\(\int \frac {\csc (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx\) [533]

Optimal result

Integrand size = 19, antiderivative size = 46 \[ \int \frac {\csc (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\frac {2 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{d f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.98 \[ \int \frac {\csc (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=-\frac {2 E\left (\left .\frac {1}{4} (-2 e+\pi -2 f x)\right |2\right )}{d f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \] Input:

Integrate[Csc[e + f*x]/(d*Csc[e + f*x])^(3/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 46, 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 = {2030, 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.

\(\displaystyle \int \frac {\csc (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx\)

\(\Big \downarrow \) 2030

\(\displaystyle \frac {\int \frac {1}{\sqrt {d \csc (e+f x)}}dx}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {1}{\sqrt {d \csc (e+f x)}}dx}{d}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {\int \sqrt {\sin (e+f x)}dx}{d \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \sqrt {\sin (e+f x)}dx}{d \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {2 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{d f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}\)

Input:

Int[Csc[e + f*x]/(d*Csc[e + f*x])^(3/2),x]
 

Output:

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

Defintions of rubi rules used

rule 2030

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3119

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 4258

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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.78 (sec) , antiderivative size = 246, normalized size of antiderivative = 5.35

method	result	size
default	\(-\frac {\sqrt {2}\, \left (2 \sqrt {1+i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \sqrt {1-i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \sqrt {-i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \left (\cos \left (f x +e \right )+1\right ) \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {1+i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \sqrt {1-i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \sqrt {-i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \left (-\cos \left (f x +e \right )-1\right ) \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \left (-1+\cos \left (f x +e \right )\right )\right ) \csc \left (f x +e \right )}{d f \sqrt {d \csc \left (f x +e \right )}}\)	\(246\)
risch	\(-\frac {i \sqrt {2}}{f d \sqrt {\frac {i d \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}-1}}}+\frac {i \left (-\frac {2 i \left (i d \,{\mathrm e}^{2 i \left (f x +e \right )}-i d \right )}{d \sqrt {{\mathrm e}^{i \left (f x +e \right )} \left (i d \,{\mathrm e}^{2 i \left (f x +e \right )}-i d \right )}}-\frac {\sqrt {{\mathrm e}^{i \left (f x +e \right )}+1}\, \sqrt {-2 \,{\mathrm e}^{i \left (f x +e \right )}+2}\, \sqrt {-{\mathrm e}^{i \left (f x +e \right )}}\, \left (-2 \operatorname {EllipticE}\left (\sqrt {{\mathrm e}^{i \left (f x +e \right )}+1}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticF}\left (\sqrt {{\mathrm e}^{i \left (f x +e \right )}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {i d \,{\mathrm e}^{3 i \left (f x +e \right )}-i d \,{\mathrm e}^{i \left (f x +e \right )}}}\right ) \sqrt {2}\, \sqrt {i d \,{\mathrm e}^{i \left (f x +e \right )} \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}}{f d \sqrt {\frac {i d \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}-1}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}\)	\(303\)

Input:

int(csc(f*x+e)/(d*csc(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/d/f*2^(1/2)*(2*(1+I*(-csc(f*x+e)+cot(f*x+e)))^(1/2)*(1-I*(-csc(f*x+e)+c 
ot(f*x+e)))^(1/2)*(-I*(-csc(f*x+e)+cot(f*x+e)))^(1/2)*(cos(f*x+e)+1)*Ellip 
ticE((1+I*cot(f*x+e)-I*csc(f*x+e))^(1/2),1/2*2^(1/2))+(1+I*(-csc(f*x+e)+co 
t(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)*csc(f*x+e)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.35 \[ \int \frac {\csc (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\frac {\sqrt {2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + \sqrt {-2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{d^{2} f} \] Input:

integrate(csc(f*x+e)/(d*csc(f*x+e))^(3/2),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))))/(d^2*f)
 

Sympy [F]

\[ \int \frac {\csc (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\int \frac {\csc {\left (e + f x \right )}}{\left (d \csc {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(csc(f*x+e)/(d*csc(f*x+e))**(3/2),x)
 

Output:

Integral(csc(e + f*x)/(d*csc(e + f*x))**(3/2), x)
 

Maxima [F]

\[ \int \frac {\csc (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\int { \frac {\csc \left (f x + e\right )}{\left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(csc(f*x+e)/(d*csc(f*x+e))^(3/2),x, algorithm="maxima")
 

Output:

integrate(csc(f*x + e)/(d*csc(f*x + e))^(3/2), x)
 

Giac [F]

\[ \int \frac {\csc (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\int { \frac {\csc \left (f x + e\right )}{\left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(csc(f*x+e)/(d*csc(f*x+e))^(3/2),x, algorithm="giac")
 

Output:

integrate(csc(f*x + e)/(d*csc(f*x + e))^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\int \frac {1}{\sin \left (e+f\,x\right )\,{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^{3/2}} \,d x \] Input:

int(1/(sin(e + f*x)*(d/sin(e + f*x))^(3/2)),x)
 

Output:

int(1/(sin(e + f*x)*(d/sin(e + f*x))^(3/2)), x)
 

Reduce [F]

\[ \int \frac {\csc (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\frac {\sqrt {d}\, \left (\int \frac {\sqrt {\csc \left (f x +e \right )}}{\csc \left (f x +e \right )}d x \right )}{d^{2}} \] Input:

int(csc(f*x+e)/(d*csc(f*x+e))^(3/2),x)
                                                                                    
                                                                                    
 

Output:

(sqrt(d)*int(sqrt(csc(e + f*x))/csc(e + f*x),x))/d**2