\(\int \csc ^3(e+f x) \sqrt {d \csc (e+f x)} \, dx\) [514]

Optimal result

Integrand size = 21, antiderivative size = 100 \[ \int \csc ^3(e+f x) \sqrt {d \csc (e+f x)} \, dx=-\frac {6 \cos (e+f x) \sqrt {d \csc (e+f x)}}{5 f}-\frac {2 \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 d^2 f}-\frac {6 d E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{5 f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \] Output:

-6/5*cos(f*x+e)*(d*csc(f*x+e))^(1/2)/f-2/5*cos(f*x+e)*(d*csc(f*x+e))^(5/2) 
/d^2/f+6/5*d*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)
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.68 \[ \int \csc ^3(e+f x) \sqrt {d \csc (e+f x)} \, dx=-\frac {2 \sqrt {d \csc (e+f x)} \left (3 \cos (e+f x)+\cot (e+f x) \csc (e+f x)-3 E\left (\left .\frac {1}{4} (-2 e+\pi -2 f x)\right |2\right ) \sqrt {\sin (e+f x)}\right )}{5 f} \] Input:

Integrate[Csc[e + f*x]^3*Sqrt[d*Csc[e + f*x]],x]
 

Output:

(-2*Sqrt[d*Csc[e + f*x]]*(3*Cos[e + f*x] + Cot[e + f*x]*Csc[e + f*x] - 3*E 
llipticE[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]]))/(5*f)
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2030, 3042, 4255, 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.

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

\(\Big \downarrow \) 2030

\(\displaystyle \frac {\int (d \csc (e+f x))^{7/2}dx}{d^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (d \csc (e+f x))^{7/2}dx}{d^3}\)

\(\Big \downarrow \) 4255

\(\displaystyle \frac {\frac {3}{5} d^2 \int (d \csc (e+f x))^{3/2}dx-\frac {2 d \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 f}}{d^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3}{5} d^2 \int (d \csc (e+f x))^{3/2}dx-\frac {2 d \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 f}}{d^3}\)

\(\Big \downarrow \) 4255

\(\displaystyle \frac {\frac {3}{5} d^2 \left (d^2 \left (-\int \frac {1}{\sqrt {d \csc (e+f x)}}dx\right )-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}\right )-\frac {2 d \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 f}}{d^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3}{5} d^2 \left (d^2 \left (-\int \frac {1}{\sqrt {d \csc (e+f x)}}dx\right )-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}\right )-\frac {2 d \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 f}}{d^3}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {\frac {3}{5} d^2 \left (-\frac {d^2 \int \sqrt {\sin (e+f x)}dx}{\sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}\right )-\frac {2 d \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 f}}{d^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3}{5} d^2 \left (-\frac {d^2 \int \sqrt {\sin (e+f x)}dx}{\sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}\right )-\frac {2 d \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 f}}{d^3}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\frac {3}{5} d^2 \left (-\frac {2 d^2 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 d \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}\right )-\frac {2 d \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 f}}{d^3}\)

Input:

Int[Csc[e + f*x]^3*Sqrt[d*Csc[e + f*x]],x]
 

Output:

((-2*d*Cos[e + f*x]*(d*Csc[e + f*x])^(5/2))/(5*f) + (3*d^2*((-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]])))/5)/d^3
 

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 4255

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]
 

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 = 0.84 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.46

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {d \csc \left (f x +e \right )}\, \left (\left (6 \cos \left (f x +e \right )+6\right ) \sqrt {i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1-i \cot \left (f x +e \right )+i \csc \left (f x +e \right )}\, \sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}+\left (-3 \cos \left (f x +e \right )-3\right ) \sqrt {i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}\, \sqrt {1-i \cot \left (f x +e \right )+i \csc \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}+\sqrt {2}\, \left (-3-\csc \left (f x +e \right ) \cot \left (f x +e \right )\right )\right )}{5 f}\)	\(246\)

Input:

int(csc(f*x+e)^3*(d*csc(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/5/f*2^(1/2)*(d*csc(f*x+e))^(1/2)*((6*cos(f*x+e)+6)*(I*(csc(f*x+e)-cot(f* 
x+e)))^(1/2)*EllipticE((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)+(-3* 
cos(f*x+e)-3)*(I*(csc(f*x+e)-cot(f*x+e)))^(1/2)*(1-I*cot(f*x+e)+I*csc(f*x+ 
e))^(1/2)*EllipticF((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)+2^(1/2)*(-3-csc(f*x+e)*cot(f*x+e)))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.28 \[ \int \csc ^3(e+f x) \sqrt {d \csc (e+f x)} \, dx=-\frac {3 \, {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 3 \, {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {-2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (3 \, \cos \left (f x + e\right )^{3} - 4 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {d}{\sin \left (f x + e\right )}}}{5 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )}} \] Input:

integrate(csc(f*x+e)^3*(d*csc(f*x+e))^(1/2),x, algorithm="fricas")
 

Output:

-1/5*(3*(cos(f*x + e)^2 - 1)*sqrt(2*I*d)*weierstrassZeta(4, 0, weierstrass 
PInverse(4, 0, cos(f*x + e) + I*sin(f*x + e))) + 3*(cos(f*x + e)^2 - 1)*sq 
rt(-2*I*d)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(f*x + e) - 
I*sin(f*x + e))) + 2*(3*cos(f*x + e)^3 - 4*cos(f*x + e))*sqrt(d/sin(f*x + 
e)))/(f*cos(f*x + e)^2 - f)
 

Sympy [F]

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

integrate(csc(f*x+e)**3*(d*csc(f*x+e))**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate(csc(f*x+e)^3*(d*csc(f*x+e))^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(csc(f*x+e)^3*(d*csc(f*x+e))^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

sqrt(d)*int(sqrt(csc(e + f*x))*csc(e + f*x)**3,x)