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

Optimal. Leaf size=192 \[ \frac{2 \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt{c+d} (b c-a d)} \]

[Out]

(2*Sqrt[a + b]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])]
, ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[
e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(Sq
rt[c + d]*(b*c - a*d)*f)

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Rubi [A]  time = 0.120762, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {2818} \[ \frac{2 \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt{c+d} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a + b]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])]
, ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[
e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(Sq
rt[c + d]*(b*c - a*d)*f)

Rule 2818

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Si
mp[(2*(c + d*Sin[e + f*x])*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c
- a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(Sqrt[a +
b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))])/(f*(b*c - a*d)*Rt[(c + d)/(a
+ b), 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[(c + d)/(a + b)]

Rubi steps

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

Mathematica [A]  time = 0.217785, size = 189, normalized size = 0.98 \[ \frac{2 \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(a d-b c) (\sin (e+f x)-1)}{(a+b) (c+d \sin (e+f x))}} \sqrt{\frac{(a d-b c) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt{c+d} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.516, size = 1228, normalized size = 6.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/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)*EllipticF(((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)*(c*sin(f*x+e)-cos(f*x+e)*(-c^2+d^2)^(1/2)+
d*cos(f*x+e)-(-c^2+d^2)^(1/2)+d)/(c*sin(f*x+e)+cos(f*x+e)*(-c^2+d^2)^(1/2)+d*cos(f*x+e)+(-c^2+d^2)^(1/2)+d))^(
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*(-c^2
+d^2)^(1/2)-c*(-a^2+b^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))^(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)*(c*sin(f*x+e)-cos(f*x+e)*(
-c^2+d^2)^(1/2)+d*cos(f*x+e)-(-c^2+d^2)^(1/2)+d)/(c*sin(f*x+e)+cos(f*x+e)*(-c^2+d^2)^(1/2)+d*cos(f*x+e)+(-c^2+
d^2)^(1/2)+d))^(1/2)*((-c^2+d^2)^(1/2)*c/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)-d*a+c*b)*(cos(f*x+e)*(-a^2+b^2
)^(1/2)+a*sin(f*x+e)+b*cos(f*x+e)+(-a^2+b^2)^(1/2)+b)/(c*sin(f*x+e)+cos(f*x+e)*(-c^2+d^2)^(1/2)+d*cos(f*x+e)+(
-c^2+d^2)^(1/2)+d))^(1/2)*((-c^2+d^2)^(1/2)*c*(a*sin(f*x+e)-cos(f*x+e)*(-a^2+b^2)^(1/2)+b*cos(f*x+e)-(-a^2+b^2
)^(1/2)+b)/(a*(-c^2+d^2)^(1/2)-c*(-a^2+b^2)^(1/2)-d*a+c*b)/(c*sin(f*x+e)+cos(f*x+e)*(-c^2+d^2)^(1/2)+d*cos(f*x
+e)+(-c^2+d^2)^(1/2)+d))^(1/2)*(a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(1/2)*(cos(f*x+e)+1)^2*(-1+cos(f*x+e))^
2*(c*(-c^2+d^2)^(1/2)*(-a^2+b^2)^(1/2)*sin(f*x+e)+b*c*(-c^2+d^2)^(1/2)*sin(f*x+e)+c*d*(-a^2+b^2)^(1/2)*sin(f*x
+e)-a*c^2*sin(f*x+e)+b*c*d*sin(f*x+e)+cos(f*x+e)*(-c^2+d^2)^(1/2)*(-a^2+b^2)^(1/2)*d-cos(f*x+e)*(-c^2+d^2)^(1/
2)*a*c+cos(f*x+e)*(-c^2+d^2)^(1/2)*b*d-cos(f*x+e)*(-a^2+b^2)^(1/2)*c^2+cos(f*x+e)*(-a^2+b^2)^(1/2)*d^2-c^2*b*c
os(f*x+e)+cos(f*x+e)*b*d^2+d*(-c^2+d^2)^(1/2)*(-a^2+b^2)^(1/2)+b*d*(-c^2+d^2)^(1/2)+d^2*(-a^2+b^2)^(1/2)-a*c*d
+d^2*b)/sin(f*x+e)^4/(-b*cos(f*x+e)^2*d+a*d*sin(f*x+e)+b*c*sin(f*x+e)+c*a+b*d)

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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} \sqrt{d \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(b*d*cos(f*x + e)^2 - a*c - b*d - (b*c + a*d)*sin(
f*x + e)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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} \sqrt{d \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)), x)