∫ 1_________________∘ g sin(e+fx) ∘_________ a+b sin(e+fx) (c+d sin(e+fx)) dx

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

*(a*(1-csc(f*x+e))/(a+b))^(1/2)*(a*(1+csc(f*x+e))/(a-b))^(1/2)*tan(f*x+e)/a/c/f/g^(1/2)-2*d*EllipticPi(1/2*(1-

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

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

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Rubi [A] time = 0.52, antiderivative size = 246, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [B] time = 30.06, 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)*(-1/2*(a*(

b + Sqrt[-a^2 + b^2])*(b*c - a*d)*Sqrt[-c^2 + d^2]*Sec[(e + f*x)/2]^2)/(Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[(b + Sqr

t[-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])]) - 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 +

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*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 + 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 + 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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fricas [F] time = 10.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b \sin \left (f x + e\right ) + a} \sqrt {g \sin \left (f x + e\right )}}{{\left (b c + a d\right )} g \cos \left (f x + e\right )^{2} - {\left (b c + a d\right )} g + {\left (b d g \cos \left (f x + e\right )^{2} - {\left (a c + b d\right )} g\right )} \sin \left (f x + e\right )}, x\right ) \]

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]

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

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

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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )} \sqrt {g \sin \left (f x + e\right )}}\,{d x} \]

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)

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maple [B] time = 0.60, size = 3690, normalized size = 15.00 \[ \text {output too large to display} \]

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

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

2)*(-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)/(b+(-a^2+b^2)^(1/

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

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

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

*(-c^2+d^2)^(1/2)+c*(-a^2+b^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)/(b+(-a^2+b^2)^(1/2

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

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

)*(-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)/(b+(-a^2+b^2)^(1/2)

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

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

(1/2))*c/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^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)/(b+(-a^2+b^2)^(1/2))/sin

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

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

(f*x+e))^(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)*(a*(-1+cos(f*x+e))/(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)*s

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

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

1/2)-d*a+c*b)/(-c^2+d^2)^(1/2)

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )} \sqrt {g \sin \left (f x + e\right )}}\,{d x} \]

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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\sqrt {g\,\sin \left (e+f\,x\right )}\,\sqrt {a+b\,\sin \left (e+f\,x\right )}\,\left (c+d\,\sin \left (e+f\,x\right )\right )} \,d x \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {g \sin {\left (e + f x \right )}} \sqrt {a + b \sin {\left (e + f x \right )}} \left (c + d \sin {\left (e + f x \right )}\right )}\, dx \]

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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