Integrand size = 19, antiderivative size = 44 \[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\frac {2 d E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \] Output:
-2*d*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2))/f/(d*csc(f*x+e))^(1/2)/s in(f*x+e)^(1/2)
Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=-\frac {2 d E\left (\left .\frac {1}{4} (-2 e+\pi -2 f x)\right |2\right )}{f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \] Input:
Integrate[Sqrt[d*Csc[e + f*x]]*Sin[e + f*x],x]
Output:
(-2*d*EllipticE[(-2*e + Pi - 2*f*x)/4, 2])/(f*Sqrt[d*Csc[e + f*x]]*Sqrt[Si n[e + f*x]])
Time = 0.43 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3042, 2030, 4258, 3042, 3119}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sin (e+f x) \sqrt {d \csc (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {d \csc (e+f x)}}{\csc (e+f x)}dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle d \int \frac {1}{\sqrt {d \csc (e+f x)}}dx\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {d \int \sqrt {\sin (e+f x)}dx}{\sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {d \int \sqrt {\sin (e+f x)}dx}{\sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {2 d E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}\) |
Input:
Int[Sqrt[d*Csc[e + f*x]]*Sin[e + f*x],x]
Output:
(2*d*EllipticE[(e - Pi/2 + f*x)/2, 2])/(f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])
Int[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m Int[(b*v) ^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
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]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(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]
Result contains complex when optimal does not.
Time = 0.89 (sec) , antiderivative size = 237, normalized size of antiderivative = 5.39
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (2 \sqrt {1+i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \sqrt {1-i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \sqrt {-i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \left (\cos \left (f x +e \right )+1\right ) \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {1+i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \sqrt {1-i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \sqrt {-i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \left (-\cos \left (f x +e \right )-1\right ) \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \left (-1+\cos \left (f x +e \right )\right )\right ) \sqrt {d \csc \left (f x +e \right )}}{f}\) | \(237\) |
risch | \(-\frac {\left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right ) \sqrt {2}\, \sqrt {\frac {i d \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}-1}}\, {\mathrm e}^{-i \left (f x +e \right )}}{f}+\frac {\left (-\frac {2 i \left (i d \,{\mathrm e}^{2 i \left (f x +e \right )}-i d \right )}{d \sqrt {{\mathrm e}^{i \left (f x +e \right )} \left (i d \,{\mathrm e}^{2 i \left (f x +e \right )}-i d \right )}}-\frac {\sqrt {{\mathrm e}^{i \left (f x +e \right )}+1}\, \sqrt {-2 \,{\mathrm e}^{i \left (f x +e \right )}+2}\, \sqrt {-{\mathrm e}^{i \left (f x +e \right )}}\, \left (-2 \operatorname {EllipticE}\left (\sqrt {{\mathrm e}^{i \left (f x +e \right )}+1}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticF}\left (\sqrt {{\mathrm e}^{i \left (f x +e \right )}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {i d \,{\mathrm e}^{3 i \left (f x +e \right )}-i d \,{\mathrm e}^{i \left (f x +e \right )}}}\right ) \sqrt {2}\, \sqrt {\frac {i d \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}-1}}\, \sqrt {i d \,{\mathrm e}^{i \left (f x +e \right )} \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}\, {\mathrm e}^{-i \left (f x +e \right )}}{f}\) | \(314\) |
Input:
int((d*csc(f*x+e))^(1/2)*sin(f*x+e),x,method=_RETURNVERBOSE)
Output:
-1/f*2^(1/2)*(2*(1+I*(-csc(f*x+e)+cot(f*x+e)))^(1/2)*(1-I*(-csc(f*x+e)+cot (f*x+e)))^(1/2)*(-I*(-csc(f*x+e)+cot(f*x+e)))^(1/2)*(cos(f*x+e)+1)*Ellipti cE((1+I*cot(f*x+e)-I*csc(f*x+e))^(1/2),1/2*2^(1/2))+(1+I*(-csc(f*x+e)+cot( f*x+e)))^(1/2)*(1-I*(-csc(f*x+e)+cot(f*x+e)))^(1/2)*(-I*(-csc(f*x+e)+cot(f *x+e)))^(1/2)*(-cos(f*x+e)-1)*EllipticF((1+I*cot(f*x+e)-I*csc(f*x+e))^(1/2 ),1/2*2^(1/2))+2^(1/2)*(-1+cos(f*x+e)))*(d*csc(f*x+e))^(1/2)
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.34 \[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\frac {\sqrt {2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + \sqrt {-2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{f} \] Input:
integrate((d*csc(f*x+e))^(1/2)*sin(f*x+e),x, algorithm="fricas")
Output:
(sqrt(2*I*d)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(f*x + e) + I*sin(f*x + e))) + sqrt(-2*I*d)*weierstrassZeta(4, 0, weierstrassPInvers e(4, 0, cos(f*x + e) - I*sin(f*x + e))))/f
\[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\int \sqrt {d \csc {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx \] Input:
integrate((d*csc(f*x+e))**(1/2)*sin(f*x+e),x)
Output:
Integral(sqrt(d*csc(e + f*x))*sin(e + f*x), x)
\[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\int { \sqrt {d \csc \left (f x + e\right )} \sin \left (f x + e\right ) \,d x } \] Input:
integrate((d*csc(f*x+e))^(1/2)*sin(f*x+e),x, algorithm="maxima")
Output:
integrate(sqrt(d*csc(f*x + e))*sin(f*x + e), x)
\[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\int { \sqrt {d \csc \left (f x + e\right )} \sin \left (f x + e\right ) \,d x } \] Input:
integrate((d*csc(f*x+e))^(1/2)*sin(f*x+e),x, algorithm="giac")
Output:
integrate(sqrt(d*csc(f*x + e))*sin(f*x + e), x)
Timed out. \[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\int \sin \left (e+f\,x\right )\,\sqrt {\frac {d}{\sin \left (e+f\,x\right )}} \,d x \] Input:
int(sin(e + f*x)*(d/sin(e + f*x))^(1/2),x)
Output:
int(sin(e + f*x)*(d/sin(e + f*x))^(1/2), x)
\[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\sqrt {d}\, \left (\int \sqrt {\csc \left (f x +e \right )}\, \sin \left (f x +e \right )d x \right ) \] Input:
int((d*csc(f*x+e))^(1/2)*sin(f*x+e),x)
Output:
sqrt(d)*int(sqrt(csc(e + f*x))*sin(e + f*x),x)