\(\int \frac {\csc ^3(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx\) [535]

Optimal result

Integrand size = 21, antiderivative size = 73 \[ \int \frac {\csc ^3(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=-\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{d^2 f}-\frac {2 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{d f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \]

[Out]

-2*cos(f*x+e)*(d*csc(f*x+e))^(1/2)/d^2/f+2*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*Ellip
ticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2))/d/f/(d*csc(f*x+e))^(1/2)/sin(f*x+e)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 3853, 3856, 2719} \[ \int \frac {\csc ^3(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=-\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{d^2 f}-\frac {2 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{d f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}} \]

[In]

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

[Out]

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

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 2719

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 3853

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] + Dist[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 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (d \csc (e+f x))^{3/2} \, dx}{d^3} \\ & = -\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{d^2 f}-\frac {\int \frac {1}{\sqrt {d \csc (e+f x)}} \, dx}{d} \\ & = -\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{d^2 f}-\frac {\int \sqrt {\sin (e+f x)} \, dx}{d \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \\ & = -\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{d^2 f}-\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)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.75 \[ \int \frac {\csc ^3(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\frac {-2 \cot (e+f x)+\frac {2 E\left (\left .\frac {1}{4} (-2 e+\pi -2 f x)\right |2\right )}{\sqrt {\sin (e+f x)}}}{d f \sqrt {d \csc (e+f x)}} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.92 (sec) , antiderivative size = 421, normalized size of antiderivative = 5.77

method	result	size
default	\(\frac {\sqrt {2}\, \left (2 \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )-\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+2 \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\right ) \csc \left (f x +e \right )}{f \sqrt {d \csc \left (f x +e \right )}\, d}\)	\(421\)

[In]

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

[Out]

1/f*2^(1/2)*(2*(-I*(I-cot(f*x+e)+csc(f*x+e)))^(1/2)*(-I*(I+cot(f*x+e)-csc(f*x+e)))^(1/2)*(I*(-cot(f*x+e)+csc(f
*x+e)))^(1/2)*EllipticE((-I*(I-cot(f*x+e)+csc(f*x+e)))^(1/2),1/2*2^(1/2))*cos(f*x+e)-(-I*(I-cot(f*x+e)+csc(f*x
+e)))^(1/2)*(-I*(I+cot(f*x+e)-csc(f*x+e)))^(1/2)*(I*(-cot(f*x+e)+csc(f*x+e)))^(1/2)*EllipticF((-I*(I-cot(f*x+e
)+csc(f*x+e)))^(1/2),1/2*2^(1/2))*cos(f*x+e)+2*(-I*(I-cot(f*x+e)+csc(f*x+e)))^(1/2)*(-I*(I+cot(f*x+e)-csc(f*x+
e)))^(1/2)*(I*(-cot(f*x+e)+csc(f*x+e)))^(1/2)*EllipticE((-I*(I-cot(f*x+e)+csc(f*x+e)))^(1/2),1/2*2^(1/2))-(-I*
(I-cot(f*x+e)+csc(f*x+e)))^(1/2)*(-I*(I+cot(f*x+e)-csc(f*x+e)))^(1/2)*(I*(-cot(f*x+e)+csc(f*x+e)))^(1/2)*Ellip
ticF((-I*(I-cot(f*x+e)+csc(f*x+e)))^(1/2),1/2*2^(1/2))-2^(1/2))/(d*csc(f*x+e))^(1/2)/d*csc(f*x+e)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.14 \[ \int \frac {\csc ^3(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=-\frac {2 \, \sqrt {\frac {d}{\sin \left (f x + e\right )}} \cos \left (f x + e\right ) + \sqrt {2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + \sqrt {-2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{d^{2} f} \]

[In]

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

[Out]

-(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)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(f*x + e) - I*sin(f*x
+ e))))/(d^2*f)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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