\(\int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{3/2}} \, dx\) [211]
Optimal result
Integrand size = 29, antiderivative size = 744 \[
\int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{3/2}} \, dx=-\frac {2 (a-b) \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c (c-d) \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 \sqrt {a+b} (b c-a (2 c-d)) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c^2 (c-d) \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 a \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c^2 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}
\]
[Out]
-2*(a-b)*(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/(c-d)/f/(c+d)^(1/2)/(
b+a*cos(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2)-2*(b*c-a*(2*c-d))*(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^2/(c-d)/f/(c+d)^(1/2)/(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^2/f/(c+d)^(1/2)/(b
+a*cos(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2)
Rubi [A] (verified)
Time = 1.35 (sec) , antiderivative size = 744, normalized size of antiderivative = 1.00, number of
steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {4027, 2877, 2890, 3077, 2897,
3075} \[
\int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{3/2}} \, dx=-\frac {2 \sqrt {a+b} (b c-a (2 c-d)) \csc (e+f x) \sqrt {a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f (c-d) \sqrt {c+d} \sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}-\frac {2 a \sqrt {a+b} \csc (e+f x) \sqrt {a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f \sqrt {c+d} \sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}-\frac {2 (a-b) \sqrt {a+b} \csc (e+f x) \sqrt {a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c f (c-d) \sqrt {c+d} \sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}
\]
[In]
Int[(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^(3/2),x]
[Out]
(-2*(a - b)*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]*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]])/(c*(c - d)*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f
*x]]) - (2*Sqrt[a + b]*(b*c - a*(2*c - d))*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]])/(c^2*(c - d)*Sqrt[c + 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)*C
sc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqr
t[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^2*Sqrt[c + d]*f*Sqr
t[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])
Rule 2877
Int[((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)/((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2), x_Symbol] :> D
ist[d^2/b^2, Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[(b*c - a*d)/b^2, Int[Simp[b*
c + a*d + 2*b*d*Sin[e + f*x], 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}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
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 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 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*}
\text {integral}& = \frac {\left (\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}}{(d+c \cos (e+f x))^{3/2}} \, dx}{\sqrt {b+a \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^2 \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {\left ((b c-a d) \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {b c+a d+2 a c \cos (e+f x)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}} \, dx}{c^2 \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) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\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^2 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {\left ((b c-a (2 c-d)) (b c-a d) \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}{c^2 (c-d) \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {\left ((b c-a d)^2 \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}{c (c-d) \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}} \\ & = -\frac {2 (a-b) \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) E\left (\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 (c-d) \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 \sqrt {a+b} (b c-a (2 c-d)) \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) \operatorname {EllipticF}\left (\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^2 (c-d) \sqrt {c+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) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\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^2 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}} \\
\end{align*}
Mathematica [B] (warning: unable to verify)
Leaf count is larger than twice the leaf count of optimal. \(1750\) vs. \(2(744)=1488\).
Time = 19.22 (sec) , antiderivative size = 1750, normalized size of antiderivative = 2.35
\[
\int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{3/2}} \, dx=\frac {2 (d+c \cos (e+f x)) (a+b \sec (e+f x))^{3/2} (-b c \sin (e+f x)+a d \sin (e+f x))}{\left (-c^2+d^2\right ) f (b+a \cos (e+f x)) (c+d \sec (e+f x))^{3/2}}+\frac {(d+c \cos (e+f x))^{3/2} (a+b \sec (e+f x))^{3/2} \left (\frac {4 (b c-a d) \left (a b c-b^2 d\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) \operatorname {EllipticF}\left (\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 \left (a^2 c-b^2 c\right ) (b c-a d) \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) \operatorname {EllipticF}\left (\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) \operatorname {EllipticPi}\left (\frac {b c-a d}{(a+b) c},\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 (-a b c+a^2 d\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 (\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) \operatorname {EllipticF}\left (\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) \operatorname {EllipticPi}\left (\frac {b c-a d}{(a+b) c},\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 )}{(c-d) (c+d) f (b+a \cos (e+f x))^{3/2} (c+d \sec (e+f x))^{3/2}}
\]
[In]
Integrate[(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^(3/2),x]
[Out]
(2*(d + c*Cos[e + f*x])*(a + b*Sec[e + f*x])^(3/2)*(-(b*c*Sin[e + f*x]) + a*d*Sin[e + f*x]))/((-c^2 + d^2)*f*(
b + a*Cos[e + f*x])*(c + d*Sec[e + f*x])^(3/2)) + ((d + c*Cos[e + f*x])^(3/2)*(a + b*Sec[e + f*x])^(3/2)*((4*(
b*c - a*d)*(a*b*c - b^2*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]]) +
4*(a^2*c - b^2*c)*(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]*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*(-(a*b*c) + a^2*d)*((Sqrt[(-a + b)/(a + b)]*(a + b)*Cos[(e + f*x)/2]*Sqrt[d + c*Cos[e + f*x]]*EllipticE[A
rcSin[(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]*Elliptic
F[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]]))))/((c - d)*(c + d)*f*(b + a*Cos[
e + f*x])^(3/2)*(c + d*Sec[e + f*x])^(3/2))
Maple [B] (warning: unable to verify)
Leaf count of result is larger than twice the leaf count of optimal. \(3568\) vs. \(2(675)=1350\).
