\(\int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx\) [21]

Optimal result

Integrand size = 12, antiderivative size = 43 \[ \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx=\frac {2 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{b \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \]

[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)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3856, 2719} \[ \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx=\frac {2 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {\sin (a+b x)} \sqrt {c \csc (a+b x)}} \]

[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 2719

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

Rule 3856

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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx=-\frac {2 E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right )}{b \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \]

[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]])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 3.92 (sec) , antiderivative size = 297, normalized size of antiderivative = 6.91

method	result	size
risch	\(-\frac {i \sqrt {2}}{b \sqrt {\frac {i c \,{\mathrm e}^{i \left (x b +a \right )}}{{\mathrm e}^{2 i \left (x b +a \right )}-1}}}+\frac {i \left (-\frac {2 i \left (i c \,{\mathrm e}^{2 i \left (x b +a \right )}-i c \right )}{c \sqrt {{\mathrm e}^{i \left (x b +a \right )} \left (i c \,{\mathrm e}^{2 i \left (x b +a \right )}-i c \right )}}-\frac {\sqrt {{\mathrm e}^{i \left (x b +a \right )}+1}\, \sqrt {-2 \,{\mathrm e}^{i \left (x b +a \right )}+2}\, \sqrt {-{\mathrm e}^{i \left (x b +a \right )}}\, \left (-2 \operatorname {EllipticE}\left (\sqrt {{\mathrm e}^{i \left (x b +a \right )}+1}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticF}\left (\sqrt {{\mathrm e}^{i \left (x b +a \right )}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {i c \,{\mathrm e}^{3 i \left (x b +a \right )}-i c \,{\mathrm e}^{i \left (x b +a \right )}}}\right ) \sqrt {2}\, \sqrt {i c \,{\mathrm e}^{i \left (x b +a \right )} \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}}{b \sqrt {\frac {i c \,{\mathrm e}^{i \left (x b +a \right )}}{{\mathrm e}^{2 i \left (x b +a \right )}-1}}\, \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}\)	\(297\)
default	\(-\frac {\csc \left (x b +a \right ) \left (2 \cos \left (x b +a \right ) \sqrt {-i \left (i+\cot \left (x b +a \right )-\csc \left (x b +a \right )\right )}\, \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )-\cos \left (x b +a \right ) \sqrt {-i \left (i+\cot \left (x b +a \right )-\csc \left (x b +a \right )\right )}\, \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )+2 \sqrt {-i \left (i+\cot \left (x b +a \right )-\csc \left (x b +a \right )\right )}\, \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {-i \left (i+\cot \left (x b +a \right )-\csc \left (x b +a \right )\right )}\, \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )+\cos \left (x b +a \right ) \sqrt {2}-\sqrt {2}\right ) \sqrt {2}}{b \sqrt {c \csc \left (x b +a \right )}}\)	\(429\)

[In]

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

[Out]

-I/b*2^(1/2)/(I*c*exp(I*(b*x+a))/(exp(I*(b*x+a))^2-1))^(1/2)+I/b*(-2*I*(I*c*exp(I*(b*x+a))^2-I*c)/c/(exp(I*(b*
x+a))*(I*c*exp(I*(b*x+a))^2-I*c))^(1/2)-(exp(I*(b*x+a))+1)^(1/2)*(-2*exp(I*(b*x+a))+2)^(1/2)*(-exp(I*(b*x+a)))
^(1/2)/(I*c*exp(I*(b*x+a))^3-I*c*exp(I*(b*x+a)))^(1/2)*(-2*EllipticE((exp(I*(b*x+a))+1)^(1/2),1/2*2^(1/2))+Ell
ipticF((exp(I*(b*x+a))+1)^(1/2),1/2*2^(1/2))))*2^(1/2)/(I*c*exp(I*(b*x+a))/(exp(I*(b*x+a))^2-1))^(1/2)*(I*c*ex
p(I*(b*x+a))*(exp(I*(b*x+a))^2-1))^(1/2)/(exp(I*(b*x+a))^2-1)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx=\frac {\sqrt {2 i \, c} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + \sqrt {-2 i \, c} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}{b c} \]

[In]

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

[Out]

(sqrt(2*I*c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*x + a) + I*sin(b*x + a))) + sqrt(-2*I*c)*we
ierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*x + a) - I*sin(b*x + a))))/(b*c)

Sympy [F]

\[ \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx=\int \frac {1}{\sqrt {c \csc {\left (a + b x \right )}}}\, dx \]

[In]

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

[Out]

Integral(1/sqrt(c*csc(a + b*x)), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx=\int { \frac {1}{\sqrt {c \csc \left (b x + a\right )}} \,d x } \]

[In]

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

[Out]

integrate(1/sqrt(c*csc(b*x + a)), x)

Giac [F]

\[ \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx=\int { \frac {1}{\sqrt {c \csc \left (b x + a\right )}} \,d x } \]

[In]

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

[Out]

integrate(1/sqrt(c*csc(b*x + a)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx=\int \frac {1}{\sqrt {\frac {c}{\sin \left (a+b\,x\right )}}} \,d x \]

[In]

int(1/(c/sin(a + b*x))^(1/2),x)

[Out]

int(1/(c/sin(a + b*x))^(1/2), x)