3.216 \(\int \frac {(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{9/2}} \, dx\)

Optimal. Leaf size=1428 \[ -\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)};\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)} a^2}{c^5 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 \left (-\left (\left (35 c^6+67 d^2 c^4-6 d^4 c^2\right ) b^2\right )+2 a c d \left (91 c^4-2 d^2 c^2+7 d^4\right ) b-a^2 d^2 \left (162 c^4-101 d^2 c^2+35 d^4\right )\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{105 c^3 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}-\frac {2 d \left (14 b c^3-19 a d c^2-2 b d^2 c+7 a d^3\right ) (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{35 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}+\frac {2 d^2 (b+a \cos (e+f x))^2 \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{7 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3 \sqrt {c+d \sec (e+f x)}}+\frac {2 (a-b) \sqrt {a+b} \left (-\left (\left (-105 d^8+392 c^2 d^6-485 c^4 d^4+582 c^6 d^2\right ) a^3\right )+2 b c d \left (406 c^6+73 d^2 c^4+132 d^4 c^2-35 d^6\right ) a^2-b^2 c^2 \left (245 c^6+852 d^2 c^4+41 d^4 c^2+14 d^6\right ) a+2 b^3 c^3 d \left (133 c^4+62 d^2 c^2-3 d^4\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 (\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)}}{105 c^4 (c-d)^4 (c+d)^{7/2} (b c-a d)^2 f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 \sqrt {a+b} \left (b^3 \left (35 c^4+231 d c^3+67 d^2 c^2+57 d^3 c-6 d^4\right ) c^4-a b^2 \left (245 c^5+413 d c^4+439 d^2 c^3+53 d^3 c^2-12 d^4 c+14 d^5\right ) c^3+a^2 b \left (315 c^6+497 d c^5+219 d^2 c^4-73 d^3 c^3+208 d^4 c^2+56 d^5 c-70 d^6\right ) c^2-a^3 d \left (525 c^7+57 d c^6-699 d^2 c^5+214 d^3 c^4+672 d^4 c^3-280 d^5 c^2-210 d^6 c+105 d^7\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 (\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)}}{105 c^5 (c-d)^4 (c+d)^{7/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}} \]

[Out]

