\(\int \frac {\sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sec ^{\frac {5}{2}}(c+d x)} \, dx\) [1035]

Optimal result

Integrand size = 45, antiderivative size = 273 \[ \int \frac {\sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {2 \left (a^2-b^2\right ) (2 A b-5 a B) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{15 a^2 d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (2 A b^2-5 a b B-3 a^2 (3 A+5 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{15 a^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (A b+5 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a d \sqrt {\sec (c+d x)}} \]

[Out]

-2/15*(a^2-b^2)*(2*A*b-5*B*a)*(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^2/d/(a+b*sec(d*x+c))^(1/2)+2/5*A*sin(d
*x+c)*(a+b*sec(d*x+c))^(1/2)/d/sec(d*x+c)^(3/2)+2/15*(A*b+5*B*a)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a/d/sec(d*x
+c)^(1/2)-2/15*(2*A*b^2-5*B*a*b-3*a^2*(3*A+5*C))*(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^2/d/((b+a*cos(d*x+c))/(a+b))^(1/2)/sec(d*x+c
)^(1/2)

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 273, 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 = {4179, 4189, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \frac {\sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {2 \left (-3 a^2 (3 A+5 C)-5 a b B+2 A b^2\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{15 a^2 d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {2 \left (a^2-b^2\right ) (2 A b-5 a B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{15 a^2 d \sqrt {a+b \sec (c+d x)}}+\frac {2 (5 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{15 a d \sqrt {\sec (c+d x)}}+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)} \]

[In]

Int[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]

[Out]

(-2*(a^2 - b^2)*(2*A*b - 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[
Sec[c + d*x]])/(15*a^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(2*A*b^2 - 5*a*b*B - 3*a^2*(3*A + 5*C))*EllipticE[(c +
d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*
x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(A*b + 5*a*B)*Sqrt[a + b*Sec[
c + d*x]]*Sin[c + d*x])/(15*a*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 4179

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*
Csc[e + f*x])^n/(f*n)), x] - Dist[1/(d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[A*b*
m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Csc[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]

Rule 4189

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1
)*((d*Csc[e + f*x])^n/(a*f*n)), x] + Dist[1/(a*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[
a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ
[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2}{5} \int \frac {\frac {1}{2} (A b+5 a B)+\frac {1}{2} (3 a A+5 b B+5 a C) \sec (c+d x)+\frac {1}{2} b (2 A+5 C) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (A b+5 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a d \sqrt {\sec (c+d x)}}-\frac {4 \int \frac {\frac {1}{4} \left (2 A b^2-5 a b B-3 a^2 (3 A+5 C)\right )-\frac {1}{4} a (7 A b+5 a B+15 b C) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{15 a} \\ & = \frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (A b+5 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a d \sqrt {\sec (c+d x)}}-\frac {\left (\left (a^2-b^2\right ) (2 A b-5 a B)\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{15 a^2}-\frac {\left (2 A b^2-5 a b B-3 a^2 (3 A+5 C)\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{15 a^2} \\ & = \frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (A b+5 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a d \sqrt {\sec (c+d x)}}-\frac {\left (\left (a^2-b^2\right ) (2 A b-5 a B) \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{15 a^2 \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (2 A b^2-5 a b B-3 a^2 (3 A+5 C)\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{15 a^2 \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \\ & = \frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (A b+5 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a d \sqrt {\sec (c+d x)}}-\frac {\left (\left (a^2-b^2\right ) (2 A b-5 a B) \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}{15 a^2 \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (2 A b^2-5 a b B-3 a^2 (3 A+5 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}{15 a^2 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}} \\ & = -\frac {2 \left (a^2-b^2\right ) (2 A b-5 a B) \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)}}{15 a^2 d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (2 A b^2-5 a b B-3 a^2 (3 A+5 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{15 a^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (A b+5 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a 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 = 7.34 (sec) , antiderivative size = 3426, normalized size of antiderivative = 12.55 \[ \int \frac {\sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx=\text {Result too large to show} \]

[In]

Integrate[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]

[Out]

