3.45 \(\int \frac{1}{\sqrt{g \sin (e+f x)} \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx\)

Optimal. Leaf size=246 \[ -\frac{2 d \tan (e+f x) \sqrt{-\cot ^2(e+f x)} \sqrt{g \sin (e+f x)} \sqrt{\frac{a \csc (e+f x)+b}{a+b}} \Pi \left (\frac{2 c}{c+d};\sin ^{-1}\left (\frac{\sqrt{1-\csc (e+f x)}}{\sqrt{2}}\right )|\frac{2 a}{a+b}\right )}{c f g (c+d) \sqrt{a+b \sin (e+f x)}}-\frac{2 \sqrt{a+b} \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{g} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{g \sin (e+f x)}}\right )|-\frac{a+b}{a-b}\right )}{a c f \sqrt{g}} \]

[Out]

(-2*Sqrt[a + b]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sq
rt[g]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(a*c*f*
Sqrt[g]) - (2*d*Sqrt[-Cot[e + f*x]^2]*Sqrt[(b + a*Csc[e + f*x])/(a + b)]*EllipticPi[(2*c)/(c + d), ArcSin[Sqrt
[1 - Csc[e + f*x]]/Sqrt[2]], (2*a)/(a + b)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/(c*(c + d)*f*g*Sqrt[a + b*Sin[e
+ f*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.523502, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2939, 2816, 2937} \[ -\frac{2 d \tan (e+f x) \sqrt{-\cot ^2(e+f x)} \sqrt{g \sin (e+f x)} \sqrt{\frac{a \csc (e+f x)+b}{a+b}} \Pi \left (\frac{2 c}{c+d};\sin ^{-1}\left (\frac{\sqrt{1-\csc (e+f x)}}{\sqrt{2}}\right )|\frac{2 a}{a+b}\right )}{c f g (c+d) \sqrt{a+b \sin (e+f x)}}-\frac{2 \sqrt{a+b} \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{g} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{g \sin (e+f x)}}\right )|-\frac{a+b}{a-b}\right )}{a c f \sqrt{g}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]

[Out]

(-2*Sqrt[a + b]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sq
rt[g]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(a*c*f*
Sqrt[g]) - (2*d*Sqrt[-Cot[e + f*x]^2]*Sqrt[(b + a*Csc[e + f*x])/(a + b)]*EllipticPi[(2*c)/(c + d), ArcSin[Sqrt
[1 - Csc[e + f*x]]/Sqrt[2]], (2*a)/(a + b)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/(c*(c + d)*f*g*Sqrt[a + b*Sin[e
+ f*x]])

Rule 2939

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

Rule 2816

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

Rule 2937

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{g \sin (e+f x)} \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx &=\frac{\int \frac{1}{\sqrt{g \sin (e+f x)} \sqrt{a+b \sin (e+f x)}} \, dx}{c}-\frac{d \int \frac{\sqrt{g \sin (e+f x)}}{\sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx}{c g}\\ &=-\frac{2 \sqrt{a+b} \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (1+\csc (e+f x))}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{g} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{g \sin (e+f x)}}\right )|-\frac{a+b}{a-b}\right ) \tan (e+f x)}{a c f \sqrt{g}}-\frac{2 d \sqrt{-\cot ^2(e+f x)} \sqrt{\frac{b+a \csc (e+f x)}{a+b}} \Pi \left (\frac{2 c}{c+d};\sin ^{-1}\left (\frac{\sqrt{1-\csc (e+f x)}}{\sqrt{2}}\right )|\frac{2 a}{a+b}\right ) \sqrt{g \sin (e+f x)} \tan (e+f x)}{c (c+d) f g \sqrt{a+b \sin (e+f x)}}\\ \end{align*}

Mathematica [B]  time = 29.9797, size = 4935, normalized size = 20.06 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]

[Out]

