))^(1/2)*(((a-b)/(a+b))^(1/2)*(1-cos(d*x+c))^3*b*csc(d*x+c)^3-EllipticF(((a-b)/(a+b))^
(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*(-(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-(1-cos(d*x+c))^2*b*c
sc(d*x+c)^2-a-b)/(a+b))^(1/2)*((1-cos(d*x+c))^2*csc(d*x+c)^2+1)^(1/2)-(-(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-(1-co
s(d*x+c))^2*b*csc(d*x+c)^2-a-b)/(a+b))^(1/2)*((1-cos(d*x+c))^2*csc(d*x+c)^2+1)^(1/2)*EllipticE(((a-b)/(a+b))^(
1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*b+((a-b)/(a+b))^(1/2)*b*(-cot(d*x+c)+csc(d*x+c)))/((1-cos(
d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c))^4*b*csc(d*x+c)^4-2*(1-cos(d*x+c))^2*b*csc(d*x+c)^2-a-b)

Fricas [C] (verification not implemented)

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

Time = 0.17 (sec) , antiderivative size = 860, normalized size of antiderivative = 2.46 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/9*(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 + (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(
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^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 + (-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)*c
os(d*x + c))*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*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 + (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(a)*weierstrassZe
ta(-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^4*b + I*(5*A
- 3*C)*a^3*b^2 + 6*I*B*a^2*b^3 - 8*I*A*a*b^4 + (-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(a)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPIn
verse(-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)) - 6*((A*a^5 - A*a^3*b^2)*cos(d*x + c)^2 + ((A - 3*C)*a^4*b + 3*B*a^3*b^2 - 4*A*a^2*b^3)*cos(d*x + c))*sqrt
((a*cos(d*x + c) + b)/cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/((a^7 - a^5*b^2)*d*cos(d*x + c) + (a^6*b
- a^4*b^3)*d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F]

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

[In]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

[In]

int((A + B/cos(c + d*x) + C/cos(c + d*x)^2)/((a + b/cos(c + d*x))^(3/2)*(1/cos(c + d*x))^(3/2)),x)

[Out]

int((A + B/cos(c + d*x) + C/cos(c + d*x)^2)/((a + b/cos(c + d*x))^(3/2)*(1/cos(c + d*x))^(3/2)), x)

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