3.13 \(\int \frac {1}{\sqrt {\csc (a+b x)}} \, dx\)

Optimal. Leaf size=41 \[ \frac {2 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3771, 2639} \[ \frac {2 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Csc[a + b*x]],x]

[Out]

(2*Sqrt[Csc[a + b*x]]*EllipticE[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/b

Rule 2639

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

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 40, normalized size = 0.98 \[ -\frac {2 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[Csc[a + b*x]],x]

[Out]

(-2*Sqrt[Csc[a + b*x]]*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]])/b

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fricas [F]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {\csc \left (b x + a\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(csc(b*x + a)), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\csc \left (b x + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(csc(b*x + a)), x)

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maple [A]  time = 2.65, size = 91, normalized size = 2.22 \[ -\frac {\sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \left (2 \EllipticE \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-\EllipticF \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )\right )}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/csc(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)*(2*EllipticE((sin(b*x+a)+1)^(1/2),1/2*2^(1/2
))-EllipticF((sin(b*x+a)+1)^(1/2),1/2*2^(1/2)))/cos(b*x+a)/sin(b*x+a)^(1/2)/b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\csc \left (b x + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(csc(b*x + a)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {\frac {1}{\sin \left (a+b\,x\right )}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/sin(a + b*x))^(1/2),x)

[Out]

int(1/(1/sin(a + b*x))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\csc {\left (a + b x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(csc(a + b*x)), x)

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