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

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

[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]])

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Rubi [A]  time = 0.525648, antiderivative size = 250, 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 = {2933, 2816, 2937} \[ \frac{2 (b c-a d) \tan (e+f x) \sqrt{-\cot ^2(e+f x)} \sqrt{g \sin (e+f x)} \sqrt{\frac{a \csc (e+f x)+b}{a+b}} \Pi \left (\frac{2 c}{c+d};\sin ^{-1}\left (\frac{\sqrt{1-\csc (e+f x)}}{\sqrt{2}}\right )|\frac{2 a}{a+b}\right )}{c f g (c+d) \sqrt{a+b \sin (e+f x)}}-\frac{2 \sqrt{a+b} \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{g} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{g \sin (e+f x)}}\right )|-\frac{a+b}{a-b}\right )}{c f \sqrt{g}} \]

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 2933

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 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 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{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*}

Mathematica [B]  time = 29.3622, size = 8202, normalized size = 32.81 \[ \text{Result too large to show} \]

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]  time = 0.309, size = 3290, normalized size = 13.2 \begin{align*} \text{output too large to display} \end{align*}

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)

[Out]

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

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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}}{{\left (d \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((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., 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 )}}{d g \cos \left (f x + e\right )^{2} - c g \sin \left (f x + e\right ) - d g}, x\right ) \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \sin \left (f x + e\right ) + a}}{{\left (d \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((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)