3.3.12 \(\int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{5/2}} \, dx\) [212]

Optimal. Leaf size=919 \[ -\frac {2 (a-b) \sqrt {a+b} \left (3 b c^3-7 a c^2 d+b c d^2+3 a d^3\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left (\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{3 c^2 (c-d)^2 (c+d)^{3/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 \sqrt {a+b} \left (b^2 c^3 (3 c+d)-2 a b c^2 \left (3 c^2+2 c d-d^2\right )+a^2 d \left (9 c^3-2 c^2 d-6 c d^2+3 d^3\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left (\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{3 c^3 (c-d)^2 (c+d)^{3/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 a \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left (\frac {(a+b) c}{a (c+d)};\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c^3 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 d (b c-a d) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}} \]

[Out]

-2/3*d*(-a*d+b*c)*sin(f*x+e)*(a+b*sec(f*x+e))^(1/2)/c/(c^2-d^2)/f/(d+c*cos(f*x+e))/(c+d*sec(f*x+e))^(1/2)-2/3*
(a-b)*(-7*a*c^2*d+3*a*d^3+3*b*c^3+b*c*d^2)*(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-co
s(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^2/(c-d)^2/(c+d)^(3/2)/(-a*d+b*c)/f/(b+a*cos(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2)-2/3*(b^2*c^3*(3*
c+d)-2*a*b*c^2*(3*c^2+2*c*d-d^2)+a^2*d*(9*c^3-2*c^2*d-6*c*d^2+3*d^3))*(d+c*cos(f*x+e))^(3/2)*csc(f*x+e)*Ellipt
icF((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)^2/(c+d)^(3/2)/(-a*d+b*c)/f/(b+a*cos(f*x+e))^(1/2)/(c+d*sec(f*x
+e))^(1/2)-2*a*(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^3/f/(c+d)^(1/2)/(b+a*cos(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.45, antiderivative size = 919, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {4027, 3068, 3132, 2890, 3077, 2897, 3075} \begin {gather*} -\frac {2 (a-b) \sqrt {a+b} \left (3 b c^3-7 a d c^2+b d^2 c+3 a d^3\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))}} \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)} (d+c \cos (e+f x))^{3/2}}{3 c^2 (c-d)^2 (c+d)^{3/2} (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^2 (3 c+d) c^3-2 a b \left (3 c^2+2 d c-d^2\right ) c^2+a^2 d \left (9 c^3-2 d c^2-6 d^2 c+3 d^3\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))}} \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)} (d+c \cos (e+f x))^{3/2}}{3 c^3 (c-d)^2 (c+d)^{3/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 a \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))}} \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)} (d+c \cos (e+f x))^{3/2}}{c^3 \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) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f \sqrt {c+d \sec (e+f x)} (d+c \cos (e+f x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^(5/2),x]

[Out]

