3.3.15 \(\int \frac {(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{7/2}} \, dx\) [215]

Optimal. Leaf size=1150 \[ \frac {2 (a-b) \sqrt {a+b} \left (b^2 c^2 d \left (29 c^2+3 d^2\right )-a b c \left (35 c^4+34 c^2 d^2-5 d^4\right )+a^2 \left (58 c^4 d-41 c^2 d^3+15 d^5\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left (\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{15 c^3 (c-d)^3 (c+d)^{5/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 \sqrt {a+b} \left (b^3 c^4 \left (5 c^2+24 c d+3 d^2\right )-a b^2 c^3 \left (35 c^3+42 c^2 d+21 c d^2-2 d^3\right )+a^2 b c^2 \left (45 c^4+48 c^3 d+c^2 d^2-8 c d^3+10 d^4\right )-a^3 d \left (60 c^5-2 c^4 d-66 c^3 d^2+25 c^2 d^3+30 c d^4-15 d^5\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left (\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{15 c^4 (c-d)^3 (c+d)^{5/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 a^2 \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left (\frac {(a+b) c}{a (c+d)};\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c^4 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 d (b c-a d) (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}+\frac {2 (b c-a d) \left (5 b c^3-13 a c^2 d+3 b c d^2+5 a d^3\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{15 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}} \]

[Out]

-2/5*d*(-a*d+b*c)*(b+a*cos(f*x+e))*sin(f*x+e)*(a+b*sec(f*x+e))^(1/2)/c/(c^2-d^2)/f/(d+c*cos(f*x+e))^2/(c+d*sec
(f*x+e))^(1/2)+2/15*(-a*d+b*c)*(-13*a*c^2*d+5*a*d^3+5*b*c^3+3*b*c*d^2)*sin(f*x+e)*(a+b*sec(f*x+e))^(1/2)/c^2/(
c^2-d^2)^2/f/(d+c*cos(f*x+e))/(c+d*sec(f*x+e))^(1/2)+2/15*(a-b)*(b^2*c^2*d*(29*c^2+3*d^2)-a*b*c*(35*c^4+34*c^2
*d^2-5*d^4)+a^2*(58*c^4*d-41*c^2*d^3+15*d^5))*(d+c*cos(f*x+e))^(3/2)*csc(f*x+e)*EllipticE((c+d)^(1/2)*(b+a*cos
(f*x+e))^(1/2)/(a+b)^(1/2)/(d+c*cos(f*x+e))^(1/2),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(a+b)^(1/2)*(-(-a*d+b*c)*(1
-cos(f*x+e))/(a+b)/(d+c*cos(f*x+e)))^(1/2)*(-(-a*d+b*c)*(1+cos(f*x+e))/(a-b)/(d+c*cos(f*x+e)))^(1/2)*(a+b*sec(
f*x+e))^(1/2)/c^3/(c-d)^3/(c+d)^(5/2)/(-a*d+b*c)/f/(b+a*cos(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2)+2/15*(b^3*c^4
*(5*c^2+24*c*d+3*d^2)-a*b^2*c^3*(35*c^3+42*c^2*d+21*c*d^2-2*d^3)+a^2*b*c^2*(45*c^4+48*c^3*d+c^2*d^2-8*c*d^3+10
*d^4)-a^3*d*(60*c^5-2*c^4*d-66*c^3*d^2+25*c^2*d^3+30*c*d^4-15*d^5))*(d+c*cos(f*x+e))^(3/2)*csc(f*x+e)*Elliptic
F((c+d)^(1/2)*(b+a*cos(f*x+e))^(1/2)/(a+b)^(1/2)/(d+c*cos(f*x+e))^(1/2),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(a+b)
^(1/2)*(-(-a*d+b*c)*(1-cos(f*x+e))/(a+b)/(d+c*cos(f*x+e)))^(1/2)*(-(-a*d+b*c)*(1+cos(f*x+e))/(a-b)/(d+c*cos(f*
x+e)))^(1/2)*(a+b*sec(f*x+e))^(1/2)/c^4/(c-d)^3/(c+d)^(5/2)/(-a*d+b*c)/f/(b+a*cos(f*x+e))^(1/2)/(c+d*sec(f*x+e
))^(1/2)-2*a^2*(d+c*cos(f*x+e))^(3/2)*csc(f*x+e)*EllipticPi((c+d)^(1/2)*(b+a*cos(f*x+e))^(1/2)/(a+b)^(1/2)/(d+
c*cos(f*x+e))^(1/2),(a+b)*c/a/(c+d),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(a+b)^(1/2)*(-(-a*d+b*c)*(1-cos(f*x+e))/(
a+b)/(d+c*cos(f*x+e)))^(1/2)*(-(-a*d+b*c)*(1+cos(f*x+e))/(a-b)/(d+c*cos(f*x+e)))^(1/2)*(a+b*sec(f*x+e))^(1/2)/
c^4/f/(c+d)^(1/2)/(b+a*cos(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 2.36, antiderivative size = 1150, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {4027, 3068, 3126, 3132, 2890, 3077, 2897, 3075} \begin {gather*} -\frac {2 \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left (\frac {(a+b) c}{a (c+d)};\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)} a^2}{c^4 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 (b c-a d) \left (5 b c^3-13 a d c^2+3 b d^2 c+5 a d^3\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{15 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}-\frac {2 d (b c-a d) (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}+\frac {2 (a-b) \sqrt {a+b} \left (\left (15 d^5-41 c^2 d^3+58 c^4 d\right ) a^2-b c \left (35 c^4+34 d^2 c^2-5 d^4\right ) a+b^2 c^2 d \left (29 c^2+3 d^2\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left (\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{15 c^3 (c-d)^3 (c+d)^{5/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 \sqrt {a+b} \left (b^3 \left (5 c^2+24 d c+3 d^2\right ) c^4-a b^2 \left (35 c^3+42 d c^2+21 d^2 c-2 d^3\right ) c^3+a^2 b \left (45 c^4+48 d c^3+d^2 c^2-8 d^3 c+10 d^4\right ) c^2-a^3 d \left (60 c^5-2 d c^4-66 d^2 c^3+25 d^3 c^2+30 d^4 c-15 d^5\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left (\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{15 c^4 (c-d)^3 (c+d)^{5/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^(7/2),x]