(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(9*a^2*A - 2*A*b^2 + 5*a*b*B + 15*a^2*C
)*Cot[c])/(15*a^2*d) + (4*(A*b + 5*a*B)*Cos[d*x]*Sin[c])/(15*a*d) + (2*A*Cos[2*d*x]*Sin[2*c])/(5*d) + (4*(A*b
+ 5*a*B)*Cos[c]*Sin[d*x])/(15*a*d) + (2*A*Cos[2*c]*Sin[2*d*x])/(5*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*
c + 2*d*x])*Sec[c + d*x]^(5/2)) - (28*A*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*S
qrt[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]*Sqrt[a + b*Sec[c + d*x]]*(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])]*Sq
rt[(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]]]])/(15*a*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2
*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(5/2)) - (4*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])))]*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(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 - ArcTan[Cot[c]]]])/(3*
d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(
5/2)) - (4*b*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]*Si
n[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Csc[c]*Sqrt[a + b*S
ec[c + d*x]]*(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 - ArcTan[Cot[c]]]])/(a*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c +
2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(5/2)) - (6*a*A*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(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]]]*Sq
rt[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 + T
an[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 + ArcT
an[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]]]*Sq
rt[1 + Tan[c]^2]]))/(5*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*
x]^(5/2)) + (4*A*b^2*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(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))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*a*d*Sqrt[b
+ a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2)) - (2*b*B*Csc[c]*Sqrt[a
+ b*Sec[c + d*x]]*(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 + T
an[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*Co
s[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*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[
c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2)) - (2*a*C*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(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]))))]*S
in[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))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]
]*Sqrt[1 + Tan[c]^2]]))/(d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c +
d*x]^(5/2))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(4449\) vs. \(2(303)=606\).

Time = 11.11 (sec) , antiderivative size = 4450, normalized size of antiderivative = 16.30

method	result	size
parts	\(\text {Expression too large to display}\)	\(4450\)
default	\(\text {Expression too large to display}\)	\(4802\)

[In]

int((A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2)/sec(d*x+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/15*A/d/a^2/((a-b)/(a+b))^(1/2)*(a+b*sec(d*x+c))^(1/2)/(b+a*cos(d*x+c))/sec(d*x+c)^(5/2)/(1+cos(d*x+c))*(-7*
(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-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*b^2+9*(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+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)*Ell
ipticE(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^2+9*(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)^2-9*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*Ellipt
icE(((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)^2-2*(1/(a+b)*(b+a*cos(d*
x+c))/(1+cos(d*x+c)))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*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)^2+18*(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)-18*(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^3*sec(d*x+c)-4*(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)-9*((a-b)/(
a+b))^(1/2)*a^2*b*tan(d*x+c)*sec(d*x+c)-((a-b)/(a+b))^(1/2)*a*b^2*tan(d*x+c)*sec(d*x+c)-4*((a-b)/(a+b))^(1/2)*
a^2*b*tan(d*x+c)+((a-b)/(a+b))^(1/2)*a*b^2*tan(d*x+c)+2*((a-b)/(a+b))^(1/2)*b^3*tan(d*x+c)*sec(d*x+c)-4*((a-b)
/(a+b))^(1/2)*a^2*b*sin(d*x+c)-3*((a-b)/(a+b))^(1/2)*a^3*cos(d*x+c)*sin(d*x+c)+9*(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^3-9*(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^3-2*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)
*(1/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*b^3-2*(
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*b^2*sec(d*x+c)^2+9*(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*sec(
d*x+c)^2+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)*EllipticE(((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)^2-14*(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)-4*(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*sec(d*x+c)+18*(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*sec(d*x+c)-7*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*Ellipt
icF(((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)^2-3*((a-b)/(a+b))^(1/2
)*a^3*sin(d*x+c)-9*((a-b)/(a+b))^(1/2)*a^3*tan(d*x+c)+4*(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*b^2*sec(d*x+
c))+2/3*B/d/a/((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))*(Ell
ipticE(((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*cos(d*x+c)-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^2*cos(d*x+c)+Ell
ipticF(((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*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*b*cos(d*x+c)+((a
-b)/(a+b))^(1/2)*a^2*cos(d*x+c)*sin(d*x+c)+2*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)*a*b-2*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^2+2*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^2-2*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+((a-b)/(a+b))^(1/2)*a^2*sin(d*x+c)+2*((a-b)/(a+b))^(1/2)*a*b*sin(d*x+c)+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)*a*b*sec(d*x+c)-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^2*sec(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^2*sec(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*b*sec(d*x+c)+((a-b)/(a+b))^(1/2)*a*b*tan(d*x
+c)+((a-b)/(a+b))^(1/2)*b^2*tan(d*x+c))+2*C/d/((a-b)/(a+b))^(1/2)*((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)*((1-cos(d*x+c))^3*a*((a-b)/(a+b))^(1/2)*cs
c(d*x+c)^3-((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*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))*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)-(-(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+(-(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))*b-(
(a-b)/(a+b))^(1/2)*a*(-cot(d*x+c)+csc(d*x+c))-((a-b)/(a+b))^(1/2)*b*(-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*cs
c(d*x+c)^2-1))^(1/2)

