\(\int \frac {1}{\sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx\) [786]

Optimal result

Integrand size = 29, antiderivative size = 189 \[ \int \frac {1}{\sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=\frac {2 \sqrt {3+b} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(b c-3 d) \sqrt {c+d} f} \]

[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)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.02, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2897} \[ \int \frac {1}{\sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=\frac {2 \sqrt {a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt {c+d} (b c-a d)} \]

[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 2897

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])/(f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]*Cos[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)))], 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*} \text {integral}& = \frac {2 \sqrt {a+b} \operatorname {EllipticF}\left (\arcsin \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] (verified)

Time = 0.24 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),-\frac {(3+b) (c-d)}{(-3+b) (c+d)}\right ) \sec (e+f x) (-1+\sin (e+f x)) \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(-3+b) (c+d \sin (e+f x))}}}{\sqrt {3+b} \sqrt {c+d} f \sqrt {-\frac {(b c-3 d) (-1+\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}}} \]

[In]

Integrate[1/(Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]),x]

[Out]

(-2*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[3 + b*Sin[e + f*x]])/(Sqrt[3 + b]*Sqrt[c + d*Sin[e + f*x]])], -(((3 + b
)*(c - d))/((-3 + b)*(c + d)))]*Sec[e + f*x]*(-1 + Sin[e + f*x])*Sqrt[((b*c - 3*d)*(1 + Sin[e + f*x]))/((-3 +
b)*(c + d*Sin[e + f*x]))])/(Sqrt[3 + b]*Sqrt[c + d]*f*Sqrt[-(((b*c - 3*d)*(-1 + Sin[e + f*x]))/((3 + b)*(c + d
*Sin[e + f*x])))])

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1104\) vs. \(2(177)=354\).

Time = 11.22 (sec) , antiderivative size = 1105, normalized size of antiderivative = 5.85

method	result	size
default	\(\text {Expression too large to display}\)	\(1105\)

[In]

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

[Out]

4/f*(c*(-a^2+b^2)^(1/2)*(-c^2+d^2)^(1/2)*sin(f*x+e)+cos(f*x+e)*(-a^2+b^2)^(1/2)*(-c^2+d^2)^(1/2)*d-cos(f*x+e)*
(-a^2+b^2)^(1/2)*c^2+c*d*(-a^2+b^2)^(1/2)*sin(f*x+e)+cos(f*x+e)*(-a^2+b^2)^(1/2)*d^2-cos(f*x+e)*(-c^2+d^2)^(1/
2)*a*c+b*c*(-c^2+d^2)^(1/2)*sin(f*x+e)+cos(f*x+e)*(-c^2+d^2)^(1/2)*b*d-a*c^2*sin(f*x+e)-c^2*b*cos(f*x+e)+b*c*d
*sin(f*x+e)+cos(f*x+e)*b*d^2+d*(-a^2+b^2)^(1/2)*(-c^2+d^2)^(1/2)+d^2*(-a^2+b^2)^(1/2)+b*d*(-c^2+d^2)^(1/2)-a*c
*d+b*d^2)*(((-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+a*cos(f*x+e)-a)*(-c^2+d^2)^(1/2)*c/(sin(f*x+e)*(-c^2+d^2)
^(1/2)+d*sin(f*x+e)-c*cos(f*x+e)+c)/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)+a*d-b*c))^(1/2)*(((-a^2+b^2)^(1/2)*
sin(f*x+e)+b*sin(f*x+e)-a*cos(f*x+e)+a)*(-c^2+d^2)^(1/2)*c/(sin(f*x+e)*(-c^2+d^2)^(1/2)+d*sin(f*x+e)-c*cos(f*x
+e)+c)/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)-a*d+b*c))^(1/2)*(-(sin(f*x+e)*(-c^2+d^2)^(1/2)-d*sin(f*x+e)+c*co
s(f*x+e)-c)*(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-a*d+b*c)/(sin(f*x+e)*(-c^2+d^2)^(1/2)+d*sin(f*x+e)-c*cos(f*
x+e)+c)/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)-a*d+b*c))^(1/2)*(a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(1/2)*E
llipticF((-(-csc(f*x+e)*c+cot(f*x+e)*c+(-c^2+d^2)^(1/2)-d)*(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-a*d+b*c)/(cs
c(f*x+e)*c-cot(f*x+e)*c+(-c^2+d^2)^(1/2)+d)/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)-a*d+b*c))^(1/2),((c*(-a^2+b
^2)^(1/2)+a*(-c^2+d^2)^(1/2)-a*d+b*c)*(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)+a*d-b*c)/(c*(-a^2+b^2)^(1/2)-a*(-
c^2+d^2)^(1/2)+a*d-b*c)/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-a*d+b*c))^(1/2))/(-b*cos(f*x+e)^2*d+sin(f*x+e)*
a*d+sin(f*x+e)*b*c+a*c+b*d)/(-c^2+d^2)^(1/2)/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-a*d+b*c)

Fricas [F]

\[ \int \frac {1}{\sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {1}{\sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {1}{\sqrt {a + b \sin {\left (e + f x \right )}} \sqrt {c + d \sin {\left (e + f x \right )}}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {1}{\sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {1}{\sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {1}{\sqrt {a+b\,\sin \left (e+f\,x\right )}\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]

[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)