[Out]

(2*(a - b)*Sqrt[a + b]*(b^2*c^2*d*(29*c^2 + 3*d^2) - a*b*c*(35*c^4 + 34*c^2*d^2 - 5*d^4) + a^2*(58*c^4*d - 41*
c^2*d^3 + 15*d^5))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d
)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[ArcSi
n[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(
c + d))]*Sqrt[a + b*Sec[e + f*x]])/(15*c^3*(c - d)^3*(c + d)^(5/2)*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt
[c + d*Sec[e + f*x]]) + (2*Sqrt[a + b]*(b^3*c^4*(5*c^2 + 24*c*d + 3*d^2) - a*b^2*c^3*(35*c^3 + 42*c^2*d + 21*c
*d^2 - 2*d^3) + a^2*b*c^2*(45*c^4 + 48*c^3*d + c^2*d^2 - 8*c*d^3 + 10*d^4) - a^3*d*(60*c^5 - 2*c^4*d - 66*c^3*
d^2 + 25*c^2*d^3 + 30*c*d^4 - 15*d^5))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))
]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e +
f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)
*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(15*c^4*(c - d)^3*(c + d)^(5/2)*(b*c - a*d)*f*Sqrt[b +
a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*a^2*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b
)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[
e + f*x])^(3/2)*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])
/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^4*
Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*d*(b*c - a*d)*(b + a*Cos[e + f*x])*Sqrt[
a + b*Sec[e + f*x]]*Sin[e + f*x])/(5*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^2*Sqrt[c + d*Sec[e + f*x]]) + (2*(b*
c - a*d)*(5*b*c^3 - 13*a*c^2*d + 3*b*c*d^2 + 5*a*d^3)*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(15*c^2*(c^2 - d^
2)^2*f*(d + c*Cos[e + f*x])*Sqrt[c + d*Sec[e + f*x]])

Rule 2890

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[
2*((a + b*Sin[e + f*x])/(d*f*Rt[(a + b)/(c + d), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c -
d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticPi
[b*((c + d)/(d*(a + b))), ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]])],
(a - b)*((c + d)/((a + b)*(c - d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2
, 0] && NeQ[c^2 - d^2, 0] && PosQ[(a + b)/(c + d)]

Rule 2897

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Si
mp[2*((c + d*Sin[e + f*x])/(f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 - Sin[e +
f*x])/((a + b)*(c + d*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 + Sin[e + f*x])/((a - b)*(c + d*Sin[e + f*x]))
)]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], (a + b)*((c -
d)/((a - b)*(c + d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c
^2 - d^2, 0] && PosQ[(c + d)/(a + b)]