2/7*d^2*(b+a*cos(f*x+e))^2*sin(f*x+e)*(a+b*sec(f*x+e))^(1/2)/c/(c^2-d^2)/f/(d+c*cos(f*x+e))^3/(c+d*sec(f*x+e))
^(1/2)-2/35*d*(-19*a*c^2*d+7*a*d^3+14*b*c^3-2*b*c*d^2)*(b+a*cos(f*x+e))*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))^2/(c+d*sec(f*x+e))^(1/2)-2/105*(2*a*b*c*d*(91*c^4-2*c^2*d^2+7*d^4)-a^2*d^2*(162
*c^4-101*c^2*d^2+35*d^4)-b^2*(35*c^6+67*c^4*d^2-6*c^2*d^4))*sin(f*x+e)*(a+b*sec(f*x+e))^(1/2)/c^3/(c^2-d^2)^3/
f/(d+c*cos(f*x+e))/(c+d*sec(f*x+e))^(1/2)+2/105*(a-b)*(2*b^3*c^3*d*(133*c^4+62*c^2*d^2-3*d^4)+2*a^2*b*c*d*(406
*c^6+73*c^4*d^2+132*c^2*d^4-35*d^6)-a*b^2*c^2*(245*c^6+852*c^4*d^2+41*c^2*d^4+14*d^6)-a^3*(582*c^6*d^2-485*c^4
*d^4+392*c^2*d^6-105*d^8))*(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^4/(
c-d)^4/(c+d)^(7/2)/(-a*d+b*c)^2/f/(b+a*cos(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2)+2/105*(b^3*c^4*(35*c^4+231*c^3
*d+67*c^2*d^2+57*c*d^3-6*d^4)-a*b^2*c^3*(245*c^5+413*c^4*d+439*c^3*d^2+53*c^2*d^3-12*c*d^4+14*d^5)+a^2*b*c^2*(
315*c^6+497*c^5*d+219*c^4*d^2-73*c^3*d^3+208*c^2*d^4+56*c*d^5-70*d^6)-a^3*d*(525*c^7+57*c^6*d-699*c^5*d^2+214*
c^4*d^3+672*c^3*d^4-280*c^2*d^5-210*c*d^6+105*d^7))*(d+c*cos(f*x+e))^(3/2)*csc(f*x+e)*EllipticF((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^5/(c-d)^4/(c+d)^(7/2)/(-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^5/f/(c+d)^(1/2
)/(b+a*cos(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 5.44, antiderivative size = 1428, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {3942, 3048, 3047, 3053, 2811, 2998, 2818, 2996} \[ -\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)};\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)} a^2}{c^5 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 \left (-\left (35 c^6+67 d^2 c^4-6 d^4 c^2\right ) b^2+2 a c d \left (91 c^4-2 d^2 c^2+7 d^4\right ) b-a^2 d^2 \left (162 c^4-101 d^2 c^2+35 d^4\right )\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{105 c^3 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}-\frac {2 d \left (14 b c^3-19 a d c^2-2 b d^2 c+7 a d^3\right ) (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{35 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}+\frac {2 d^2 (b+a \cos (e+f x))^2 \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{7 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3 \sqrt {c+d \sec (e+f x)}}+\frac {2 (a-b) \sqrt {a+b} \left (-\left (-105 d^8+392 c^2 d^6-485 c^4 d^4+582 c^6 d^2\right ) a^3+2 b c d \left (406 c^6+73 d^2 c^4+132 d^4 c^2-35 d^6\right ) a^2-b^2 c^2 \left (245 c^6+852 d^2 c^4+41 d^4 