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

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Rubi [A]  time = 0.12, antiderivative size = 192, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.24, size = 191, normalized size = 0.99 \[ \frac {2 \sqrt {a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {\frac {(a d-b c) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt {c+d} (b c-a d)} \]

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]))/(Sq
rt[c + d]*(b*c - a*d)*f)

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fricas [F]  time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{b d \cos \left (f x + e\right )^{2} - a c - b d - {\left (b c + a d\right )} \sin \left (f x + e\right )}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}\,{d x} \]

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)

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maple [B]  time = 0.69, size = 1233, normalized size = 6.42 \[ \text {result too large to display} \]

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*EllipticF((-(cos(f*x+e)*(-c^2+d^2)^(1/2)-c*sin(f*x+e)-d*cos(f*x+e)+(-c^2+d^2)^(1/2)-d)*(c*(-a^2+b^2)^(1/2
)-a*(-c^2+d^2)^(1/2)-d*a+c*b)/(cos(f*x+e)*(-c^2+d^2)^(1/2)+c*sin(f*x+e)+d*cos(f*x+e)+(-c^2+d^2)^(1/2)+d)/(a*(-
c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)-d*a+c*b))^(1/2),((a*(-c^2+d^2)^(1/2)+c*(-a^2+b^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)/(a*(-c^2+d^2)^(1/2)-c*(-a^2+b^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))^(1/2))*((cos(f*x+e)*(-c^2+d^2)^(1/2)-c*sin(f*x+e)-d*cos(f*x+e)+(-c^2+d^2)^(1/2)-d)/(co
s(f*x+e)*(-c^2+d^2)^(1/2)+c*sin(f*x+e)+d*cos(f*x+e)+(-c^2+d^2)^(1/2)+d)*(a*(-c^2+d^2)^(1/2)-c*(-a^2+b^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))^(1/2)*((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)/(cos(f*x+e)*(-c^2+d^2)^(1/2)+c*sin(f*x+e)+d*cos(f*x+e)+(-c^2+d^2)^(1/2)+d)*(-c^
2+d^2)^(1/2)*c/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)-d*a+c*b))^(1/2)*(-(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)/(cos(f*x+e)*(-c^2+d^2)^(1/2)+c*sin(f*x+e)+d*cos(f*x+e)+(-c^2+d^2)^(1/2)+d
)*(-c^2+d^2)^(1/2)*c/(a*(-c^2+d^2)^(1/2)-c*(-a^2+b^2)^(1/2)-d*a+c*b))^(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-cos(f*x+e)*b*c^2+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/(-cos(f*x+e)^2*b*d+a*d*sin(f*x+e)+b*c*sin(f*
x+e)+c*a+b*d)/(-c^2+d^2)^(1/2)/(a*(-c^2+d^2)^(1/2)-c*(-a^2+b^2)^(1/2)+d*a-c*b)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {a+b\,\sin \left (e+f\,x\right )}\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b \sin {\left (e + f x \right )}} \sqrt {c + d \sin {\left (e + f x \right )}}}\, dx \]

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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