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

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

[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]
time = 0.07, 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 = {2897} \begin {gather*} \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 (\text {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)} \end {gather*}

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 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*} \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 = 0.15, size = 191, normalized size = 0.99 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1235\) vs. \(2(177)=354\).
time = 12.99, size = 1236, normalized size = 6.44

method	result	size
default	\(-\frac {4 \EllipticF \left (\sqrt {-\frac {\left (\sqrt {-c^{2}+d^{2}}\, \cos \left (f x +e \right )-c \sin \left (f x +e \right )-d \cos \left (f x +e \right )+\sqrt {-c^{2}+d^{2}}-d \right ) \left (-a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}-a d +b c \right )}{\left (\sqrt {-c^{2}+d^{2}}\, \cos \left (f x +e \right )+c \sin \left (f x +e \right )+d \cos \left (f x +e \right )+\sqrt {-c^{2}+d^{2}}+d \right ) \left (a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}-a d +b c \right )}}, \sqrt {\frac {\left (a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}-a d +b c \right ) \left (a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}+a d -b c \right )}{\left (-a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}+a d -b c \right ) \left (-a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}-a d +b c \right )}}\right ) \sqrt {\frac {\left (\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}-a \sin \left (f x +e \right )-\cos \left (f x +e \right ) b +\sqrt {-a^{2}+b^{2}}-b \right ) \sqrt {-c^{2}+d^{2}}\, c}{\left (\sqrt {-c^{2}+d^{2}}\, \cos \left (f x +e \right )+c \sin \left (f x +e \right )+d \cos \left (f x +e \right )+\sqrt {-c^{2}+d^{2}}+d \right ) \left (-a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}+a d -b c \right )}}\, \sqrt {\frac {\left (\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}+a \sin \left (f x +e \right )+\cos \left (f x +e \right ) b +\sqrt {-a^{2}+b^{2}}+b \right ) \sqrt {-c^{2}+d^{2}}\, c}{\left (\sqrt {-c^{2}+d^{2}}\, \cos \left (f x +e \right )+c \sin \left (f x +e \right )+d \cos \left (f x +e \right )+\sqrt {-c^{2}+d^{2}}+d \right ) \left (a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}-a d +b c \right )}}\, \sqrt {-\frac {\left (\sqrt {-c^{2}+d^{2}}\, \cos \left (f x +e \right )-c \sin \left (f x +e \right )-d \cos \left (f x +e \right )+\sqrt {-c^{2}+d^{2}}-d \right ) \left (-a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}-a d +b c \right )}{\left (\sqrt {-c^{2}+d^{2}}\, \cos \left (f x +e \right )+c \sin \left (f x +e \right )+d \cos \left (f x +e \right )+\sqrt {-c^{2}+d^{2}}+d \right ) \left (a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}-a d +b c \right )}}\, \sqrt {a +b \sin \left (f x +e \right )}\, \sqrt {c +d \sin \left (f x +e \right )}\, \left (\cos \left (f x +e \right )+1\right )^{2} \left (\cos \left (f x +e \right )-1\right )^{2} \left (\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}\, \sqrt {-c^{2}+d^{2}}\, d -\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}\, c^{2}+\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}\, d^{2}-\cos \left (f x +e \right ) \sqrt {-c^{2}+d^{2}}\, a c +\cos \left (f x +e \right ) \sqrt {-c^{2}+d^{2}}\, b d -\cos \left (f x +e \right ) b \,c^{2}+\cos \left (f x +e \right ) b \,d^{2}+c \sqrt {-a^{2}+b^{2}}\, \sqrt {-c^{2}+d^{2}}\, \sin \left (f x +e \right )+c d \sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )+b c \sqrt {-c^{2}+d^{2}}\, \sin \left (f x +e \right )-a \,c^{2} \sin \left (f x +e \right )+b c d \sin \left (f x +e \right )+d \sqrt {-a^{2}+b^{2}}\, \sqrt {-c^{2}+d^{2}}+d^{2} \sqrt {-a^{2}+b^{2}}+b d \sqrt {-c^{2}+d^{2}}-a c d +d^{2} b \right )}{f \sin \left (f x +e \right )^{4} \left (\left (\cos ^{2}\left (f x +e \right )\right ) b d -\sin \left (f x +e \right ) a d -\sin \left (f x +e \right ) b c -a c -b d \right ) \sqrt {-c^{2}+d^{2}}\, \left (-a \sqrt {-c^{2}+d^{2}}+c \sqrt {-a^{2}+b^{2}}-a d +b c \right )}\)	\(1236\)

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________