(-4*Sqrt[-a^2 + b^2]*Cos[(e + f*x)/2]^4*(-2*(b + Sqrt[-a^2 + b^2])*(b*c - a*d)*Sqrt[-c^2 + d^2]*EllipticF[ArcS
in[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt
[-a^2 + b^2])] - a*d*((a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)
/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2]
)/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b^2])*
(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])
), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b
+ Sqrt[-a^2 + b^2])]))*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-((a*Tan[(e + f*x)/2])/
(b + Sqrt[-a^2 + b^2])))^(3/2))/(a^2*c*(-(b*c) + a*d)*Sqrt[-c^2 + d^2]*f*Sin[e + f*x]^(3/2)*Sqrt[g*Sin[e + f*x
]]*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])*((3*Sqrt[-a^2 + b^2]*Cos[(e + f*x)/2]^2*(-2*(b + Sqrt[-a^2 + b^2]
)*(b*c - a*d)*Sqrt[-c^2 + d^2]*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b
^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] - a*d*((a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c
^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sq
rt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2
+ b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c +
Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-
a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]))*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e +
f*x]))/(a^2 - b^2)]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))])/(a*(b + Sqrt[-a^2 + b^2])*c*(-(b*c)
+ a*d)*Sqrt[-c^2 + d^2]*Sin[e + f*x]^(3/2)*Sqrt[a + b*Sin[e + f*x]]) + (2*b*Sqrt[-a^2 + b^2]*Cos[(e + f*x)/2]
^4*Cos[e + f*x]*(-2*(b + Sqrt[-a^2 + b^2])*(b*c - a*d)*Sqrt[-c^2 + d^2]*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 +
b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] - a*d*((a
*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]
*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt
[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*
EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(b + Sqrt
[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]))*
Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])))
^(3/2))/(a^2*c*(-(b*c) + a*d)*Sqrt[-c^2 + d^2]*Sin[e + f*x]^(3/2)*(a + b*Sin[e + f*x])^(3/2)) + (6*Sqrt[-a^2 +
b^2]*Cos[(e + f*x)/2]^4*Cos[e + f*x]*(-2*(b + Sqrt[-a^2 + b^2])*(b*c - a*d)*Sqrt[-c^2 + d^2]*EllipticF[ArcSin
[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-
a^2 + b^2])] - a*d*((a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(
b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/
Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b^2])*(d
+ Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])),
ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b +
Sqrt[-a^2 + b^2])]))*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-((a*Tan[(e + f*x)/2])/(b
+ Sqrt[-a^2 + b^2])))^(3/2))/(a^2*c*(-(b*c) + a*d)*Sqrt[-c^2 + d^2]*Sin[e + f*x]^(5/2)*Sqrt[a + b*Sin[e + f*x
]]) + (8*Sqrt[-a^2 + b^2]*Cos[(e + f*x)/2]^3*(-2*(b + Sqrt[-a^2 + b^2])*(b*c - a*d)*Sqrt[-c^2 + d^2]*EllipticF
[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b +
Sqrt[-a^2 + b^2])] - a*d*((a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^
2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*
x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b
^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 +
d^2])), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2
])/(b + Sqrt[-a^2 + b^2])]))*Sin[(e + f*x)/2]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-
((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])))^(3/2))/(a^2*c*(-(b*c) + a*d)*Sqrt[-c^2 + d^2]*Sin[e + f*x]^(3/2
)*Sqrt[a + b*Sin[e + f*x]]) - (2*Sqrt[-a^2 + b^2]*Cos[(e + f*x)/2]^4*(-2*(b + Sqrt[-a^2 + b^2])*(b*c - a*d)*Sq
rt[-c^2 + d^2]*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (
2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] - a*d*((a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*Elli
pticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sqrt[(b + Sqrt[-a^
2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] + (-(a
*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]
*c - a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt
[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]))*(-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])))^(3/2)*((
a*b*Cos[e + f*x]*Sec[(e + f*x)/2]^2)/(a^2 - b^2) + (a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x])*Tan[(e + f*x)/2]
)/(a^2 - b^2)))/(a^2*c*(-(b*c) + a*d)*Sqrt[-c^2 + d^2]*Sin[e + f*x]^(3/2)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[(a*Sec
[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]) - (4*Sqrt[-a^2 + b^2]*Cos[(e + f*x)/2]^4*Sqrt[(a*Sec[(e +
f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])))^(3/2)*(-(a*(b +
Sqrt[-a^2 + b^2])*(b*c - a*d)*Sqrt[-c^2 + d^2]*Sec[(e + f*x)/2]^2)/(2*Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[(b + Sqrt[
-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(2*Sq
rt[-a^2 + b^2])]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])]) - a*d*((a*(a*c
+ (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*Sec[(e + f*x)/2]^2)/(4*Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[(b + Sq
rt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(2
*Sqrt[-a^2 + b^2])]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])]*(1 - (c*(b +
Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2]))/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]))) + (a*(-(a*c) +
(b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*Sec[(e + f*x)/2]^2)/(4*Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[(b + Sqrt
[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(2*S
qrt[-a^2 + b^2])]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])]*(1 - (c*(b + Sq
rt[-a^2 + b^2] + a*Tan[(e + f*x)/2]))/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])))))))/(a^2*c*(-(b*c
) + a*d)*Sqrt[-c^2 + d^2]*Sin[e + f*x]^(3/2)*Sqrt[a + b*Sin[e + f*x]])))

