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

Optimal. Leaf size=254 \[ \frac{2 \sqrt{g} \sqrt{a+b} \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{g} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{g \sin (e+f x)}}\right )|-\frac{a+b}{a-b}\right )}{d f}-\frac{2 (b c-a d) \tan (e+f x) \sqrt{-\cot ^2(e+f x)} \sqrt{g \sin (e+f x)} \sqrt{\frac{a \csc (e+f x)+b}{a+b}} \Pi \left (\frac{2 c}{c+d};\sin ^{-1}\left (\frac{\sqrt{1-\csc (e+f x)}}{\sqrt{2}}\right )|\frac{2 a}{a+b}\right )}{d f (c+d) \sqrt{a+b \sin (e+f x)}} \]

[Out]

(2*Sqrt[a + b]*Sqrt[g]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticPi[(a
+ b)/b, ArcSin[(Sqrt[g]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])], -((a + b)/(a - b))]*Ta
n[e + f*x])/(d*f) - (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), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], (2*a)/(a + b)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/(d*(c + d)*f
*Sqrt[a + b*Sin[e + f*x]])

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Rubi [A]  time = 0.520263, antiderivative size = 254, 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 = {2929, 2809, 2937} \[ \frac{2 \sqrt{g} \sqrt{a+b} \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{g} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{g \sin (e+f x)}}\right )|-\frac{a+b}{a-b}\right )}{d f}-\frac{2 (b c-a d) \tan (e+f x) \sqrt{-\cot ^2(e+f x)} \sqrt{g \sin (e+f x)} \sqrt{\frac{a \csc (e+f x)+b}{a+b}} \Pi \left (\frac{2 c}{c+d};\sin ^{-1}\left (\frac{\sqrt{1-\csc (e+f x)}}{\sqrt{2}}\right )|\frac{2 a}{a+b}\right )}{d f (c+d) \sqrt{a+b \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[(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[g]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticPi[(a
+ b)/b, ArcSin[(Sqrt[g]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])], -((a + b)/(a - b))]*Ta
n[e + f*x])/(d*f) - (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), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], (2*a)/(a + b)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/(d*(c + d)*f
*Sqrt[a + b*Sin[e + f*x]])

Rule 2929

Int[(Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]])/((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)]), x_Symbol] :> Dist[b/d, Int[Sqrt[g*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x], x] - Dist[(b*c -
a*d)/d, 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 2809

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

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{\sqrt{g \sin (e+f x)} \sqrt{a+b \sin (e+f x)}}{c+d \sin (e+f x)} \, dx &=\frac{b \int \frac{\sqrt{g \sin (e+f x)}}{\sqrt{a+b \sin (e+f x)}} \, dx}{d}-\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}{d}\\ &=\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}} \Pi \left (\frac{a+b}{b};\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)}{d f}-\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)}{d (c+d) f \sqrt{a+b \sin (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 29.6299, size = 23019, normalized size = 90.63 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [C]  time = 0.349, size = 6200, normalized size = 24.4 \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+d*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 )}}{d \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+d*sin(f*x+e)),x, algorithm="maxima")

[Out]

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

[Out]

Integral(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{\sqrt{b \sin \left (f x + e\right ) + a} \sqrt{g \sin \left (f x + e\right )}}{d \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+d*sin(f*x+e)),x, algorithm="giac")

[Out]

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