3.3.8 \(\int \frac {1}{\sqrt {a+b \sin (e+f x)}} \, dx\) [208]

Optimal. Leaf size=62 \[ \frac {2 F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{f \sqrt {a+b \sin (e+f x)}} \]

[Out]

-2*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(
b/(a+b))^(1/2))*((a+b*sin(f*x+e))/(a+b))^(1/2)/f/(a+b*sin(f*x+e))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2742, 2740} \begin {gather*} \frac {2 \sqrt {\frac {a+b \sin (e+f x)}{a+b}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{f \sqrt {a+b \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b*Sin[e + f*x]],x]

[Out]

(2*EllipticF[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]
])

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.04, size = 61, normalized size = 0.98 \begin {gather*} -\frac {2 F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{f \sqrt {a+b \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + b*Sin[e + f*x]],x]

[Out]

(-2*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e +
f*x]])

________________________________________________________________________________________

Maple [A]
time = 1.98, size = 126, normalized size = 2.03

method	result	size
default	\(\frac {2 \left (a -b \right ) \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (f x +e \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )}{b \cos \left (f x +e \right ) \sqrt {a +b \sin \left (f x +e \right )}\, f}\)	\(126\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(a-b)*((a+b*sin(f*x+e))/(a-b))^(1/2)*(-(sin(f*x+e)-1)*b/(a+b))^(1/2)*(-(1+sin(f*x+e))*b/(a-b))^(1/2)*Ellipti
cF(((a+b*sin(f*x+e))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))/b/cos(f*x+e)/(a+b*sin(f*x+e))^(1/2)/f

________________________________________________________________________________________

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),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.11, size = 152, normalized size = 2.45 \begin {gather*} \frac {\sqrt {2} \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) - 3 i \, b \sin \left (f x + e\right ) - 2 i \, a}{3 \, b}\right ) + \sqrt {2} \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) + 3 i \, b \sin \left (f x + e\right ) + 2 i \, a}{3 \, b}\right )}{b f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(2)*sqrt(I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*cos
(f*x + e) - 3*I*b*sin(f*x + e) - 2*I*a)/b) + sqrt(2)*sqrt(-I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2,
-8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(f*x + e) + 3*I*b*sin(f*x + e) + 2*I*a)/b))/(b*f)

________________________________________________________________________________________

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 )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a + b*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),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 6.91, size = 55, normalized size = 0.89 \begin {gather*} -\frac {2\,\mathrm {F}\left (\frac {\pi }{4}-\frac {e}{2}-\frac {f\,x}{2}\middle |\frac {2\,b}{a+b}\right )\,\sqrt {\frac {a+b\,\sin \left (e+f\,x\right )}{a+b}}}{f\,\sqrt {a+b\,\sin \left (e+f\,x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*sin(e + f*x))^(1/2),x)

[Out]

-(2*ellipticF(pi/4 - e/2 - (f*x)/2, (2*b)/(a + b))*((a + b*sin(e + f*x))/(a + b))^(1/2))/(f*(a + b*sin(e + f*x
))^(1/2))

________________________________________________________________________________________