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Maple [B]  time = 0.382, size = 3691, normalized size = 15. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f/a/(-c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)+d*a-c*b)/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)-d*a+c*b)/(-c^2+d
^2)^(1/2)/(a+b*sin(f*x+e))^(1/2)*(2*EllipticPi((-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/si
n(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-d*a+c*b),1/
2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*b*d*(-c^2+d^2)^(1/2)*(-a^2+b^2)^(1/2)-4*EllipticF((
-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2),1/2*2^(1/2)
*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*b*d*(-c^2+d^2)^(1/2)*(-a^2+b^2)^(1/2)+4*EllipticF((-(-(-a^2+
b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-a^
2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b^2*c*(-c^2+d^2)^(1/2)*(-a^2+b^2)^(1/2)+2*EllipticPi((-(-(-a^2+b^2)^(1/
2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-
a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)-d*a+c*b),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*b*d*(-
c^2+d^2)^(1/2)*(-a^2+b^2)^(1/2)-EllipticPi((-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/sin(f*
x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-d*a+c*b),1/2*2^
(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^3*d*(-c^2+d^2)^(1/2)+2*EllipticPi((-(-(-a^2+b^2)^(1/2)*
sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2
+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-d*a+c*b),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*b^2*d*(-c
^2+d^2)^(1/2)+2*EllipticF((-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/sin(f*x+e)/(b+(-a^2+b^2
)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^3*d*(-c^2+d^2)^(1/2)-2*EllipticF(
(-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2),1/2*2^(1/2
)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^2*b*c*(-c^2+d^2)^(1/2)-4*EllipticF((-(-(-a^2+b^2)^(1/2)*sin
(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/
(-a^2+b^2)^(1/2))^(1/2))*a*b^2*d*(-c^2+d^2)^(1/2)+4*EllipticF((-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos
(f*x+e)*a-a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))
*b^3*c*(-c^2+d^2)^(1/2)-EllipticPi((-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/sin(f*x+e)/(b+
(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)-d*a+c*b),1/2*2^(1/2)*((
b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^3*d*(-c^2+d^2)^(1/2)+2*EllipticPi((-(-(-a^2+b^2)^(1/2)*sin(f*x+
e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1
/2)+a*(-c^2+d^2)^(1/2)-d*a+c*b),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*b^2*d*(-c^2+d^2)^
(1/2)-EllipticPi((-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))
^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-d*a+c*b),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2)
)/(-a^2+b^2)^(1/2))^(1/2))*a^2*c*d*(-a^2+b^2)^(1/2)+2*EllipticPi((-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+
cos(f*x+e)*a-a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)
^(1/2)-d*a+c*b),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*b*d^2*(-a^2+b^2)^(1/2)+EllipticPi
((-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+
b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)-d*a+c*b),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/
2))^(1/2))*a^2*c*d*(-a^2+b^2)^(1/2)-2*EllipticPi((-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/
sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)-d*a+c*b),
1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*b*d^2*(-a^2+b^2)^(1/2)-EllipticPi((-(-(-a^2+b^2)^
(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c
*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-d*a+c*b),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^3*d
^2-EllipticPi((-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1
/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-d*a+c*b),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(
-a^2+b^2)^(1/2))^(1/2))*a^2*b*c*d+2*EllipticPi((-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/si
n(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-d*a+c*b),1/
2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*b^2*d^2+EllipticPi((-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-
b*sin(f*x+e)+cos(f*x+e)*a-a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)
+a*(-c^2+d^2)^(1/2)-d*a+c*b),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^3*d^2+EllipticPi((-(
-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)
^(1/2))*c/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)-d*a+c*b),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^
(1/2))*a^2*b*c*d-2*EllipticPi((-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/sin(f*x+e)/(b+(-a^2
+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)-d*a+c*b),1/2*2^(1/2)*((b+(-a
^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*b^2*d^2)*((-1+cos(f*x+e))*a/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*(
((-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(-a^2+b^2)^(1/2)/sin(f*x+e))^(1/2)*(-(-(-a^2+b^2)^(1
/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/sin(f*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2)*sin(f*x+e)^2*2^(1/2)/(g*sin
(f*x+e))^(1/2)/(-1+cos(f*x+e))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sin \left (f x + e\right ) + a}{\left (d \sin \left (f x + e\right ) + c\right )} \sqrt{g \sin \left (f x + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+d*sin(f*x+e))/(g*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*sin(f*x + e) + a)*(d*sin(f*x + e) + c)*sqrt(g*sin(f*x + e))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+d*sin(f*x+e))/(g*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{g \sin{\left (e + f x \right )}} \sqrt{a + b \sin{\left (e + f x \right )}} \left (c + d \sin{\left (e + f x \right )}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+d*sin(f*x+e))/(g*sin(f*x+e))**(1/2)/(a+b*sin(f*x+e))**(1/2),x)

[Out]

Integral(1/(sqrt(g*sin(e + f*x))*sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sin \left (f x + e\right ) + a}{\left (d \sin \left (f x + e\right ) + c\right )} \sqrt{g \sin \left (f x + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+d*sin(f*x+e))/(g*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*sin(f*x + e) + a)*(d*sin(f*x + e) + c)*sqrt(g*sin(f*x + e))), x)