c^2+14 d^6\right ) a+2 b^3 c^3 d \left (133 c^4+62 d^2 c^2-3 d^4\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 (\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)}}{105 c^4 (c-d)^4 (c+d)^{7/2} (b c-a d)^2 f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 \sqrt {a+b} \left (b^3 \left (35 c^4+231 d c^3+67 d^2 c^2+57 d^3 c-6 d^4\right ) c^4-a b^2 \left (245 c^5+413 d c^4+439 d^2 c^3+53 d^3 c^2-12 d^4 c+14 d^5\right ) c^3+a^2 b \left (315 c^6+497 d c^5+219 d^2 c^4-73 d^3 c^3+208 d^4 c^2+56 d^5 c-70 d^6\right ) c^2-a^3 d \left (525 c^7+57 d c^6-699 d^2 c^5+214 d^3 c^4+672 d^4 c^3-280 d^5 c^2-210 d^6 c+105 d^7\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 (\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)}}{105 c^5 (c-d)^4 (c+d)^{7/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^(9/2),x]

[Out]

(2*(a - b)*Sqrt[a + b]*(2*b^3*c^3*d*(133*c^4 + 62*c^2*d^2 - 3*d^4) + 2*a^2*b*c*d*(406*c^6 + 73*c^4*d^2 + 132*c
^2*d^4 - 35*d^6) - a*b^2*c^2*(245*c^6 + 852*c^4*d^2 + 41*c^2*d^4 + 14*d^6) - a^3*(582*c^6*d^2 - 485*c^4*d^4 +
392*c^2*d^6 - 105*d^8))*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[
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]])/(105*c^4*(c - d)^4*(c + d)^(7/2)*(b*c - a*d)^2*f*Sqrt[b + a*Cos[e + f*
x]]*Sqrt[c + d*Sec[e + f*x]]) + (2*Sqrt[a + b]*(b^3*c^4*(35*c^4 + 231*c^3*d + 67*c^2*d^2 + 57*c*d^3 - 6*d^4) -
a*b^2*c^3*(245*c^5 + 413*c^4*d + 439*c^3*d^2 + 53*c^2*d^3 - 12*c*d^4 + 14*d^5) + a^2*b*c^2*(315*c^6 + 497*c^5
*d + 219*c^4*d^2 - 73*c^3*d^3 + 208*c^2*d^4 + 56*c*d^5 - 70*d^6) - a^3*d*(525*c^7 + 57*c^6*d - 699*c^5*d^2 + 2
14*c^4*d^3 + 672*c^3*d^4 - 280*c^2*d^5 - 210*c*d^6 + 105*d^7))*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*C
os[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(105*c^5*(c - d)^4*(c + d)^(7/2
)*(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^5*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) + (2*d^2*(b + a*Cos[
e + f*x])^2*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(7*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^3*Sqrt[c + d*Sec[e
+ f*x]]) - (2*d*(14*b*c^3 - 19*a*c^2*d - 2*b*c*d^2 + 7*a*d^3)*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Si
n[e + f*x])/(35*c^2*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x])^2*Sqrt[c + d*Sec[e + f*x]]) - (2*(2*a*b*c*d*(91*c^4 -
2*c^2*d^2 + 7*d^4) - a^2*d^2*(162*c^4 - 101*c^2*d^2 + 35*d^4) - b^2*(35*c^6 + 67*c^4*d^2 - 6*c^2*d^4))*Sqrt[a
+ b*Sec[e + f*x]]*Sin[e + f*x])/(105*c^3*(c^2 - d^2)^3*f*(d + c*Cos[e + f*x])*Sqrt[c + d*Sec[e + f*x]])