(-2*(a - b)*Sqrt[a + b]*(3*b*c^3 - 7*a*c^2*d + b*c*d^2 + 3*a*d^3)*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*C
os[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]])/(3*c^2*(c - d)^2*(c + d)^(3/
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^2*c^3*(3*c + d) - 2*a*
b*c^2*(3*c^2 + 2*c*d - d^2) + a^2*d*(9*c^3 - 2*c^2*d - 6*c*d^2 + 3*d^3))*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]*Sq
rt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(3*c^3*(c - d)^2*(c +
d)^(3/2)*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*a*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^3*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*d*(b*c -
a*d)*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(3*c*(c^2 - d^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 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))^{3/2}}{(c+d \sec (e+f x))^{5/2}} \, 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))^{3/2}}{(d+c \cos (e+f x))^{5/2}} \, dx}{\sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\\ &=-\frac {2 d (b c-a d) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x)) \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 {\frac {1}{2} (b c-a d) (3 b c-a d)-\frac {1}{2} \left (3 a^2 c d+b^2 c d-2 a b \left (3 c^2-d^2\right )\right ) \cos (e+f x)+\frac {3}{2} a^2 \left (c^2-d^2\right ) \cos ^2(e+f x)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}} \, dx}{3 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) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}+\frac {\left (a^2 \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^3 \sqrt {b+a \cos (e+f x)} \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 {\frac {1}{2} c^2 (b c-a d) (3 b c-a d)-\frac {3}{2} a^2 d^2 \left (c^2-d^2\right )+c \left (-3 a^2 d \left (c^2-d^2\right )+\frac {1}{2} c \left (-3 a^2 c d-b^2 c d+2 a b \left (3 c^2-d^2\right )\right )\right ) \cos (e+f x)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}} \, dx}{3 c^3 \left (c^2-d^2\right ) \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\\ &=-\frac {2 a \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^3 \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) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}+\frac {\left ((b c-a d) \left (3 b c^3-7 a c^2 d+b c d^2+3 a d^3\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}{3 c^2 (c-d) \left (c^2-d^2\right ) \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {\left (\left (b^2 c^3 (3 c+d)-2 a b c^2 \left (3 c^2+2 c d-d^2\right )+a^2 d \left (9 c^3-2 c^2 d-6 c d^2+3 d^3\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}{3 c^3 (c-d) \left (c^2-d^2\right ) \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\\ &=-\frac {2 (a-b) \sqrt {a+b} \left (3 b c^3-7 a c^2 d+b c d^2+3 a d^3\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)}}{3 c^2 (c-d)^2 (c+d)^{3/2} (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^2 c^3 (3 c+d)-2 a b c^2 \left (3 c^2+2 c d-d^2\right )+a^2 d \left (9 c^3-2 c^2 d-6 c d^2+3 d^3\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)}}{3 c^3 (c-d)^2 (c+d)^{3/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 a \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^3 \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) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1960\) vs. \(2(919)=1838\).
time = 6.71, size = 1960, normalized size = 2.13 \begin {gather*} \frac {(d+c \cos (e+f x))^3 \sec (e+f x) (a+b \sec (e+f x))^{3/2} \left (\frac {2 \left (-b c d \sin (e+f x)+a d^2 \sin (e+f x)\right )}{3 c \left (c^2-d^2\right ) (d+c \cos (e+f x))^2}+\frac {2 \left (3 b c^3 \sin (e+f x)-7 a c^2 d \sin (e+f x)+b c d^2 \sin (e+f x)+3 a d^3 \sin (e+f x)\right )}{3 c \left (c^2-d^2\right )^2 (d+c \cos (e+f x))}\right )}{f (b+a \cos (e+f x)) (c+d \sec (e+f x))^{5/2}}+\frac {(d+c \cos (e+f x))^{5/2} \sec (e+f x) (a+b \sec (e+f x))^{3/2} \left (\frac {4 (b c-a d) \left (3 a b c^3+a^2 c^2 d-4 b^2 c^2 d+a b c d^2-a^2 d^3\right ) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} (e+f x)\right )}{c-d}} \sqrt {\frac {(c+d) (b+a \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \csc (e+f x) F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}}}{\sqrt {2}}\right )|\frac {2 (b c-a d)}{(a+b) (c-d)}\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{(a+b) (c+d) \sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}+4 (b c-a d) \left (3 a^2 c^3-3 b^2 c^3+4 a b c^2 d+a^2 c d^2-b^2 c d^2-4 a b d^3\right ) \left (\frac {\sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} (e+f x)\right )}{c-d}} \sqrt {\frac {(c+d) (b+a \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \csc (e+f x) F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}}}{\sqrt {2}}\right )|\frac {2 (b c-a d)}{(a+b) (c-d)}\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{(a+b) (c+d) \sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}-\frac {\sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} (e+f x)\right )}{c-d}} \sqrt {\frac {(c+d) (b+a \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \csc (e+f x) \Pi \left (\frac {b c-a d}{(a+b) c};\text {ArcSin}\left (\frac {\sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}}}{\sqrt {2}}\right )|\frac {2 (b c-a d)}{(a+b) (c-d)}\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{(a+b) c \sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}\right )+2 \left (-3 a b c^3+7 a^2 c^2 d-a b c d^2-3 a^2 d^3\right ) \left (\frac {\sqrt {\frac {-a+b}{a+b}} (a+b) \cos \left (\frac {1}{2} (e+f x)\right ) \sqrt {d+c \cos (e+f x)} E\left (\text {ArcSin}\left (\frac {\sqrt {\frac {-a+b}{a+b}} \sin \left (\frac {1}{2} (e+f x)\right )}{\sqrt {\frac {b+a \cos (e+f x)}{a+b}}}\right )|\frac {2 (b c-a d)}{(-a+b) (c+d)}\right )}{a c \sqrt {\frac {(a+b) \cos ^2\left (\frac {1}{2} (e+f x)\right )}{b+a \cos (e+f x)}} \sqrt {b+a \cos (e+f x)} \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \sqrt {\frac {(a+b) (d+c \cos (e+f x))}{(c+d) (b+a \cos (e+f x))}}}-\frac {2 (b c-a d) \left (\frac {(b c+(a+b) d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} (e+f x)\right )}{c-d}} \sqrt {\frac {(c+d) (b+a \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \csc (e+f x) F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}}}{\sqrt {2}}\right )|\frac {2 (b c-a d)}{(a+b) (c-d)}\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{(a+b) (c+d) \sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}-\frac {(b c+a d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} (e+f x)\right )}{c-d}} \sqrt {\frac {(c+d) (b+a \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \csc (e+f x) \Pi \left (\frac {b c-a d}{(a+b) c};\text {ArcSin}\left (\frac {\sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}}}{\sqrt {2}}\right )|\frac {2 (b c-a d)}{(a+b) (c-d)}\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{(a+b) c \sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}\right )}{a c}+\frac {\sqrt {d+c \cos (e+f x)} \sin (e+f x)}{c \sqrt {b+a \cos (e+f x)}}\right )\right )}{3 c (c-d)^2 (c+d)^2 f (b+a \cos (e+f x))^{3/2} (c+d \sec (e+f x))^{5/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^(5/2),x]