Time = 13.97 (sec) , antiderivative size = 3569, normalized size of antiderivative =
4.80
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method | result | size |
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default |
\(\text {Expression too large to display}\) |
\(3569\) |
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[In]
int((a+b*sec(f*x+e))^(3/2)/(c+d*sec(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
[Out]
-2/f/((a-b)/(a+b))^(1/2)/(c-d)/(c+d)/c*((a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/
((1-cos(f*x+e))^2*csc(f*x+e)^2-1))^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)*((c*(1-cos(f*x+e))^2*csc(f*x+e)^2-d
*(1-cos(f*x+e))^2*csc(f*x+e)^2-c-d)/((1-cos(f*x+e))^2*csc(f*x+e)^2-1))^(1/2)*(-2*((a-b)/(a+b))^(1/2)*a*b*c*d*(
-cot(f*x+e)+csc(f*x+e))+(-(a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*(
-(c*(1-cos(f*x+e))^2*csc(f*x+e)^2-d*(1-cos(f*x+e))^2*csc(f*x+e)^2-c-d)/(c+d))^(1/2)*EllipticE(((a-b)/(a+b))^(1
/2)*(-cot(f*x+e)+csc(f*x+e)),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a^2*c*d-(-(a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-
cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*(-(c*(1-cos(f*x+e))^2*csc(f*x+e)^2-d*(1-cos(f*x+e))^2*csc(f*x+e)^
2-c-d)/(c+d))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(f*x+e)+csc(f*x+e)),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a*
b*c^2-(-(a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*(-(c*(1-cos(f*x+e))
^2*csc(f*x+e)^2-d*(1-cos(f*x+e))^2*csc(f*x+e)^2-c-d)/(c+d))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(f*x+e)+c
sc(f*x+e)),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a*b*d^2+(-(a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(
f*x+e)^2-a-b)/(a+b))^(1/2)*(-(c*(1-cos(f*x+e))^2*csc(f*x+e)^2-d*(1-cos(f*x+e))^2*csc(f*x+e)^2-c-d)/(c+d))^(1/2
)*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(f*x+e)+csc(f*x+e)),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*b^2*c*d-(-(a*(1-cos(
f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*(-(c*(1-cos(f*x+e))^2*csc(f*x+e)^2-d*
(1-cos(f*x+e))^2*csc(f*x+e)^2-c-d)/(c+d))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(-cot(f*x+e)+csc(f*x+e)),((a+b)*
(c-d)/(a-b)/(c+d))^(1/2))*a^2*c*d+2*(-(a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a
+b))^(1/2)*(-(c*(1-cos(f*x+e))^2*csc(f*x+e)^2-d*(1-cos(f*x+e))^2*csc(f*x+e)^2-c-d)/(c+d))^(1/2)*EllipticF(((a-
b)/(a+b))^(1/2)*(-cot(f*x+e)+csc(f*x+e)),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a*b*c^2-(-(a*(1-cos(f*x+e))^2*csc(f*
x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*(-(c*(1-cos(f*x+e))^2*csc(f*x+e)^2-d*(1-cos(f*x+e))^2
*csc(f*x+e)^2-c-d)/(c+d))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(-cot(f*x+e)+csc(f*x+e)),((a+b)*(c-d)/(a-b)/(c+d
))^(1/2))*b^2*c*d+((a-b)/(a+b))^(1/2)*a^2*c*d*(1-cos(f*x+e))^3*csc(f*x+e)^3-((a-b)/(a+b))^(1/2)*a*b*c^2*(1-cos
(f*x+e))^3*csc(f*x+e)^3+((a-b)/(a+b))^(1/2)*a*b*d^2*(1-cos(f*x+e))^3*csc(f*x+e)^3-((a-b)/(a+b))^(1/2)*b^2*c*d*
(1-cos(f*x+e))^3*csc(f*x+e)^3+(-(a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(
1/2)*(-(c*(1-cos(f*x+e))^2*csc(f*x+e)^2-d*(1-cos(f*x+e))^2*csc(f*x+e)^2-c-d)/(c+d))^(1/2)*EllipticE(((a-b)/(a+
b))^(1/2)*(-cot(f*x+e)+csc(f*x+e)),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*b^2*c^2-((a-b)/(a+b))^(1/2)*a^2*c*d*(-cot(
f*x+e)+csc(f*x+e))+((a-b)/(a+b))^(1/2)*a*b*c^2*(-cot(f*x+e)+csc(f*x+e))+((a-b)/(a+b))^(1/2)*a*b*d^2*(-cot(f*x+
e)+csc(f*x+e))-((a-b)/(a+b))^(1/2)*b^2*c*d*(-cot(f*x+e)+csc(f*x+e))-(-(a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-co
s(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*(-(c*(1-cos(f*x+e))^2*csc(f*x+e)^2-d*(1-cos(f*x+e))^2*csc(f*x+e)^2-
