\(\int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx\) [510]

Optimal result

Integrand size = 19, antiderivative size = 44 \[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\frac {2 d E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \]

[Out]

-2*d*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)
)/f/(d*csc(f*x+e))^(1/2)/sin(f*x+e)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {16, 3856, 2719} \[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\frac {2 d E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}} \]

[In]

Int[Sqrt[d*Csc[e + f*x]]*Sin[e + f*x],x]

[Out]

(2*d*EllipticE[(e - Pi/2 + f*x)/2, 2])/(f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=-\frac {2 d E\left (\left .\frac {1}{4} (-2 e+\pi -2 f x)\right |2\right )}{f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \]

[In]

Integrate[Sqrt[d*Csc[e + f*x]]*Sin[e + f*x],x]

[Out]

(-2*d*EllipticE[(-2*e + Pi - 2*f*x)/4, 2])/(f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.95 (sec) , antiderivative size = 314, normalized size of antiderivative = 7.14

method	result	size
risch	\(-\frac {\left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right ) \sqrt {2}\, \sqrt {\frac {i d \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}-1}}\, {\mathrm e}^{-i \left (f x +e \right )}}{f}+\frac {\left (-\frac {2 i \left (i d \,{\mathrm e}^{2 i \left (f x +e \right )}-i d \right )}{d \sqrt {{\mathrm e}^{i \left (f x +e \right )} \left (i d \,{\mathrm e}^{2 i \left (f x +e \right )}-i d \right )}}-\frac {\sqrt {{\mathrm e}^{i \left (f x +e \right )}+1}\, \sqrt {-2 \,{\mathrm e}^{i \left (f x +e \right )}+2}\, \sqrt {-{\mathrm e}^{i \left (f x +e \right )}}\, \left (-2 E\left (\sqrt {{\mathrm e}^{i \left (f x +e \right )}+1}, \frac {\sqrt {2}}{2}\right )+F\left (\sqrt {{\mathrm e}^{i \left (f x +e \right )}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {i d \,{\mathrm e}^{3 i \left (f x +e \right )}-i d \,{\mathrm e}^{i \left (f x +e \right )}}}\right ) \sqrt {2}\, \sqrt {\frac {i d \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}-1}}\, \sqrt {i d \,{\mathrm e}^{i \left (f x +e \right )} \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}\, {\mathrm e}^{-i \left (f x +e \right )}}{f}\)	\(314\)
default	\(-\frac {\sqrt {2}\, \sqrt {d \csc \left (f x +e \right )}\, \left (2 \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )-\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+2 \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \cos \left (f x +e \right )-\sqrt {2}\right )}{f}\)	\(423\)

[In]

int(sin(f*x+e)*(d*csc(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-(exp(I*(f*x+e))^2-1)/f*2^(1/2)*(I*d*exp(I*(f*x+e))/(exp(I*(f*x+e))^2-1))^(1/2)/exp(I*(f*x+e))+1/f*(-2*I*(I*d*
exp(I*(f*x+e))^2-I*d)/d/(exp(I*(f*x+e))*(I*d*exp(I*(f*x+e))^2-I*d))^(1/2)-(exp(I*(f*x+e))+1)^(1/2)*(-2*exp(I*(
f*x+e))+2)^(1/2)*(-exp(I*(f*x+e)))^(1/2)/(I*d*exp(I*(f*x+e))^3-I*d*exp(I*(f*x+e)))^(1/2)*(-2*EllipticE((exp(I*
(f*x+e))+1)^(1/2),1/2*2^(1/2))+EllipticF((exp(I*(f*x+e))+1)^(1/2),1/2*2^(1/2))))*2^(1/2)*(I*d*exp(I*(f*x+e))/(
exp(I*(f*x+e))^2-1))^(1/2)*(I*d*exp(I*(f*x+e))*(exp(I*(f*x+e))^2-1))^(1/2)/exp(I*(f*x+e))

Fricas [C] (verification not implemented)

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

Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.34 \[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\frac {\sqrt {2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + \sqrt {-2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{f} \]

[In]

integrate(sin(f*x+e)*(d*csc(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

(sqrt(2*I*d)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(f*x + e) + I*sin(f*x + e))) + sqrt(-2*I*d)*we
ierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(f*x + e) - I*sin(f*x + e))))/f

Sympy [F]

\[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\int \sqrt {d \csc {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx \]

[In]

integrate(sin(f*x+e)*(d*csc(f*x+e))**(1/2),x)

[Out]

Integral(sqrt(d*csc(e + f*x))*sin(e + f*x), x)

Maxima [F]

\[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\int { \sqrt {d \csc \left (f x + e\right )} \sin \left (f x + e\right ) \,d x } \]

[In]

integrate(sin(f*x+e)*(d*csc(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(d*csc(f*x + e))*sin(f*x + e), x)

Giac [F]

\[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\int { \sqrt {d \csc \left (f x + e\right )} \sin \left (f x + e\right ) \,d x } \]

[In]

integrate(sin(f*x+e)*(d*csc(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(d*csc(f*x + e))*sin(f*x + e), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\int \sin \left (e+f\,x\right )\,\sqrt {\frac {d}{\sin \left (e+f\,x\right )}} \,d x \]

[In]

int(sin(e + f*x)*(d/sin(e + f*x))^(1/2),x)

[Out]

int(sin(e + f*x)*(d/sin(e + f*x))^(1/2), x)