Rule 3068

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b*c - a*d))*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1
)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Si
n[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c -
(A*b + a*B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) - a*(b*c - a*d)*(B*c - A*
d)*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*Sin[e + f*x]^
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2
, 0] && GtQ[m, 1] && LtQ[n, -1]

Rule 3075

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin
[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A*(c - d)*((a + b*Sin[e + f*x])/(f*(b*c - a*d)^2*Rt[(a + b)/(c +
d), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d)
)*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticE[ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin
[e + f*x]]/Sqrt[a + b*Sin[e + f*x]])], (a - b)*((c + d)/((a + b)*(c - d)))], x] /; FreeQ[{a, b, c, d, e, f, A,
B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]

Rule 3077

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*s
in[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[(A - B)/(a - b), Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e
+ f*x]]), x], x] - Dist[(A*b - a*B)/(a - b), Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin
[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2
- d^2, 0] && NeQ[A, B]

Rule 3126

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

Rule 3132

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(((a_.) + (b_.)*sin[(e_.) + (f_.
)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[C/b^2, Int[Sqrt[a + b*Sin[e + f
*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[1/b^2, Int[(A*b^2 - a^2*C + b*(b*B - 2*a*C)*Sin[e + f*x])/((a + b
*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a
*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 4027

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_), x_Symbol] :> Dist
[Sqrt[d + c*Sin[e + f*x]]*(Sqrt[a + b*Csc[e + f*x]]/(Sqrt[b + a*Sin[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])), Int[
(b + a*Sin[e + f*x])^m*((d + c*Sin[e + f*x])^n/Sin[e + f*x]^(m + n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}
, x] && NeQ[b*c - a*d, 0] && IntegerQ[m + 1/2] && IntegerQ[n + 1/2] && LeQ[-2, m + n, 0]