Rule 2811

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])*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))])/(d*f*
Rt[(a + b)/(c + d), 2]*Cos[e + f*x]), 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 2818

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])*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))])/(f*(b*c - a*d)*Rt[(c + d)/(a
+ b), 2]*Cos[e + f*x]), 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 2996

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])*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))])/(f*(b*c - a*d)^2*Rt[(a + b)/(c + d), 2]*Cos[e + f*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] && EqQ[A, B] && PosQ[(a + b)/(c + d)]

Rule 2998

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 3047

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 3048

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C + 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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n +
2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(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, 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 3053

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 3942

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))^{9/2}} \, dx &=\frac {\left (\sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {\cos ^2(e+f x) (b+a \cos (e+f x))^{5/2}}{(d+c \cos (e+f x))^{9/2}} \, dx}{\sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\\ &=\frac {2 d^2 (b+a \cos (e+f x))^2 \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{7 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3 \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 {(b+a \cos (e+f x))^{3/2} \left (-\frac {1}{2} d (7 b c-5 a d)+\frac {1}{2} \left (7 b c^2-7 a c d-2 b d^2\right ) \cos (e+f x)+\frac {7}{2} a \left (c^2-d^2\right ) \cos ^2(e+f x)\right )}{(d+c \cos (e+f x))^{7/2}} \, dx}{7 c \left (c^2-d^2\right ) \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\\ &=\frac {2 d^2 (b+a \cos (e+f x))^2 \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{7 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3 \sqrt {c+d \sec (e+f x)}}-\frac {2 d \left (14 b c^3-19 a c^2 d-2 b c d^2+7 a d^3\right ) (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{35 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^2 \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 {\sqrt {b+a \cos (e+f x)} \left (\frac {1}{4} \left (3 a^2 d^2 \left (19 c^2-7 d^2\right )-16 a b c d \left (7 c^2-d^2\right )+5 b^2 \left (7 c^4+5 c^2 d^2\right )\right )-\frac {1}{2} \left (3 b^2 c d \left (7 c^2-d^2\right )+5 a^2 \left (7 c^3 d-c d^3\right )-a b \left (35 c^4+6 c^2 d^2+7 d^4\right )\right ) \cos (e+f x)+\frac {35}{4} a^2 \left (c^2-d^2\right )^2 \cos ^2(e+f x)\right )}{(d+c \cos (e+f x))^{5/2}} \, dx}{35 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^2 (b+a \cos (e+f x))^2 \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{7 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3 \sqrt {c+d \sec (e+f x)}}-\frac {2 d \left (14 b c^3-19 a c^2 d-2 b c d^2+7 a d^3\right ) (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{35 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}-\frac {2 \left (2 a b c d \left (91 c^4-2 c^2 d^2+7 d^4\right )-a^2 d^2 \left (162 c^4-101 c^2 d^2+35 d^4\right )-b^2 \left (35 c^6+67 c^4 d^2-6 c^2 d^4\right )\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{105 c^3 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}+\frac {\left (8 \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {\frac {1}{8} \left (-3 b^3 c^3 d \left (77 c^2+19 d^2\right )+a b^2 c^2 \left (245 c^4+439 c^2 d^2-12 d^4\right )-a^2 b c d \left (497 c^4-73 c^2 d^2+56 d^4\right )+a^3 \left (162 c^4 d^2-101 c^2 d^4+35 d^6\right )\right )+\frac {1}{8} \left (b^3 c^2 \left (35 c^4+67 c^2 d^2-6 d^4\right )-a b^2 c d \left (413 c^4+53 c^2 d^2+14 d^4\right )-a^3 \left (315 c^5 d-69 c^3 d^3+42 c d^5\right )+a^2 b \left (315 c^6+219 c^4 d^2+208 c^2 d^4-70 d^6\right )\right ) \cos (e+f x)+\frac {105}{8} a^3 \left (c^2-d^2\right )^3 \cos ^2(e+f x)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}} \, dx}{105 c^3 \left (c^2-d^2\right )^3 \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\\ &=\frac {2 d^2 (b+a \cos (e+f x))^2 \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{7 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3 \sqrt {c+d \sec (e+f x)}}-\frac {2 d \left (14 