\(\int \sqrt {\csc (a+b x)} \, dx\) [12]

Optimal result

Integrand size = 10, antiderivative size = 41 \[ \int \sqrt {\csc (a+b x)} \, dx=\frac {2 \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{b} \]

[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)*EllipticF(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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3856, 2720} \[ \int \sqrt {\csc (a+b x)} \, dx=\frac {2 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{b} \]

[In]

Int[Sqrt[Csc[a + b*x]],x]

[Out]

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

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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}& = \left (\sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx \\ & = \frac {2 \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.98 \[ \int \sqrt {\csc (a+b x)} \, dx=-\frac {2 \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right ) \sqrt {\sin (a+b x)}}{b} \]

[In]

Integrate[Sqrt[Csc[a + b*x]],x]

[Out]

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

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.68

method	result	size
default	\(\frac {\sqrt {\sin \left (x b +a \right )+1}\, \sqrt {-2 \sin \left (x b +a \right )+2}\, \sqrt {-\sin \left (x b +a \right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (x b +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{\cos \left (x b +a \right ) \sqrt {\sin \left (x b +a \right )}\, b}\)	\(69\)

[In]

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

[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

Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.24 \[ \int \sqrt {\csc (a+b x)} \, dx=\frac {-i \, \sqrt {2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + i \, \sqrt {-2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{b} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(csc(a + b*x)), x)

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(csc(b*x + a)), x)

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(csc(b*x + a)), x)

Mupad [B] (verification not implemented)

Time = 21.91 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.49 \[ \int \sqrt {\csc (a+b x)} \, dx=-\frac {2\,\sqrt {\sin \left (a+b\,x\right )}\,\mathrm {F}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (a+b\,x\right )}}{2}\right )\middle |2\right )\,\sqrt {{\cos \left (a+b\,x\right )}^2}\,\sqrt {\frac {1}{\sin \left (a+b\,x\right )}}}{b\,\cos \left (a+b\,x\right )} \]

[In]

int((1/sin(a + b*x))^(1/2),x)

[Out]

-(2*sin(a + b*x)^(1/2)*ellipticF(asin((2^(1/2)*(1 - sin(a + b*x))^(1/2))/2), 2)*(cos(a + b*x)^2)^(1/2)*(1/sin(
a + b*x))^(1/2))/(b*cos(a + b*x))