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

Optimal. Leaf size=43 \[ \frac {2 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {\sin (a+b x)} \sqrt {c \csc (a+b x)}} \]

[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))/
b/(c*csc(b*x+a))^(1/2)/sin(b*x+a)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3771, 2639} \[ \frac {2 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {\sin (a+b x)} \sqrt {c \csc (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[c*Csc[a + b*x]],x]

[Out]

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

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

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Mathematica [A]  time = 0.03, size = 42, normalized size = 0.98 \[ -\frac {2 E\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )}{b \sqrt {\sin (a+b x)} \sqrt {c \csc (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[c*Csc[a + b*x]],x]

[Out]

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

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fricas [F]  time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c \csc \left (b x + a\right )}}{c \csc \left (b x + a\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*csc(b*x + a))/(c*csc(b*x + a)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(c*csc(b*x + a)), x)

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maple [C]  time = 1.07, size = 533, normalized size = 12.40 \[ -\frac {\left (2 \cos \left (b x +a \right ) \sqrt {-\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}}\, \sqrt {-\frac {i \cos \left (b x +a \right )-i-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticE \left (\sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}-\cos \left (b x +a \right ) \sqrt {-\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}}\, \sqrt {-\frac {i \cos \left (b x +a \right )-i-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}+2 \sqrt {-\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {-\frac {i \cos \left (b x +a \right )-i-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticE \left (\sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )-\sqrt {-\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {-\frac {i \cos \left (b x +a \right )-i-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+\cos \left (b x +a \right ) \sqrt {2}-\sqrt {2}\right ) \sqrt {2}}{b \sqrt {\frac {c}{\sin \left (b x +a \right )}}\, \sin \left (b x +a \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/b*(2*cos(b*x+a)*(-I*(-1+cos(b*x+a))/sin(b*x+a))^(1/2)*(-(I*cos(b*x+a)-I-sin(b*x+a))/sin(b*x+a))^(1/2)*Ellip
ticE(((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))*((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2
)-cos(b*x+a)*(-I*(-1+cos(b*x+a))/sin(b*x+a))^(1/2)*(-(I*cos(b*x+a)-I-sin(b*x+a))/sin(b*x+a))^(1/2)*EllipticF((
(I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))*((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2)+2*(-
I*(-1+cos(b*x+a))/sin(b*x+a))^(1/2)*((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2)*(-(I*cos(b*x+a)-I-sin(b*x+a
))/sin(b*x+a))^(1/2)*EllipticE(((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))-(-I*(-1+cos(b*x+a))
/sin(b*x+a))^(1/2)*((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2)*(-(I*cos(b*x+a)-I-sin(b*x+a))/sin(b*x+a))^(1
/2)*EllipticF(((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))+cos(b*x+a)*2^(1/2)-2^(1/2))/(c/sin(b
*x+a))^(1/2)/sin(b*x+a)*2^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(c*csc(b*x + a)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c/sin(a + b*x))^(1/2),x)

[Out]

int(1/(c/sin(a + b*x))^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(c*csc(a + b*x)), x)

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