3.1.13 \(\int \frac {1}{\sqrt {\sin (b x)}} \, dx\) [13]

Optimal. Leaf size=19 \[ -\frac {2 F\left (\left .\frac {\pi }{4}-\frac {b x}{2}\right |2\right )}{b} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2720} \begin {gather*} -\frac {2 F\left (\left .\frac {\pi }{4}-\frac {b x}{2}\right |2\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Sin[b*x]],x]

[Out]

(-2*EllipticF[Pi/4 - (b*x)/2, 2])/b

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {\sin (b x)}} \, dx &=-\frac {2 F\left (\left .\frac {\pi }{4}-\frac {b x}{2}\right |2\right )}{b}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 21, normalized size = 1.11 \begin {gather*} -\frac {2 F\left (\left .\frac {1}{2} \left (\frac {\pi }{2}-b x\right )\right |2\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[Sin[b*x]],x]

[Out]

(-2*EllipticF[(Pi/2 - b*x)/2, 2])/b

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Maple [A]
time = 0.05, size = 57, normalized size = 3.00

method	result	size
default	\(\frac {\sqrt {\sin \left (b x \right )+1}\, \sqrt {-2 \sin \left (b x \right )+2}\, \sqrt {-\sin \left (b x \right )}\, \EllipticF \left (\sqrt {\sin \left (b x \right )+1}, \frac {\sqrt {2}}{2}\right )}{\cos \left (b x \right ) \sqrt {\sin \left (b x \right )}\, b}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(b*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

(sin(b*x)+1)^(1/2)*(-2*sin(b*x)+2)^(1/2)*(-sin(b*x))^(1/2)*EllipticF((sin(b*x)+1)^(1/2),1/2*2^(1/2))/cos(b*x)/
sin(b*x)^(1/2)/b

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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/sin(b*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(sin(b*x)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 47, normalized size = 2.47 \begin {gather*} \frac {\sqrt {2} \sqrt {-i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x\right ) + i \, \sin \left (b x\right )\right ) + \sqrt {2} \sqrt {i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x\right ) - i \, \sin \left (b x\right )\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(2)*sqrt(-I)*weierstrassPInverse(4, 0, cos(b*x) + I*sin(b*x)) + sqrt(2)*sqrt(I)*weierstrassPInverse(4, 0,
cos(b*x) - I*sin(b*x)))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(b*x)**(1/2),x)

[Out]

Integral(1/sqrt(sin(b*x)), x)

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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/sin(b*x)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(sin(b*x)), x)

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Mupad [B]
time = 0.39, size = 15, normalized size = 0.79 \begin {gather*} -\frac {2\,\mathrm {F}\left (\frac {\pi }{4}-\frac {b\,x}{2}\middle |2\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(b*x)^(1/2),x)

[Out]

-(2*ellipticF(pi/4 - (b*x)/2, 2))/b

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