b c^3-19 a c^2 d-2 b c d^2+7 a d^3\right ) (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{35 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}-\frac {2 \left (2 a b c d \left (91 c^4-2 c^2 d^2+7 d^4\right )-a^2 d^2 \left (162 c^4-101 c^2 d^2+35 d^4\right )-b^2 \left (35 c^6+67 c^4 d^2-6 c^2 d^4\right )\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{105 c^3 \left (c^2-d^2\right )^3 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^5 \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {\left (8 \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {-\frac {105}{8} a^3 d^2 \left (c^2-d^2\right )^3+\frac {1}{8} c^2 \left (-3 b^3 c^3 d \left (77 c^2+19 d^2\right )+a b^2 c^2 \left (245 c^4+439 c^2 d^2-12 d^4\right )-a^2 b c d \left (497 c^4-73 c^2 d^2+56 d^4\right )+a^3 \left (162 c^4 d^2-101 c^2 d^4+35 d^6\right )\right )+c \left (-\frac {105}{4} a^3 d \left (c^2-d^2\right )^3+\frac {1}{8} c \left (b^3 c^2 \left (35 c^4+67 c^2 d^2-6 d^4\right )-a b^2 c d \left (413 c^4+53 c^2 d^2+14 d^4\right )-a^3 \left (315 c^5 d-69 c^3 d^3+42 c d^5\right )+a^2 b \left (315 c^6+219 c^4 d^2+208 c^2 d^4-70 d^6\right )\right )\right ) \cos (e+f x)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}} \, dx}{105 c^5 \left (c^2-d^2\right )^3 \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^5 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 d^2 (b+a \cos (e+f x))^2 \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{7 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3 \sqrt {c+d \sec (e+f x)}}-\frac {2 d \left (14 b c^3-19 a c^2 d-2 b c d^2+7 a d^3\right ) (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{35 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}-\frac {2 \left (2 a b c d \left (91 c^4-2 c^2 d^2+7 d^4\right )-a^2 d^2 \left (162 c^4-101 c^2 d^2+35 d^4\right )-b^2 \left (35 c^6+67 c^4 d^2-6 c^2 d^4\right )\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{105 c^3 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}+\frac {\left (\left (b^3 c^4 \left (35 c^4+231 c^3 d+67 c^2 d^2+57 c d^3-6 d^4\right )-a b^2 c^3 \left (245 c^5+413 c^4 d+439 c^3 d^2+53 c^2 d^3-12 c d^4+14 d^5\right )+a^2 b c^2 \left (315 c^6+497 c^5 d+219 c^4 d^2-73 c^3 d^3+208 c^2 d^4+56 c d^5-70 d^6\right )-a^3 d \left (525 c^7+57 c^6 d-699 c^5 d^2+214 c^4 d^3+672 c^3 d^4-280 c^2 d^5-210 c d^6+105 d^7\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}{105 c^5 (c-d) \left (c^2-d^2\right )^3 \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {\left (\left (2 b^3 c^3 d \left (133 c^4+62 c^2 d^2-3 d^4\right )+2 a^2 b c d \left (406 c^6+73 c^4 d^2+132 c^2 d^4-35 d^6\right )-a b^2 c^2 \left (245 c^6+852 c^4 d^2+41 c^2 d^4+14 d^6\right )-a^3 \left (582 c^6 d^2-485 c^4 d^4+392 c^2 d^6-105 d^8\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}{105 c^4 (c-d) \left (c^2-d^2\right )^3 \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\\ &=\frac {2 (a-b) \sqrt {a+b} \left (2 b^3 c^3 d \left (133 c^4+62 c^2 d^2-3 d^4\right )+2 a^2 b c d \left (406 c^6+73 c^4 d^2+132 c^2 d^4-35 d^6\right )-a b^2 c^2 \left (245 c^6+852 c^4 d^2+41 c^2 d^4+14 d^6\right )-a^3 \left (582 c^6 d^2-485 c^4 d^4+392 c^2 d^6-105 d^8\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)}}{105 c^4 (c-d)^4 (c+d)^{7/2} (b c-a d)^2 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 (35 c^4+231 c^3 d+67 c^2 d^2+57 c d^3-6 d^4\right )-a b^2 c^3 \left (245 c^5+413 c^4 d+439 c^3 d^2+53 c^2 d^3-12 c d^4+14 d^5\right )+a^2 b c^2 \left (315 c^6+497 c^5 d+219 c^4 d^2-73 c^3 d^3+208 c^2 d^4+56 c d^5-70 d^6\right )-a^3 d \left (525 c^7+57 c^6 d-699 c^5 d^2+214 c^4 d^3+672 c^3 d^4-280 c^2 d^5-210 c d^6+105 d^7\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)}}{105 c^5 (c-d)^4 (c+d)^{7/2} (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^5 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 d^2 (b+a \cos (e+f x))^2 \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{7 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3 \sqrt {c+d \sec (e+f x)}}-\frac {2 d \left (14 b c^3-19 a c^2 d-2 b c d^2+7 a d^3\right ) (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{35 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}-\frac {2 \left (2 a b c d \left (91 c^4-2 c^2 d^2+7 d^4\right )-a^2 d^2 \left (162 c^4-101 c^2 d^2+35 d^4\right )-b^2 \left (35 c^6+67 c^4 d^2-6 c^2 d^4\right )\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{105 c^3 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}\\ \end {align*}

