∫ (a+b sec(e+fx))3∕2 ____________________________

3.211 \(\int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=744 \[ -\frac {2 \sqrt {a+b} (b c-a (2 c-d)) \csc (e+f x) \sqrt {a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f (c-d) \sqrt {c+d} \sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}-\frac {2 a \sqrt {a+b} \csc (e+f x) \sqrt {a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \Pi \left (\frac {(a+b) c}{a (c+d)};\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f \sqrt {c+d} \sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}-\frac {2 (a-b) \sqrt {a+b} \csc (e+f x) \sqrt {a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} E\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c f (c-d) \sqrt {c+d} \sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}} \]

[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)

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Rubi [A] time = 1.11, antiderivative size = 744, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {3942, 2798, 2811, 2998, 2818, 2996} \[ -\frac {2 \sqrt {a+b} (b c-a (2 c-d)) \csc (e+f x) \sqrt {a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f (c-d) \sqrt {c+d} \sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}-\frac {2 a \sqrt {a+b} \csc (e+f x) \sqrt {a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \Pi \left (\frac {(a+b) c}{a (c+d)};\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f \sqrt {c+d} \sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}-\frac {2 (a-b) \sqrt {a+b} \csc (e+f x) \sqrt {a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} E\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c f (c-d) \sqrt {c+d} \sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}} \]

Antiderivative was successfully veriﬁed.

[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 2798

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 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 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))^{3/2}}{(c+d \sec (e+f x))^{3/2}} \, dx &=\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) \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^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 (\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 (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) 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)}}{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) \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^2 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\\ \end {align*}

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Mathematica [B] time = 9.72, size = 1750, normalized size = 2.35 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[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))

________________________________________________________________________________________

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

[Out]

Timed out

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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]

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))^(3/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [B] time = 2.10, size = 4298, normalized size = 5.78 \[ \text {output too large to display} \]

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))^(3/2),x)

[Out]

2/f*(b^2*c^2*((a-b)/(a+b))^(1/2)-sin(f*x+e)*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*cos(f*x+e))/(1

+cos(f*x+e))/(c+d))^(1/2)*EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(

1/2))*b^2*c^2-2*sin(f*x+e)*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*x+e))/(c+d

))^(1/2)*EllipticPi((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),-(a+b)/(a-b),((c-d)/(c+d))^(1/2)/((a-b)/(a+

b))^(1/2))*a^2*c^2+2*sin(f*x+e)*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*x+e))

/(c+d))^(1/2)*EllipticPi((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),-(a+b)/(a-b),((c-d)/(c+d))^(1/2)/((a-b

)/(a+b))^(1/2))*a^2*d^2+((a-b)/(a+b))^(1/2)*cos(f*x+e)^2*a*b*c*d-((a-b)/(a+b))^(1/2)*cos(f*x+e)*b^2*c^2-((a-b)

/(a+b))^(1/2)*cos(f*x+e)^2*a^2*d^2+((a-b)/(a+b))^(1/2)*cos(f*x+e)*a^2*d^2+sin(f*x+e)*cos(f*x+e)*((b+a*cos(f*x+

e))/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*EllipticF((-1+cos(f*x+e))*((a-b)

/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a^2*c*d+sin(f*x+e)*((b+a*cos(f*x+e))/(1+cos(f*x+e))/

(a+b))^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f

*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*b^2*c^2-2*sin(f*x+e)*cos(f*x+e)*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))

^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*EllipticPi((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e)

,-(a+b)/(a-b),((c-d)/(c+d))^(1/2)/((a-b)/(a+b))^(1/2))*a^2*c^2+2*sin(f*x+e)*cos(f*x+e)*((b+a*cos(f*x+e))/(1+co

s(f*x+e))/(a+b))^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*EllipticPi((-1+cos(f*x+e))*((a-b)/(a+b))^

(1/2)/sin(f*x+e),-(a+b)/(a-b),((c-d)/(c+d))^(1/2)/((a-b)/(a+b))^(1/2))*a^2*d^2+EllipticF((-1+cos(f*x+e))*((a-b

)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a^2*c*d*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(1/

2)*((d+c*cos(f*x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*sin(f*x+e)-2*EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/si

n(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a*b*c^2*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*cos(f*x+

e))/(1+cos(f*x+e))/(c+d))^(1/2)*sin(f*x+e)+a*b*d^2*((a-b)/(a+b))^(1/2)-b^2*c*d*((a-b)/(a+b))^(1/2)-2*sin(f*x+e

