Integrand size = 19, antiderivative size = 74 \[ \int \frac {\sin (e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=-\frac {2 \cos (e+f x)}{3 f \sqrt {d \csc (e+f x)}}+\frac {2 \sqrt {d \csc (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),2\right ) \sqrt {\sin (e+f x)}}{3 d f} \] Output:
-2/3*cos(f*x+e)/f/(d*csc(f*x+e))^(1/2)+2/3*(d*csc(f*x+e))^(1/2)*InverseJac obiAM(1/2*e-1/4*Pi+1/2*f*x,2^(1/2))*sin(f*x+e)^(1/2)/d/f
Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.86 \[ \int \frac {\sin (e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=-\frac {d \csc ^2(e+f x) \left (2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),2\right ) \sqrt {\sin (e+f x)}+\sin (2 (e+f x))\right )}{3 f (d \csc (e+f x))^{3/2}} \] Input:
Integrate[Sin[e + f*x]/Sqrt[d*Csc[e + f*x]],x]
Output:
-1/3*(d*Csc[e + f*x]^2*(2*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]] + Sin[2*(e + f*x)]))/(f*(d*Csc[e + f*x])^(3/2))
Time = 0.58 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3042, 2030, 4256, 3042, 4258, 3042, 3120}
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.
\(\displaystyle \int \frac {\sin (e+f x)}{\sqrt {d \csc (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\csc (e+f x) \sqrt {d \csc (e+f x)}}dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle d \int \frac {1}{(d \csc (e+f x))^{3/2}}dx\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle d \left (\frac {\int \sqrt {d \csc (e+f x)}dx}{3 d^2}-\frac {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle d \left (\frac {\int \sqrt {d \csc (e+f x)}dx}{3 d^2}-\frac {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}}\right )\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle d \left (\frac {\sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)} \int \frac {1}{\sqrt {\sin (e+f x)}}dx}{3 d^2}-\frac {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle d \left (\frac {\sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)} \int \frac {1}{\sqrt {\sin (e+f x)}}dx}{3 d^2}-\frac {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}}\right )\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle d \left (\frac {2 \sqrt {\sin (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),2\right ) \sqrt {d \csc (e+f x)}}{3 d^2 f}-\frac {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}}\right )\) |
Input:
Int[Sin[e + f*x]/Sqrt[d*Csc[e + f*x]],x]
Output:
d*((-2*Cos[e + f*x])/(3*d*f*Sqrt[d*Csc[e + f*x]]) + (2*Sqrt[d*Csc[e + f*x] ]*EllipticF[(e - Pi/2 + f*x)/2, 2]*Sqrt[Sin[e + f*x]])/(3*d^2*f))
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[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]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2* n]
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 = 0.76 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.88
method | result | size |
default | \(\frac {\sqrt {2}\, \left (i \sqrt {i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1-i \cot \left (f x +e \right )+i \csc \left (f x +e \right )}\, \sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-\sqrt {2}\, \cos \left (f x +e \right )\right )}{3 f \sqrt {d \csc \left (f x +e \right )}}\) | \(139\) |
Input:
int(sin(f*x+e)/(d*csc(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3/f*2^(1/2)/(d*csc(f*x+e))^(1/2)*(I*(I*(csc(f*x+e)-cot(f*x+e)))^(1/2)*El lipticF((1+I*cot(f*x+e)-I*csc(f*x+e))^(1/2),1/2*2^(1/2))*(1-I*cot(f*x+e)+I *csc(f*x+e))^(1/2)*(1+I*cot(f*x+e)-I*csc(f*x+e))^(1/2)*(cot(f*x+e)+csc(f*x +e))-2^(1/2)*cos(f*x+e))
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.15 \[ \int \frac {\sin (e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=-\frac {2 \, \sqrt {\frac {d}{\sin \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + i \, \sqrt {2 i \, d} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) - i \, \sqrt {-2 i \, d} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}{3 \, d f} \] Input:
integrate(sin(f*x+e)/(d*csc(f*x+e))^(1/2),x, algorithm="fricas")
Output:
-1/3*(2*sqrt(d/sin(f*x + e))*cos(f*x + e)*sin(f*x + e) + I*sqrt(2*I*d)*wei erstrassPInverse(4, 0, cos(f*x + e) + I*sin(f*x + e)) - I*sqrt(-2*I*d)*wei erstrassPInverse(4, 0, cos(f*x + e) - I*sin(f*x + e)))/(d*f)
\[ \int \frac {\sin (e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int \frac {\sin {\left (e + f x \right )}}{\sqrt {d \csc {\left (e + f x \right )}}}\, dx \] Input:
integrate(sin(f*x+e)/(d*csc(f*x+e))**(1/2),x)
Output:
Integral(sin(e + f*x)/sqrt(d*csc(e + f*x)), x)
\[ \int \frac {\sin (e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )}{\sqrt {d \csc \left (f x + e\right )}} \,d x } \] Input:
integrate(sin(f*x+e)/(d*csc(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate(sin(f*x + e)/sqrt(d*csc(f*x + e)), x)
\[ \int \frac {\sin (e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )}{\sqrt {d \csc \left (f x + e\right )}} \,d x } \] Input:
integrate(sin(f*x+e)/(d*csc(f*x+e))^(1/2),x, algorithm="giac")
Output:
integrate(sin(f*x + e)/sqrt(d*csc(f*x + e)), x)
Timed out. \[ \int \frac {\sin (e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int \frac {\sin \left (e+f\,x\right )}{\sqrt {\frac {d}{\sin \left (e+f\,x\right )}}} \,d x \] 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)
\[ \int \frac {\sin (e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\frac {\sqrt {d}\, \left (\int \frac {\sqrt {\csc \left (f x +e \right )}\, \sin \left (f x +e \right )}{\csc \left (f x +e \right )}d x \right )}{d} \] Input:
int(sin(f*x+e)/(d*csc(f*x+e))^(1/2),x)
Output:
(sqrt(d)*int((sqrt(csc(e + f*x))*sin(e + f*x))/csc(e + f*x),x))/d