\(\int \sqrt {c \csc (a+b x)} \, dx\) [20]

Optimal result

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

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

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, 2720} \[ \int \sqrt {c \csc (a+b x)} \, dx=\frac {2 \sqrt {\sin (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right ) \sqrt {c \csc (a+b x)}}{b} \]

[In]

Int[Sqrt[c*Csc[a + b*x]],x]

[Out]

(2*Sqrt[c*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 {c \csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx \\ & = \frac {2 \sqrt {c \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 = 42, normalized size of antiderivative = 0.98 \[ \int \sqrt {c \csc (a+b x)} \, dx=-\frac {2 \sqrt {c \csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right ) \sqrt {\sin (a+b x)}}{b} \]

[In]

Integrate[Sqrt[c*Csc[a + b*x]],x]

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 3.44 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.79

method	result	size
default	\(\frac {i \left (1+\cos \left (x b +a \right )\right ) \sqrt {2}\, \sqrt {c \csc \left (x b +a \right )}\, \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {-i \left (i+\cot \left (x b +a \right )-\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )}{b}\)	\(120\)

[In]

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

[Out]

I/b*(1+cos(b*x+a))*2^(1/2)*(c*csc(b*x+a))^(1/2)*(-I*(I-cot(b*x+a)+csc(b*x+a)))^(1/2)*(-I*(I+cot(b*x+a)-csc(b*x
+a)))^(1/2)*(I*(csc(b*x+a)-cot(b*x+a)))^(1/2)*EllipticF((-I*(I-cot(b*x+a)+csc(b*x+a)))^(1/2),1/2*2^(1/2))

Fricas [C] (verification not implemented)

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

Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.28 \[ \int \sqrt {c \csc (a+b x)} \, dx=\frac {-i \, \sqrt {2 i \, c} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + i \, \sqrt {-2 i \, c} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{b} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(c*csc(a + b*x)), x)

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(c*csc(b*x + a)), x)

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(c*csc(b*x + a)), x)

Mupad [B] (verification not implemented)

Time = 22.32 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.47 \[ \int \sqrt {c \csc (a+b x)} \, dx=-\frac {2\,\sqrt {\sin \left (a+b\,x\right )}\,\mathrm {F}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (a+b\,x\right )}}{2}\right )\middle |2\right )\,\sqrt {\frac {c}{\sin \left (a+b\,x\right )}}\,\sqrt {{\cos \left (a+b\,x\right )}^2}}{b\,\cos \left (a+b\,x\right )} \]

[In]

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