∫ ∘ __________ a + b sin(e + fx) ____∘ g sin(e + fx) (c+d sin(e+fx)) dx [43]

3.1.43 \(\int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (c+d \sin (e+f x))} \, dx\) [43]

Optimal. Leaf size=250 \[ -\frac {2 \sqrt {a+b} \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right ) \tan (e+f x)}{c f \sqrt {g}}+\frac {2 (b c-a d) \sqrt {-\cot ^2(e+f x)} \sqrt {\frac {b+a \csc (e+f x)}{a+b}} \Pi \left (\frac {2 c}{c+d};\sin ^{-1}\left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right )|\frac {2 a}{a+b}\right ) \sqrt {g \sin (e+f x)} \tan (e+f x)}{c (c+d) f g \sqrt {a+b \sin (e+f x)}} \]

[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)/c/f/g^(1/2)+2*(-a*d+b*c)*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 = 0.34, antiderivative size = 250, 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 = {3012, 2895, 3016} \begin {gather*} \frac {2 (b c-a 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};\text {ArcSin}\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 (\text {ArcSin}\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 )}{c f \sqrt {g}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

rt[g]) + (2*(b*c - a*d)*Sqrt[-Cot[e + f*x]^2]*Sqrt[(b + a*Csc[e + f*x])/(a + b)]*EllipticPi[(2*c)/(c + d), Arc

Sin[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 2895

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(

Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqrt[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]

*EllipticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2]], -(a + b)/(a - b)], x] /; Fr

eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]

Rule 3012

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) +

(f_.)*(x_)])), x_Symbol] :> Dist[a/c, Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x], x] + Dist[(b*

c - a*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 3016

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]]/(f*(c + d)*Cot[e + f*x]*Sqrt[a

+ b*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))], 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 {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (c+d \sin (e+f x))} \, dx &=\frac {a \int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx}{c}+\frac {(b c-a 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)}{c f \sqrt {g}}+\frac {2 (b c-a 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] Leaf count is larger than twice the leaf count of optimal. \(8202\) vs. \(2(250)=500\).
time = 48.67, size = 8202, normalized size = 32.81 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3297\) vs. \(2(231)=462\).
time = 0.29, size = 3298, normalized size = 13.19


method	result	size

default	\(\text {Expression too large to display}\)	\(3298\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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

/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b-4*(-c^2+d^2)^(1/2)*(-a^2+b^2)^(1/2)*EllipticF((-(cos(f*x+

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

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

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

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

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

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

)^(1/2)*EllipticF((-(cos(f*x+e)*a-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-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^2-4*(-c^2+d^2)^(1/2)*EllipticF((-(cos(f*x+

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

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

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

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

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

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

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

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

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

x+e)-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)+

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

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

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

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

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(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 \begin {gather*} \int \frac {\sqrt {a+b\,\sin \left (e+f\,x\right )}}{\sqrt {g\,\sin \left (e+f\,x\right )}\,\left (c+d\,\sin \left (e+f\,x\right )\right )} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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