Rubi steps

\begin {align*} \int \frac {(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{7/2}} \, dx &=\frac {\left (\sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {\cos (e+f x) (b+a \cos (e+f x))^{5/2}}{(d+c \cos (e+f x))^{7/2}} \, dx}{\sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\\ &=-\frac {2 d (b c-a d) (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}+\frac {\left (2 \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {\sqrt {b+a \cos (e+f x)} \left (\frac {1}{2} (5 b c-3 a d) (b c-a d)-\frac {1}{2} \left (5 a^2 c d+3 b^2 c d-2 a b \left (5 c^2-d^2\right )\right ) \cos (e+f x)+\frac {5}{2} a^2 \left (c^2-d^2\right ) \cos ^2(e+f x)\right )}{(d+c \cos (e+f x))^{5/2}} \, dx}{5 c \left (c^2-d^2\right ) \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\\ &=-\frac {2 d (b c-a d) (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}+\frac {2 (b c-a d) \left (5 b c^3-13 a c^2 d+3 b c d^2+5 a d^3\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{15 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}+\frac {\left (4 \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {\frac {1}{4} (b c-a d) \left (35 a b c^3-13 a^2 c^2 d-24 b^2 c^2 d-3 a b c d^2+5 a^2 d^3\right )-\frac {1}{4} \left (2 a b^2 c d \left (21 c^2-d^2\right )-b^3 c^2 \left (5 c^2+3 d^2\right )+6 a^3 \left (5 c^3 d-c d^3\right )-a^2 b \left (45 c^4+c^2 d^2+10 d^4\right )\right ) \cos (e+f x)+\frac {15}{4} a^3 \left (c^2-d^2\right )^2 \cos ^2(e+f x)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}} \, dx}{15 c^2 \left (c^2-d^2\right )^2 \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\\ &=-\frac {2 d (b c-a d) (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}+\frac {2 (b c-a d) \left (5 b c^3-13 a c^2 d+3 b c d^2+5 a d^3\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{15 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}+\frac {\left (a^3 \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {\sqrt {d+c \cos (e+f x)}}{\sqrt {b+a \cos (e+f x)}} \, dx}{c^4 \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {\left (4 \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {-\frac {15}{4} a^3 d^2 \left (c^2-d^2\right )^2+\frac {1}{4} c^2 (b c-a d) \left (35 a b c^3-13 a^2 c^2 d-24 b^2 c^2 d-3 a b c d^2+5 a^2 d^3\right )+c \left (-\frac {15}{2} a^3 d \left (c^2-d^2\right )^2+\frac {1}{4} c \left (-2 a b^2 c d \left (21 c^2-d^2\right )+b^3 c^2 \left (5 c^2+3 d^2\right )-6 a^3 \left (5 c^3 d-c d^3\right )+a^2 b \left (45 c^4+c^2 d^2+10 d^4\right )\right )\right ) \cos (e+f x)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}} \, dx}{15 c^4 \left (c^2-d^2\right )^2 \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\\ &=-\frac {2 a^2 \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left (\frac {(a+b) c}{a (c+d)};\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c^4 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 d (b c-a d) (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}+\frac {2 (b c-a d) \left (5 b c^3-13 a c^2 d+3 b c d^2+5 a d^3\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{15 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}+\frac {\left (\left (b^3 c^4 \left (5 c^2+24 c d+3 d^2\right )-a b^2 c^3 \left (35 c^3+42 c^2 d+21 c d^2-2 d^3\right )+a^2 b c^2 \left (45 c^4+48 c^3 d+c^2 d^2-8 c d^3+10 d^4\right )-a^3 d \left (60 c^5-2 c^4 d-66 c^3 d^2+25 c^2 d^3+30 c d^4-15 d^5\right )\right ) \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}} \, dx}{15 c^4 (c-d) \left (c^2-d^2\right )^2 \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {\left (4 \left (c \left (-\frac {15}{4} a^3 d^2 \left (c^2-d^2\right )^2+\frac {1}{4} c^2 (b c-a d) \left (35 a b c^3-13 a^2 c^2 d-24 b^2 c^2 d-3 a b c d^2+5 a^2 d^3\right )\right )-c d \left (-\frac {15}{2} a^3 d \left (c^2-d^2\right )^2+\frac {1}{4} c \left (-2 a b^2 c d \left (21 c^2-d^2\right )+b^3 c^2 \left (5 c^2+3 d^2\right )-6 a^3 \left (5 c^3 d-c d^3\right )+a^2 b \left (45 c^4+c^2 d^2+10 d^4\right )\right )\right )\right ) \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {1+\cos (e+f x)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}} \, dx}{15 c^4 (-c+d) \left (c^2-d^2\right )^2 \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\\ &=\frac {2 (a-b) \sqrt {a+b} \left (b^2 c^2 d \left (29 c^2+3 d^2\right )-a b c \left (35 c^4+34 c^2 d^2-5 d^4\right )+a^2 \left (58 c^4 d-41 c^2 d^3+15 d^5\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{15 c^3 (c-d)^3 (c+d)^{5/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 \sqrt {a+b} \left (b^3 c^4 \left (5 c^2+24 c d+3 d^2\right )-a b^2 c^3 \left (35 c^3+42 c^2 d+21 c d^2-2 d^3\right )+a^2 b c^2 \left (45 c^4+48 c^3 d+c^2 d^2-8 c d^3+10 d^4\right )-a^3 d \left (60 c^5-2 c^4 d-66 c^3 d^2+25 c^2 d^3+30 c d^4-15 d^5\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{15 c^4 (c-d)^3 (c+d)^{5/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 a^2 \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left (\frac {(a+b) c}{a (c+d)};\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c^4 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 d (b c-a d) (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}+\frac {2 (b c-a d) \left (5 b c^3-13 a c^2 d+3 b c d^2+5 a d^3\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{15 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2344\) vs. \(2(1150)=2300\).
time = 7.37, size = 2344, normalized size = 2.04 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^(7/2),x]

[Out]