c-d)/(c+d))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(-cot(f*x+e)+csc(f*x+e)),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a^2*
c^2-(-(a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*(-(c*(1-cos(f*x+e))^2
*csc(f*x+e)^2-d*(1-cos(f*x+e))^2*csc(f*x+e)^2-c-d)/(c+d))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(-cot(f*x+e)+csc
(f*x+e)),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*b^2*c^2+2*(-(a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(
f*x+e)^2-a-b)/(a+b))^(1/2)*(-(c*(1-cos(f*x+e))^2*csc(f*x+e)^2-d*(1-cos(f*x+e))^2*csc(f*x+e)^2-c-d)/(c+d))^(1/2
)*EllipticPi(((a-b)/(a+b))^(1/2)*(-cot(f*x+e)+csc(f*x+e)),-(a+b)/(a-b),((c-d)/(c+d))^(1/2)/((a-b)/(a+b))^(1/2)
)*a^2*c^2-2*(-(a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*(-(c*(1-cos(f
*x+e))^2*csc(f*x+e)^2-d*(1-cos(f*x+e))^2*csc(f*x+e)^2-c-d)/(c+d))^(1/2)*EllipticPi(((a-b)/(a+b))^(1/2)*(-cot(f
*x+e)+csc(f*x+e)),-(a+b)/(a-b),((c-d)/(c+d))^(1/2)/((a-b)/(a+b))^(1/2))*a^2*d^2+(-(a*(1-cos(f*x+e))^2*csc(f*x+
e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*(-(c*(1-cos(f*x+e))^2*csc(f*x+e)^2-d*(1-cos(f*x+e))^2*c
sc(f*x+e)^2-c-d)/(c+d))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(f*x+e)+csc(f*x+e)),((a+b)*(c-d)/(a-b)/(c+d))
^(1/2))*a^2*d^2-((a-b)/(a+b))^(1/2)*a^2*d^2*(1-cos(f*x+e))^3*csc(f*x+e)^3+((a-b)/(a+b))^(1/2)*b^2*c^2*(1-cos(f
*x+e))^3*csc(f*x+e)^3+2*(-(a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*(
-(c*(1-cos(f*x+e))^2*csc(f*x+e)^2-d*(1-cos(f*x+e))^2*csc(f*x+e)^2-c-d)/(c+d))^(1/2)*EllipticF(((a-b)/(a+b))^(1
/2)*(-cot(f*x+e)+csc(f*x+e)),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a*b*c*d-2*(-(a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(
1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*(-(c*(1-cos(f*x+e))^2*csc(f*x+e)^2-d*(1-cos(f*x+e))^2*csc(f*x+e
)^2-c-d)/(c+d))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(f*x+e)+csc(f*x+e)),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*
a*b*c*d+((a-b)/(a+b))^(1/2)*a^2*d^2*(-cot(f*x+e)+csc(f*x+e))+((a-b)/(a+b))^(1/2)*b^2*c^2*(-cot(f*x+e)+csc(f*x+
e)))/(a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(c*(1-cos(f*x+e))^2*csc(f*x+e)^2-d*
(1-cos(f*x+e))^2*csc(f*x+e)^2-c-d)
Fricas [F]
\[
\int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{3/2}} \, dx=\int { \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x }
\]
[In]
integrate((a+b*sec(f*x+e))^(3/2)/(c+d*sec(f*x+e))^(3/2),x, algorithm="fricas")
[Out]
integral((b*sec(f*x + e) + a)^(3/2)*sqrt(d*sec(f*x + e) + c)/(d^2*sec(f*x + e)^2 + 2*c*d*sec(f*x + e) + c^2),
x)
Sympy [F]
\[
\int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{3/2}} \, dx=\int \frac {\left (a + b \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\left (c + d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx
\]
[In]
integrate((a+b*sec(f*x+e))**(3/2)/(c+d*sec(f*x+e))**(3/2),x)
[Out]
Integral((a + b*sec(e + f*x))**(3/2)/(c + d*sec(e + f*x))**(3/2), x)
Maxima [F]
\[
\int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{3/2}} \, dx=\int { \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x }
\]
[In]
integrate((a+b*sec(f*x+e))^(3/2)/(c+d*sec(f*x+e))^(3/2),x, algorithm="maxima")
[Out]
integrate((b*sec(f*x + e) + a)^(3/2)/(d*sec(f*x + e) + c)^(3/2), x)
Giac [F]
\[
\int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{3/2}} \, dx=\int { \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x }
\]
[In]
integrate((a+b*sec(f*x+e))^(3/2)/(c+d*sec(f*x+e))^(3/2),x, algorithm="giac")
[Out]
integrate((b*sec(f*x + e) + a)^(3/2)/(d*sec(f*x + e) + c)^(3/2), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{3/2}} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x
\]
[In]
int((a + b/cos(e + f*x))^(3/2)/(c + d/cos(e + f*x))^(3/2),x)
[Out]
int((a + b/cos(e + f*x))^(3/2)/(c + d/cos(e + f*x))^(3/2), x)