3.34 \(\int \frac{1}{\sqrt{g \sin (e+f x)} \sqrt{a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx\)

Optimal. Leaf size=256 \[ \frac{2 b \sqrt{a+b} \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{g} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{g \sin (e+f x)}}\right )|-\frac{a+b}{a-b}\right )}{a c f \sqrt{g} (a-b)}-\frac{\sqrt{\frac{\sin (e+f x)}{\sin (e+f x)+1}} \sqrt{a+b \sin (e+f x)} E\left (\sin ^{-1}\left (\frac{\cos (e+f x)}{\sin (e+f x)+1}\right )|-\frac{a-b}{a+b}\right )}{c f (a-b) \sqrt{g \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a+b) (\sin (e+f x)+1)}}} \]

[Out]

-((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*b*Sqrt[a + b]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*Ellipti
cF[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*(a - b)*c*f*Sqrt[g])

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Rubi [A]  time = 0.519903, antiderivative size = 256, 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 = {2938, 2816, 2932} \[ \frac{2 b \sqrt{a+b} \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{g} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{g \sin (e+f x)}}\right )|-\frac{a+b}{a-b}\right )}{a c f \sqrt{g} (a-b)}-\frac{\sqrt{\frac{\sin (e+f x)}{\sin (e+f x)+1}} \sqrt{a+b \sin (e+f x)} E\left (\sin ^{-1}\left (\frac{\cos (e+f x)}{\sin (e+f x)+1}\right )|-\frac{a-b}{a+b}\right )}{c f (a-b) \sqrt{g \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a+b) (\sin (e+f x)+1)}}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]*(c + c*Sin[e + f*x])),x]

[Out]

-((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*b*Sqrt[a + b]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*Ellipti
cF[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*(a - b)*c*f*Sqrt[g])

Rule 2938

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

Mathematica [C]  time = 9.93923, size = 1659, normalized size = 6.48 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]*(c + c*Sin[e + f*x])),x]

[Out]

(-2*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/((a - b)*f*S
qrt[g*Sin[e + f*x]]*(c + c*Sin[e + f*x])) + ((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Sqrt[Sin[e + f*x]]*((4*a*
(a - b)*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*
(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Cs
c[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)
*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) + (2*a*ArcTanh[(Sqrt[b]*Sqrt[Sin[e + f*x]])/Sqrt[a + b*Sin[e + f
*x]]]*Cos[e + f*x]^2)/(Sqrt[b]*(1 - Sin[e + f*x]^2)) + 4*a^2*((Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a +
b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec
[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-
e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (Sqrt[(
(a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticPi[-(a/b), ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a +
b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e
+ Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/(b*Sqrt[Sin[e
+ f*x]]*Sqrt[a + b*Sin[e + f*x]])) - 2*b*((Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(b*Sqrt[Sin[e + f*x]]) + (I
*Cos[(-e + Pi/2 - f*x)/2]*Csc[e + f*x]*EllipticE[I*ArcSinh[Sin[(-e + Pi/2 - f*x)/2]/Sqrt[Sin[e + f*x]]], (-2*a
)/(-a - b)]*Sqrt[a + b*Sin[e + f*x]])/(b*Sqrt[Cos[(-e + Pi/2 - f*x)/2]^2*Csc[e + f*x]]*Sqrt[(Csc[e + f*x]*(a +
b*Sin[e + f*x]))/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sq
rt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2
- f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a
+ b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (a*Sqrt[((a + b)*Cot[(-e + Pi/2
- f*x)/2]^2)/(-a + b)]*EllipticPi[-(a/b), ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sq
rt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Si
n[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/(b*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Si
n[e + f*x]])))/b) + (2*b*Cot[e + f*x]*(-(a*ArcTanh[(Sqrt[b]*Sqrt[Sin[e + f*x]])/Sqrt[a + b*Sin[e + f*x]]])/(2*
b^(3/2)) + (Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/(2*b))*Sin[2*(e + f*x)])/(1 - Sin[e + f*x]^2)))/(2*(a
- b)*f*Sqrt[g*Sin[e + f*x]]*(c + c*Sin[e + f*x]))

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Maple [B]  time = 0.431, size = 9043, normalized size = 35.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c+c*sin(f*x+e))/(g*sin(f*x+e))^(1/2)/(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{1}{\sqrt{b \sin \left (f x + e\right ) + a}{\left (c \sin \left (f x + e\right ) + c\right )} \sqrt{g \sin \left (f x + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+c*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)*(c*sin(f*x + e) + c)*sqrt(g*sin(f*x + e))), 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 )}}{{\left (a + b\right )} c g \cos \left (f x + e\right )^{2} -{\left (a + b\right )} c g +{\left (b c g \cos \left (f x + e\right )^{2} -{\left (a + b\right )} c g\right )} \sin \left (f x + e\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{\sqrt{g \sin{\left (e + f x \right )}} \sqrt{a + b \sin{\left (e + f x \right )}} \sin{\left (e + f x \right )} + \sqrt{g \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(1/(c+c*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))*sin(e + f*x) + sqrt(g*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{1}{\sqrt{b \sin \left (f x + e\right ) + a}{\left (c \sin \left (f x + e\right ) + c\right )} \sqrt{g \sin \left (f x + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+c*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)*(c*sin(f*x + e) + c)*sqrt(g*sin(f*x + e))), x)