3.21 \(\int \frac{1}{\sqrt{\sin (a+b x)}} \, dx\)

Optimal. Leaf size=21 \[ \frac{2 F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{b} \]

[Out]

(2*EllipticF[(a - Pi/2 + b*x)/2, 2])/b

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Rubi [A]  time = 0.0077145, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2641} \[ \frac{2 F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Sin[a + b*x]],x]

[Out]

(2*EllipticF[(a - Pi/2 + 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 (a+b x)}} \, dx &=\frac{2 F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right )}{b}\\ \end{align*}

Mathematica [A]  time = 0.0185693, size = 24, normalized size = 1.14 \[ -\frac{2 F\left (\left .\frac{1}{2} \left (-a-b x+\frac{\pi }{2}\right )\right |2\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[Sin[a + b*x]],x]

[Out]

(-2*EllipticF[(-a + Pi/2 - b*x)/2, 2])/b

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Maple [A]  time = 0.023, size = 69, normalized size = 3.3 \begin{align*}{\frac{1}{b\cos \left ( bx+a \right ) }\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( bx+a \right ) +1},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{\sin \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin(b*x+a))^(1/2)*EllipticF((sin(b*x+a)+1)^(1/2),1/2*2^(1/2))/c
os(b*x+a)/sin(b*x+a)^(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 + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(sin(b*x + a)), 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 + a\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(sin(b*x + a)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(sin(a + 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 + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(sin(b*x + a)), x)