Integrand size = 21, antiderivative size = 70 \[ \int \frac {\csc ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=-\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{d f}-\frac {2 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \] Output:
-2*cos(f*x+e)*(d*csc(f*x+e))^(1/2)/d/f+2*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)
Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.74 \[ \int \frac {\csc ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\frac {-2 \cot (e+f x)+\frac {2 E\left (\left .\frac {1}{4} (-2 e+\pi -2 f x)\right |2\right )}{\sqrt {\sin (e+f x)}}}{f \sqrt {d \csc (e+f x)}} \] Input:
Integrate[Csc[e + f*x]^2/Sqrt[d*Csc[e + f*x]],x]
Output:
(-2*Cot[e + f*x] + (2*EllipticE[(-2*e + Pi - 2*f*x)/4, 2])/Sqrt[Sin[e + f* x]])/(f*Sqrt[d*Csc[e + f*x]])
Time = 0.58 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2030, 3042, 4255, 3042, 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.
\(\displaystyle \int \frac {\csc ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle \frac {\int (d \csc (e+f x))^{3/2}dx}{d^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int (d \csc (e+f x))^{3/2}dx}{d^2}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {d^2 \left (-\int \frac {1}{\sqrt {d \csc (e+f x)}}dx\right )-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}}{d^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {d^2 \left (-\int \frac {1}{\sqrt {d \csc (e+f x)}}dx\right )-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}}{d^2}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {-\frac {d^2 \int \sqrt {\sin (e+f x)}dx}{\sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}}{d^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {d^2 \int \sqrt {\sin (e+f x)}dx}{\sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}}{d^2}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {-\frac {2 d^2 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}}{d^2}\) |
Input:
Int[Csc[e + f*x]^2/Sqrt[d*Csc[e + f*x]],x]
Output:
((-2*d*Cos[e + f*x]*Sqrt[d*Csc[e + f*x]])/f - (2*d^2*EllipticE[(e - Pi/2 + f*x)/2, 2])/(f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]]))/d^2
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)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[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 = 0.73 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.30
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (\sqrt {2}+\left (\operatorname {EllipticF}\left (\sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )-2 \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {1+i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \sqrt {1-i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \sqrt {-i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \left (\cos \left (f x +e \right )+1\right )\right ) \csc \left (f x +e \right )}{f \sqrt {d \csc \left (f x +e \right )}}\) | \(161\) |
Input:
int(csc(f*x+e)^2/(d*csc(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/f*2^(1/2)*(2^(1/2)+(EllipticF((1+I*cot(f*x+e)-I*csc(f*x+e))^(1/2),1/2*2 ^(1/2))-2*EllipticE((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))/(d*csc(f*x+e))^(1/2)*csc(f *x+e)
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.19 \[ \int \frac {\csc ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=-\frac {2 \, \sqrt {\frac {d}{\sin \left (f x + e\right )}} \cos \left (f x + e\right ) + \sqrt {2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + \sqrt {-2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{d f} \] Input:
integrate(csc(f*x+e)^2/(d*csc(f*x+e))^(1/2),x, algorithm="fricas")
Output:
-(2*sqrt(d/sin(f*x + e))*cos(f*x + e) + sqrt(2*I*d)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(f*x + e) + I*sin(f*x + e))) + sqrt(-2*I*d)*w eierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(f*x + e) - I*sin(f*x + e))))/(d*f)
\[ \int \frac {\csc ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int \frac {\csc ^{2}{\left (e + f x \right )}}{\sqrt {d \csc {\left (e + f x \right )}}}\, dx \] Input:
integrate(csc(f*x+e)**2/(d*csc(f*x+e))**(1/2),x)
Output:
Integral(csc(e + f*x)**2/sqrt(d*csc(e + f*x)), x)
\[ \int \frac {\csc ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int { \frac {\csc \left (f x + e\right )^{2}}{\sqrt {d \csc \left (f x + e\right )}} \,d x } \] Input:
integrate(csc(f*x+e)^2/(d*csc(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate(csc(f*x + e)^2/sqrt(d*csc(f*x + e)), x)
\[ \int \frac {\csc ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int { \frac {\csc \left (f x + e\right )^{2}}{\sqrt {d \csc \left (f x + e\right )}} \,d x } \] Input:
integrate(csc(f*x+e)^2/(d*csc(f*x+e))^(1/2),x, algorithm="giac")
Output:
integrate(csc(f*x + e)^2/sqrt(d*csc(f*x + e)), x)
Timed out. \[ \int \frac {\csc ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^2\,\sqrt {\frac {d}{\sin \left (e+f\,x\right )}}} \,d x \] Input:
int(1/(sin(e + f*x)^2*(d/sin(e + f*x))^(1/2)),x)
Output:
int(1/(sin(e + f*x)^2*(d/sin(e + f*x))^(1/2)), x)
\[ \int \frac {\csc ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\frac {\sqrt {d}\, \left (\int \sqrt {\csc \left (f x +e \right )}\, \csc \left (f x +e \right )d x \right )}{d} \] Input:
int(csc(f*x+e)^2/(d*csc(f*x+e))^(1/2),x)
Output:
(sqrt(d)*int(sqrt(csc(e + f*x))*csc(e + f*x),x))/d