Fricas [C] (verification not implemented)

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

Time = 0.14 (sec) , antiderivative size = 539, normalized size of antiderivative = 1.97 \[ \int \frac {\sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx=\frac {\sqrt {2} {\left (-15 i \, B a^{3} - 3 i \, {\left (A + 5 \, C\right )} a^{2} b + 10 i \, B a b^{2} - 4 i \, A b^{3}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) + \sqrt {2} {\left (15 i \, B a^{3} + 3 i \, {\left (A + 5 \, C\right )} a^{2} b - 10 i \, B a b^{2} + 4 i \, A b^{3}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) - 3 \, \sqrt {2} {\left (-3 i \, {\left (3 \, A + 5 \, C\right )} a^{3} - 5 i \, B a^{2} b + 2 i \, A a b^{2}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) - 3 \, \sqrt {2} {\left (3 i \, {\left (3 \, A + 5 \, C\right )} a^{3} + 5 i \, B a^{2} b - 2 i \, A a b^{2}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) + \frac {6 \, {\left (3 \, A a^{3} \cos \left (d x + c\right )^{2} + {\left (5 \, B a^{3} + A a^{2} b\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{45 \, a^{3} d} \]

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2)/sec(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

1/45*(sqrt(2)*(-15*I*B*a^3 - 3*I*(A + 5*C)*a^2*b + 10*I*B*a*b^2 - 4*I*A*b^3)*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
)*(15*I*B*a^3 + 3*I*(A + 5*C)*a^2*b - 10*I*B*a*b^2 + 4*I*A*b^3)*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*(3*
A + 5*C)*a^3 - 5*I*B*a^2*b + 2*I*A*a*b^2)*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*(3*A + 5*C)*a^3 + 5*I*B*a^2*b - 2*I*A*a*b^2)*sqrt(a)*weierstrassZ
eta(-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)) + 6*(3*A*a^3*cos(d*x + c)^2 + (5*B
*a^3 + A*a^2*b)*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^3*d)

Sympy [F]

\[ \int \frac {\sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {\sqrt {a + b \sec {\left (c + d x \right )}} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right )}{\sec ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \]

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)**2)*(a+b*sec(d*x+c))**(1/2)/sec(d*x+c)**(5/2),x)

[Out]

Integral(sqrt(a + b*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)**2)/sec(c + d*x)**(5/2), x)

Maxima [F]

\[ \int \frac {\sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt {b \sec \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2)/sec(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*sqrt(b*sec(d*x + c) + a)/sec(d*x + c)^(5/2), x)

Giac [F]

\[ \int \frac {\sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt {b \sec \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2)/sec(d*x+c)^(5/2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*sqrt(b*sec(d*x + c) + a)/sec(d*x + c)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]

[In]

int(((a + b/cos(c + d*x))^(1/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/cos(c + d*x))^(5/2),x)

[Out]

int(((a + b/cos(c + d*x))^(1/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/cos(c + d*x))^(5/2), x)