((d + c*Cos[e + f*x])^4*Sec[e + f*x]*(a + b*Sec[e + f*x])^(5/2)*((-2*(b^2*c^2*d*Sin[e + f*x] - 2*a*b*c*d^2*Sin
[e + f*x] + a^2*d^3*Sin[e + f*x]))/(5*c^2*(c^2 - d^2)*(d + c*Cos[e + f*x])^3) + (2*(5*b^2*c^4*Sin[e + f*x] - 2
1*a*b*c^3*d*Sin[e + f*x] + 16*a^2*c^2*d^2*Sin[e + f*x] + 3*b^2*c^2*d^2*Sin[e + f*x] + 5*a*b*c*d^3*Sin[e + f*x]
- 8*a^2*d^4*Sin[e + f*x]))/(15*c^2*(c^2 - d^2)^2*(d + c*Cos[e + f*x])^2) + (2*(35*a*b*c^5*Sin[e + f*x] - 58*a
^2*c^4*d*Sin[e + f*x] - 29*b^2*c^4*d*Sin[e + f*x] + 34*a*b*c^3*d^2*Sin[e + f*x] + 41*a^2*c^2*d^3*Sin[e + f*x]
- 3*b^2*c^2*d^3*Sin[e + f*x] - 5*a*b*c*d^4*Sin[e + f*x] - 15*a^2*d^5*Sin[e + f*x]))/(15*c^2*(c^2 - d^2)^3*(d +
c*Cos[e + f*x]))))/(f*(b + a*Cos[e + f*x])^2*(c + d*Sec[e + f*x])^(7/2)) + ((d + c*Cos[e + f*x])^(7/2)*Sec[e
+ f*x]*(a + b*Sec[e + f*x])^(5/2)*((4*(b*c - a*d)*(10*a^2*b*c^5 + 5*b^3*c^5 + 13*a^3*c^4*d - 48*a*b^2*c^4*d +
15*a^2*b*c^3*d^2 + 27*b^3*c^3*d^2 - 18*a^3*c^2*d^3 - 16*a*b^2*c^2*d^3 + 7*a^2*b*c*d^4 + 5*a^3*d^5)*Sqrt[((c +
d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a
- b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c
*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^
4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) + 4*(b*c - a*d)*(15*a^3*c^5 - 35*a*b^2*
c^5 + 23*a^2*b*c^4*d + 29*b^3*c^4*d + 13*a^3*c^3*d^2 - 5*a*b^2*c^3*d^2 - 75*a^2*b*c^2*d^3 + 3*b^3*c^2*d^3 + 4*
a^3*c*d^4 + 8*a*b^2*c*d^4 + 20*a^2*b*d^5)*((Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Co
s[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*
d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]
], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c
*Cos[e + f*x]]) - (Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)
/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*Elliptic
Pi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2
]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[
e + f*x]])) + 2*(-35*a^2*b*c^5 + 58*a^3*c^4*d + 29*a*b^2*c^4*d - 34*a^2*b*c^3*d^2 - 41*a^3*c^2*d^3 + 3*a*b^2*c
^2*d^3 + 5*a^2*b*c*d^4 + 15*a^3*d^5)*((Sqrt[(-a + b)/(a + b)]*(a + b)*Cos[(e + f*x)/2]*Sqrt[d + c*Cos[e + f*x]
]*EllipticE[ArcSin[(Sqrt[(-a + b)/(a + b)]*Sin[(e + f*x)/2])/Sqrt[(b + a*Cos[e + f*x])/(a + b)]], (2*(b*c - a*
d))/((-a + b)*(c + d))])/(a*c*Sqrt[((a + b)*Cos[(e + f*x)/2]^2)/(b + a*Cos[e + f*x])]*Sqrt[b + a*Cos[e + f*x]]
*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*Sqrt[((a + b)*(d + c*Cos[e + f*x]))/((c + d)*(b + a*Cos[e + f*x]))]) - (2*
(b*c - a*d)*(((b*c + (a + b)*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*
Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e +
f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c -
a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x
]]) - ((b*c + a*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)
/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*Elliptic
Pi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2
]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[
e + f*x]])))/(a*c) + (Sqrt[d + c*Cos[e + f*x]]*Sin[e + f*x])/(c*Sqrt[b + a*Cos[e + f*x]]))))/(15*c^2*(c - d)^3
*(c + d)^3*f*(b + a*Cos[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^(7/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(32282\) vs. \(2(1065)=2130\).
time = 3.40, size = 32283, normalized size = 28.07

method	result	size
default	\(\text {Expression too large to display}\)	\(32283\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sec(f*x+e))^(5/2)/(c+d*sec(f*x+e))^(7/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))^(5/2)/(c+d*sec(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e) + a)^(5/2)/(d*sec(f*x + e) + c)^(7/2), x)

________________________________________________________________________________________

Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))^(5/2)/(c+d*sec(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))**(5/2)/(c+d*sec(f*x+e))**(7/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6190 deep

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))^(5/2)/(c+d*sec(f*x+e))^(7/2),x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e) + a)^(5/2)/(d*sec(f*x + e) + c)^(7/2), x)

________________________________________________________________________________________

Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/cos(e + f*x))^(5/2)/(c + d/cos(e + f*x))^(7/2),x)

[Out]

\text{Hanged}

________________________________________________________________________________________