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

Optimal. Leaf size=252 \[ \frac{g \sqrt{\frac{\sin (e+f x)}{\sin (e+f x)+1}} \sqrt{a+b \sin (e+f x)} E\left (\sin ^{-1}\left (\frac{\cos (e+f x)}{\sin (e+f x)+1}\right )|-\frac{a-b}{a+b}\right )}{c f (a-b) \sqrt{g \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a+b) (\sin (e+f x)+1)}}}-\frac{2 \sqrt{g} \sqrt{a+b} \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{g} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{g \sin (e+f x)}}\right )|-\frac{a+b}{a-b}\right )}{c f (a-b)} \]

[Out]

(g*EllipticE[ArcSin[Cos[e + f*x]/(1 + Sin[e + f*x])], -((a - b)/(a + b))]*Sqrt[Sin[e + f*x]/(1 + Sin[e + f*x])
]*Sqrt[a + b*Sin[e + f*x]])/((a - b)*c*f*Sqrt[g*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a + b)*(1 + Sin[e +
f*x]))]) - (2*Sqrt[a + b]*Sqrt[g]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*El
lipticF[ArcSin[(Sqrt[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 - b)*c*f)

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Rubi [A]  time = 0.511218, antiderivative size = 252, 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 = {2936, 2816, 2932} \[ \frac{g \sqrt{\frac{\sin (e+f x)}{\sin (e+f x)+1}} \sqrt{a+b \sin (e+f x)} E\left (\sin ^{-1}\left (\frac{\cos (e+f x)}{\sin (e+f x)+1}\right )|-\frac{a-b}{a+b}\right )}{c f (a-b) \sqrt{g \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a+b) (\sin (e+f x)+1)}}}-\frac{2 \sqrt{g} \sqrt{a+b} \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{g} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{g \sin (e+f x)}}\right )|-\frac{a+b}{a-b}\right )}{c f (a-b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g*EllipticE[ArcSin[Cos[e + f*x]/(1 + Sin[e + f*x])], -((a - b)/(a + b))]*Sqrt[Sin[e + f*x]/(1 + Sin[e + f*x])
]*Sqrt[a + b*Sin[e + f*x]])/((a - b)*c*f*Sqrt[g*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a + b)*(1 + Sin[e +
f*x]))]) - (2*Sqrt[a + b]*Sqrt[g]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*El
lipticF[ArcSin[(Sqrt[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 - b)*c*f)

Rule 2936

Int[Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)])), x_Symbol] :> -Dist[(a*g)/(b*c - a*d), Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x]
, x] + Dist[(c*g)/(b*c - a*d), Int[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[g*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] && (EqQ[a^2 - b^2, 0] || EqQ[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 2932

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)])), x_Symbol] :> -Simp[(Sqrt[a + b*Sin[e + f*x]]*Sqrt[(d*Sin[e + f*x])/(c + d*Sin[e + f*x])]*Ellipt
icE[ArcSin[(c*Cos[e + f*x])/(c + d*Sin[e + f*x])], (b*c - a*d)/(b*c + a*d)])/(d*f*Sqrt[g*Sin[e + f*x]]*Sqrt[(c
^2*(a + b*Sin[e + f*x]))/((a*c + b*d)*(c + d*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] && EqQ[c^2 - d^2, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{g \sin (e+f x)}}{\sqrt{a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx &=-\frac{g \int \frac{\sqrt{a+b \sin (e+f x)}}{\sqrt{g \sin (e+f x)} (c+c \sin (e+f x))} \, dx}{a-b}+\frac{(a g) \int \frac{1}{\sqrt{g \sin (e+f x)} \sqrt{a+b \sin (e+f x)}} \, dx}{(a-b) c}\\ &=\frac{g E\left (\sin ^{-1}\left (\frac{\cos (e+f x)}{1+\sin (e+f x)}\right )|-\frac{a-b}{a+b}\right ) \sqrt{\frac{\sin (e+f x)}{1+\sin (e+f x)}} \sqrt{a+b \sin (e+f x)}}{(a-b) c f \sqrt{g \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a+b) (1+\sin (e+f x))}}}-\frac{2 \sqrt{a+b} \sqrt{g} \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-b) c f}\\ \end{align*}

Mathematica [B]  time = 33.5851, size = 5708, normalized size = 22.65 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 0.442, size = 6817, normalized size = 27.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{b \sin \left (f x + e\right ) + a} \sqrt{g \sin \left (f x + e\right )}}{b c \cos \left (f x + e\right )^{2} -{\left (a + b\right )} c \sin \left (f x + e\right ) -{\left (a + b\right )} c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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