[Out]

((d + c*Cos[e + f*x])^3*Sec[e + f*x]*(a + b*Sec[e + f*x])^(3/2)*((2*(-(b*c*d*Sin[e + f*x]) + a*d^2*Sin[e + f*x
]))/(3*c*(c^2 - d^2)*(d + c*Cos[e + f*x])^2) + (2*(3*b*c^3*Sin[e + f*x] - 7*a*c^2*d*Sin[e + f*x] + b*c*d^2*Sin
[e + f*x] + 3*a*d^3*Sin[e + f*x]))/(3*c*(c^2 - d^2)^2*(d + c*Cos[e + f*x]))))/(f*(b + a*Cos[e + f*x])*(c + d*S
ec[e + f*x])^(5/2)) + ((d + c*Cos[e + f*x])^(5/2)*Sec[e + f*x]*(a + b*Sec[e + f*x])^(3/2)*((4*(b*c - a*d)*(3*a
*b*c^3 + a^2*c^2*d - 4*b^2*c^2*d + a*b*c*d^2 - a^2*d^3)*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)*(3*a^2*c^3 - 3*b^2*c^3 + 4*a*b*c^2*d + a^2*c*d^2 - b^2*c*d^2 - 4*a
*b*d^3)*((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)*C
ot[(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*(-3*a*b*c^3 + 7*a^
2*c^2*d - a*b*c*d^2 - 3*a^2*d^3)*((Sqrt[(-a + b)/(a + b)]*(a + b)*Cos[(e + f*x)/2]*Sqrt[d + c*Cos[e + f*x]]*El
lipticE[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]]*Sqr
t[(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]*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]]))))/(3*c*(c - d)^2*(c + d
)^2*f*(b + a*Cos[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(13059\) vs. \(2(840)=1680\).
time = 2.52, size = 13060, normalized size = 14.21

method	result	size
default	\(\text {Expression too large to display}\)	\(13060\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sec(f*x+e))^(3/2)/(c+d*sec(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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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))^(3/2)/(c+d*sec(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e) + a)^(3/2)/(d*sec(f*x + e) + c)^(5/2), x)

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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))^(3/2)/(c+d*sec(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\left (c + d \sec {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))**(3/2)/(c+d*sec(f*x+e))**(5/2),x)

[Out]

Integral((a + b*sec(e + f*x))**(3/2)/(c + d*sec(e + f*x))**(5/2), x)

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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))^(3/2)/(c+d*sec(f*x+e))^(5/2),x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e) + a)^(3/2)/(d*sec(f*x + e) + c)^(5/2), x)

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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))^(3/2)/(c + d/cos(e + f*x))^(5/2),x)

[Out]

\text{Hanged}

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