Integrand size = 21, antiderivative size = 74 \[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=-\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d f}+\frac {2 \sqrt {d \csc (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),2\right ) \sqrt {\sin (e+f x)}}{3 f} \] Output:
-2/3*cos(f*x+e)*(d*csc(f*x+e))^(3/2)/d/f+2/3*(d*csc(f*x+e))^(1/2)*InverseJ acobiAM(1/2*e-1/4*Pi+1/2*f*x,2^(1/2))*sin(f*x+e)^(1/2)/f
Time = 0.23 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.74 \[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=-\frac {2 (d \csc (e+f x))^{3/2} \left (\cos (e+f x)+\operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),2\right ) \sin ^{\frac {3}{2}}(e+f x)\right )}{3 d f} \] Input:
Integrate[Csc[e + f*x]^2*Sqrt[d*Csc[e + f*x]],x]
Output:
(-2*(d*Csc[e + f*x])^(3/2)*(Cos[e + f*x] + EllipticF[(-2*e + Pi - 2*f*x)/4 , 2]*Sin[e + f*x]^(3/2)))/(3*d*f)
Time = 0.58 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2030, 3042, 4255, 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.
\(\displaystyle \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle \frac {\int (d \csc (e+f x))^{5/2}dx}{d^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int (d \csc (e+f x))^{5/2}dx}{d^2}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {\frac {1}{3} d^2 \int \sqrt {d \csc (e+f x)}dx-\frac {2 d \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 f}}{d^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{3} d^2 \int \sqrt {d \csc (e+f x)}dx-\frac {2 d \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 f}}{d^2}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {\frac {1}{3} d^2 \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)} \int \frac {1}{\sqrt {\sin (e+f x)}}dx-\frac {2 d \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 f}}{d^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{3} d^2 \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)} \int \frac {1}{\sqrt {\sin (e+f x)}}dx-\frac {2 d \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 f}}{d^2}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {\frac {2 d^2 \sqrt {\sin (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),2\right ) \sqrt {d \csc (e+f x)}}{3 f}-\frac {2 d \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 f}}{d^2}\) |
Input:
Int[Csc[e + f*x]^2*Sqrt[d*Csc[e + f*x]],x]
Output:
((-2*d*Cos[e + f*x]*(d*Csc[e + f*x])^(3/2))/(3*f) + (2*d^2*Sqrt[d*Csc[e + f*x]]*EllipticF[(e - Pi/2 + f*x)/2, 2]*Sqrt[Sin[e + f*x]])/(3*f))/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[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[(-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 = 0.75 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.82
method | result | size |
default | \(-\frac {\sqrt {2}\, \sqrt {d \csc \left (f x +e \right )}\, \left (i \left (-\cos \left (f x +e \right )-1\right ) \sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}\, \sqrt {1-i \cot \left (f x +e \right )+i \csc \left (f x +e \right )}\, \sqrt {-i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \cot \left (f x +e \right )\right )}{3 f}\) | \(135\) |
Input:
int(csc(f*x+e)^2*(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*(-cos(f*x+e)-1)*(1+I*cot(f*x+e)-I*c sc(f*x+e))^(1/2)*(1-I*cot(f*x+e)+I*csc(f*x+e))^(1/2)*(-I*(-csc(f*x+e)+cot( f*x+e)))^(1/2)*EllipticF((1+I*cot(f*x+e)-I*csc(f*x+e))^(1/2),1/2*2^(1/2))+ 2^(1/2)*cot(f*x+e))
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.30 \[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=\frac {-i \, \sqrt {2 i \, d} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + i \, \sqrt {-2 i \, d} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 2 \, \sqrt {\frac {d}{\sin \left (f x + e\right )}} \cos \left (f x + e\right )}{3 \, f \sin \left (f x + e\right )} \] Input:
integrate(csc(f*x+e)^2*(d*csc(f*x+e))^(1/2),x, algorithm="fricas")
Output:
1/3*(-I*sqrt(2*I*d)*sin(f*x + e)*weierstrassPInverse(4, 0, cos(f*x + e) + I*sin(f*x + e)) + I*sqrt(-2*I*d)*sin(f*x + e)*weierstrassPInverse(4, 0, co s(f*x + e) - I*sin(f*x + e)) - 2*sqrt(d/sin(f*x + e))*cos(f*x + e))/(f*sin (f*x + e))
\[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=\int \sqrt {d \csc {\left (e + f x \right )}} \csc ^{2}{\left (e + f x \right )}\, dx \] Input:
integrate(csc(f*x+e)**2*(d*csc(f*x+e))**(1/2),x)
Output:
Integral(sqrt(d*csc(e + f*x))*csc(e + f*x)**2, x)
\[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=\int { \sqrt {d \csc \left (f x + e\right )} \csc \left (f x + e\right )^{2} \,d x } \] Input:
integrate(csc(f*x+e)^2*(d*csc(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(d*csc(f*x + e))*csc(f*x + e)^2, x)
\[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=\int { \sqrt {d \csc \left (f x + e\right )} \csc \left (f x + e\right )^{2} \,d x } \] Input:
integrate(csc(f*x+e)^2*(d*csc(f*x+e))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(d*csc(f*x + e))*csc(f*x + e)^2, x)
Timed out. \[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=\int \frac {\sqrt {\frac {d}{\sin \left (e+f\,x\right )}}}{{\sin \left (e+f\,x\right )}^2} \,d x \] Input:
int((d/sin(e + f*x))^(1/2)/sin(e + f*x)^2,x)
Output:
int((d/sin(e + f*x))^(1/2)/sin(e + f*x)^2, x)
\[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=\sqrt {d}\, \left (\int \sqrt {\csc \left (f x +e \right )}\, \csc \left (f x +e \right )^{2}d x \right ) \] 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)**2,x)