)*cos(f*x+e)*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*Ellip

ticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a*b*c*d+2*sin(f*x+e)*cos(

f*x+e)*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*EllipticE((

-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a*b*c*d-((a-b)/(a+b))^(1/2)*cos

(f*x+e)^2*a*b*c^2-((a-b)/(a+b))^(1/2)*cos(f*x+e)*a^2*c*d+((a-b)/(a+b))^(1/2)*cos(f*x+e)*a*b*c^2-((a-b)/(a+b))^

(1/2)*cos(f*x+e)*a*b*d^2+((a-b)/(a+b))^(1/2)*cos(f*x+e)*b^2*c*d+((a-b)/(a+b))^(1/2)*cos(f*x+e)^2*a^2*c*d-sin(f

*x+e)*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*EllipticE((-

1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a^2*d^2+2*EllipticE((-1+cos(f*x+

e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a*b*c*d*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(

a+b))^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*sin(f*x+e)-2*sin(f*x+e)*cos(f*x+e)*((b+a*cos(f*x+e))

/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*EllipticF((-1+cos(f*x+e))*((a-b)/(a

+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a*b*c^2+sin(f*x+e)*cos(f*x+e)*((b+a*cos(f*x+e))/(1+cos(

f*x+e))/(a+b))^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/

2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*b^2*c*d-sin(f*x+e)*cos(f*x+e)*((b+a*cos(f*x+e))/(1+cos(f*x+e))/

(a+b))^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f

*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a^2*c*d+sin(f*x+e)*cos(f*x+e)*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(

1/2)*((d+c*cos(f*x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((

a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a*b*c^2+sin(f*x+e)*cos(f*x+e)*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(1/2)*((d

+c*cos(f*x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-

d)/(a-b)/(c+d))^(1/2))*a*b*d^2-sin(f*x+e)*cos(f*x+e)*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*cos(f

*x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)

/(c+d))^(1/2))*b^2*c*d-2*sin(f*x+e)*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*x

+e))/(c+d))^(1/2)*EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a*

b*c*d-a*b*c*d*((a-b)/(a+b))^(1/2)+sin(f*x+e)*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*cos(f*x+e))/(

1+cos(f*x+e))/(c+d))^(1/2)*EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^

(1/2))*a^2*c^2+EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*b^2*c

*d*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*sin(f*x+e)-Elli

pticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a^2*c*d*((b+a*cos(f*x+e)

)/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*sin(f*x+e)+EllipticE((-1+cos(f*x+e

))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a*b*c^2*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a

+b))^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*sin(f*x+e)+EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1

/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a*b*d^2*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*co

s(f*x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*sin(f*x+e)-EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+

b)*(c-d)/(a-b)/(c+d))^(1/2))*b^2*c*d*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*

x+e))/(c+d))^(1/2)*sin(f*x+e)+sin(f*x+e)*cos(f*x+e)*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*cos(f*

x+e))/(1+cos(f*x+e))/(c+d))^(1/2)*EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/

(c+d))^(1/2))*a^2*c^2+sin(f*x+e)*cos(f*x+e)*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*cos(f*x+e))/(1

+cos(f*x+e))/(c+d))^(1/2)*EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(

1/2))*b^2*c^2-sin(f*x+e)*cos(f*x+e)*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*x

+e))/(c+d))^(1/2)*EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a^

2*d^2-sin(f*x+e)*cos(f*x+e)*((b+a*cos(f*x+e))/(1+cos(f*x+e))/(a+b))^(1/2)*((d+c*cos(f*x+e))/(1+cos(f*x+e))/(c+

d))^(1/2)*EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*b^2*c^2)*c

os(f*x+e)*((b+a*cos(f*x+e))/cos(f*x+e))^(1/2)*((d+c*cos(f*x+e))/cos(f*x+e))^(1/2)/sin(f*x+e)/(d+c*cos(f*x+e))/

(b+a*cos(f*x+e))/c/(c-d)/((a-b)/(a+b))^(1/2)/(c+d)

________________________________________________________________________________________

maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]

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))^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]

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))^(3/2),x)

[Out]

int((a + b/cos(e + f*x))^(3/2)/(c + d/cos(e + f*x))^(3/2), x)

________________________________________________________________________________________

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

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))**(3/2),x)

[Out]

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

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