3.13 \(\int \frac{1}{\sqrt{\sin (b x)}} \, dx\)

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

[Out]

(-2*EllipticF[Pi/4 - (b*x)/2, 2])/b

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Rubi [A]  time = 0.008046, antiderivative size = 19, normalized size of antiderivative = 1., 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 = {2641} \[ -\frac{2 F\left (\left .\frac{\pi }{4}-\frac{b x}{2}\right |2\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*EllipticF[Pi/4 - (b*x)/2, 2])/b

Rule 2641

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

Mathematica [A]  time = 0.024718, size = 21, normalized size = 1.11 \[ -\frac{2 F\left (\left .\frac{1}{2} \left (\frac{\pi }{2}-b x\right )\right |2\right )}{b} \]

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 = 0.034, size = 57, normalized size = 3. \begin{align*}{\frac{1}{b\cos \left ( bx \right ) }\sqrt{\sin \left ( bx \right ) +1}\sqrt{-2\,\sin \left ( bx \right ) +2}\sqrt{-\sin \left ( bx \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( bx \right ) +1},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{\sin \left ( bx \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\sin \left (b x\right )}}\,{d x} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{\sin \left (b x\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(sin(b*x)), x)

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

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\sin \left (b x\right )}}\,{d x} \end{align*}

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)