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Mathematica [B]  time = 8.29, size = 2979, normalized size = 2.09 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^(9/2),x]

[Out]

((d + c*Cos[e + f*x])^5*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^(5/2)*((2*(b^2*c^2*d^2*Sin[e + f*x] - 2*a*b*c*d^3*
Sin[e + f*x] + a^2*d^4*Sin[e + f*x]))/(7*c^3*(c^2 - d^2)*(d + c*Cos[e + f*x])^4) + (2*(-14*b^2*c^4*d*Sin[e + f
*x] + 43*a*b*c^3*d^2*Sin[e + f*x] - 29*a^2*c^2*d^3*Sin[e + f*x] + 2*b^2*c^2*d^3*Sin[e + f*x] - 19*a*b*c*d^4*Si
n[e + f*x] + 17*a^2*d^5*Sin[e + f*x]))/(35*c^3*(c^2 - d^2)^2*(d + c*Cos[e + f*x])^3) + (2*(35*b^2*c^6*Sin[e +
f*x] - 224*a*b*c^5*d*Sin[e + f*x] + 234*a^2*c^4*d^2*Sin[e + f*x] + 67*b^2*c^4*d^2*Sin[e + f*x] + 52*a*b*c^3*d^
3*Sin[e + f*x] - 209*a^2*c^2*d^4*Sin[e + f*x] - 6*b^2*c^2*d^4*Sin[e + f*x] - 20*a*b*c*d^5*Sin[e + f*x] + 71*a^
2*d^6*Sin[e + f*x]))/(105*c^3*(c^2 - d^2)^3*(d + c*Cos[e + f*x])^2) + (2*(245*a*b^2*c^8*Sin[e + f*x] - 812*a^2
*b*c^7*d*Sin[e + f*x] - 266*b^3*c^7*d*Sin[e + f*x] + 582*a^3*c^6*d^2*Sin[e + f*x] + 852*a*b^2*c^6*d^2*Sin[e +
f*x] - 146*a^2*b*c^5*d^3*Sin[e + f*x] - 124*b^3*c^5*d^3*Sin[e + f*x] - 485*a^3*c^4*d^4*Sin[e + f*x] + 41*a*b^2
*c^4*d^4*Sin[e + f*x] - 264*a^2*b*c^3*d^5*Sin[e + f*x] + 6*b^3*c^3*d^5*Sin[e + f*x] + 392*a^3*c^2*d^6*Sin[e +
f*x] + 14*a*b^2*c^2*d^6*Sin[e + f*x] + 70*a^2*b*c*d^7*Sin[e + f*x] - 105*a^3*d^8*Sin[e + f*x]))/(105*c^3*(b*c
- a*d)*(c^2 - d^2)^4*(d + c*Cos[e + f*x]))))/(f*(b + a*Cos[e + f*x])^2*(c + d*Sec[e + f*x])^(9/2)) + ((d + c*C
os[e + f*x])^(9/2)*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^(5/2)*((4*(b*c - a*d)*(-70*a^2*b^2*c^8 - 35*b^4*c^8 - 7
7*a^3*b*c^7*d + 427*a*b^3*c^7*d + 162*a^4*c^6*d^2 - 522*a^2*b^2*c^6*d^2 - 298*b^4*c^6*d^2 + 348*a^3*b*c^5*d^3
+ 666*a*b^3*c^5*d^3 - 263*a^4*c^4*d^4 - 586*a^2*b^2*c^4*d^4 - 51*b^4*c^4*d^4 + 127*a^3*b*c^3*d^5 + 59*a*b^3*c^
3*d^5 + 136*a^4*c^2*d^6 + 26*a^2*b^2*c^2*d^6 - 14*a^3*b*c*d^7 - 35*a^4*d^8)*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)*Sqr
t[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) + 4*(b*c - a*d)*(-105*a^3*b*c^8 + 245*a*b^3*c^8 + 105*a^4*c^7*
d - 567*a^2*b^2*c^7*d - 266*b^4*c^7*d + 190*a^3*b*c^6*d^2 + 586*a*b^3*c^6*d^2 + 162*a^4*c^5*d^3 + 706*a^2*b^2*
c^5*d^3 - 124*b^4*c^5*d^3 - 1261*a^3*b*c^4*d^4 - 83*a*b^3*c^4*d^4 + 145*a^4*c^3*d^5 - 223*a^2*b^2*c^3*d^5 + 6*
b^4*c^3*d^5 + 548*a^3*b*c^2*d^6 + 20*a*b^3*c^2*d^6 - 28*a^4*c*d^7 + 84*a^2*b^2*c*d^7 - 140*a^3*b*d^8)*((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]]) - (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]*EllipticPi[(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*(245*a^2*b^2*c^8 - 812*a^3*b*c^7*d
- 266*a*b^3*c^7*d + 582*a^4*c^6*d^2 + 852*a^2*b^2*c^6*d^2 - 146*a^3*b*c^5*d^3 - 124*a*b^3*c^5*d^3 - 485*a^4*c
^4*d^4 + 41*a^2*b^2*c^4*d^4 - 264*a^3*b*c^3*d^5 + 6*a*b^3*c^3*d^5 + 392*a^4*c^2*d^6 + 14*a^2*b^2*c^2*d^6 + 70*
a^3*b*c*d^7 - 105*a^4*d^8)*((Sqrt[(-a + b)/(a + b)]*(a + b)*Cos[(e + f*x)/2]*Sqrt[d + c*Cos[e + f*x]]*Elliptic
E[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]*Ellip
ticF[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]*EllipticPi[(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]]))))/(105*c^3*(c - d)^4*(c + d)^
4*(-(b*c) + a*d)*f*(b + a*Cos[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^(9/2))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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))^(9/2),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {9}{2}}}\,{d x} \]

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))^(9/2),x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e) + a)^(5/2)/(d*sec(f*x + e) + c)^(9/2), x)

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maple [B]  time = 4.94, size = 75468, normalized size = 52.85 \[ \text {output too large to display} \]

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))^(9/2),x)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {9}{2}}}\,{d x} \]

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))^(9/2),x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e) + a)^(5/2)/(d*sec(f*x + e) + c)^(9/2), x)

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]

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))^(9/2),x)

[Out]

\text{Hanged}

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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))**(9/2),x)

[Out]

Timed out

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