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

Optimal. Leaf size=267 \[ \frac{g \sqrt{\frac{\sin (e+f x)}{\sin (e+f x)+1}} \sqrt{a+b \sin (e+f x)} E\left (\sin ^{-1}\left (\frac{\cos (e+f x)}{\sin (e+f x)+1}\right )|-\frac{a-b}{a+b}\right )}{c f \sqrt{g \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a+b) (\sin (e+f x)+1)}}}+\frac{2 \sqrt{g} \sec (e+f x) \sqrt{\frac{a (1-\sin (e+f x))}{a+b \sin (e+f x)}} \sqrt{\frac{a (\sin (e+f x)+1)}{a+b \sin (e+f x)}} (a+b \sin (e+f x)) \Pi \left (\frac{b}{a+b};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{g \sin (e+f x)}}{\sqrt{g} \sqrt{a+b \sin (e+f x)}}\right )|-\frac{a-b}{a+b}\right )}{c f \sqrt{a+b}} \]

[Out]

(2*Sqrt[g]*EllipticPi[b/(a + b), ArcSin[(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])/(Sqrt[g]*Sqrt[a + b*Sin[e + f*x]])]
, -((a - b)/(a + b))]*Sec[e + f*x]*Sqrt[(a*(1 - Sin[e + f*x]))/(a + b*Sin[e + f*x])]*Sqrt[(a*(1 + Sin[e + f*x]
))/(a + b*Sin[e + f*x])]*(a + b*Sin[e + f*x]))/(Sqrt[a + b]*c*f) + (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]])/(c*f*Sqrt[g*Sin
[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a + b)*(1 + Sin[e + f*x]))])

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Rubi [A]  time = 0.499523, antiderivative size = 267, 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 = {2928, 2811, 2932} \[ \frac{g \sqrt{\frac{\sin (e+f x)}{\sin (e+f x)+1}} \sqrt{a+b \sin (e+f x)} E\left (\sin ^{-1}\left (\frac{\cos (e+f x)}{\sin (e+f x)+1}\right )|-\frac{a-b}{a+b}\right )}{c f \sqrt{g \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a+b) (\sin (e+f x)+1)}}}+\frac{2 \sqrt{g} \sec (e+f x) \sqrt{\frac{a (1-\sin (e+f x))}{a+b \sin (e+f x)}} \sqrt{\frac{a (\sin (e+f x)+1)}{a+b \sin (e+f x)}} (a+b \sin (e+f x)) \Pi \left (\frac{b}{a+b};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{g \sin (e+f x)}}{\sqrt{g} \sqrt{a+b \sin (e+f x)}}\right )|-\frac{a-b}{a+b}\right )}{c f \sqrt{a+b}} \]

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]

(2*Sqrt[g]*EllipticPi[b/(a + b), ArcSin[(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])/(Sqrt[g]*Sqrt[a + b*Sin[e + f*x]])]
, -((a - b)/(a + b))]*Sec[e + f*x]*Sqrt[(a*(1 - Sin[e + f*x]))/(a + b*Sin[e + f*x])]*Sqrt[(a*(1 + Sin[e + f*x]
))/(a + b*Sin[e + f*x])]*(a + b*Sin[e + f*x]))/(Sqrt[a + b]*c*f) + (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]])/(c*f*Sqrt[g*Sin
[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a + b)*(1 + Sin[e + f*x]))])

Rule 2928

Int[(Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]])/((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)]), x_Symbol] :> Dist[g/d, Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[g*Sin[e + f*x]], x], x] - Dist[(c*g)/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 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 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 &=-\left (g \int \frac{\sqrt{a+b \sin (e+f x)}}{\sqrt{g \sin (e+f x)} (c+c \sin (e+f x))} \, dx\right )+\frac{g \int \frac{\sqrt{a+b \sin (e+f x)}}{\sqrt{g \sin (e+f x)}} \, dx}{c}\\ &=\frac{2 \sqrt{g} \Pi \left (\frac{b}{a+b};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{g \sin (e+f x)}}{\sqrt{g} \sqrt{a+b \sin (e+f x)}}\right )|-\frac{a-b}{a+b}\right ) \sec (e+f x) \sqrt{\frac{a (1-\sin (e+f x))}{a+b \sin (e+f x)}} \sqrt{\frac{a (1+\sin (e+f x))}{a+b \sin (e+f x)}} (a+b \sin (e+f x))}{\sqrt{a+b} c f}+\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)}}{c f \sqrt{g \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a+b) (1+\sin (e+f x))}}}\\ \end{align*}

Mathematica [C]  time = 34.3669, size = 13199, normalized size = 49.43 \[ \text{Result too large to show} \]

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 [C]  time = 0.803, size = 22961, normalized size = 86. \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)*(a+b*sin(f*x+e))^(1/2)/(c+c*sin(f*x+e)),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{b \sin \left (f x + e\right ) + a} \sqrt{g \sin \left (f x + e\right )}}{c \sin \left (f x + e\right ) + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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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 )} + 1}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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