\(\int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx\) [513]

Optimal result

Integrand size = 21, antiderivative size = 74 \[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=-\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d f}+\frac {2 \sqrt {d \csc (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),2\right ) \sqrt {\sin (e+f x)}}{3 f} \]

[Out]

-2/3*cos(f*x+e)*(d*csc(f*x+e))^(3/2)/d/f-2/3*(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)*Ell
ipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2))*(d*csc(f*x+e))^(1/2)*sin(f*x+e)^(1/2)/f

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 74, 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, 2720} \[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=\frac {2 \sqrt {\sin (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),2\right ) \sqrt {d \csc (e+f x)}}{3 f}-\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d f} \]

[In]

Int[Csc[e + f*x]^2*Sqrt[d*Csc[e + f*x]],x]

[Out]

(-2*Cos[e + f*x]*(d*Csc[e + f*x])^(3/2))/(3*d*f) + (2*Sqrt[d*Csc[e + f*x]]*EllipticF[(e - Pi/2 + f*x)/2, 2]*Sq
rt[Sin[e + f*x]])/(3*f)

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 2720

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]

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))^{5/2} \, dx}{d^2} \\ & = -\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d f}+\frac {1}{3} \int \sqrt {d \csc (e+f x)} \, dx \\ & = -\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d f}+\frac {1}{3} \left (\sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {1}{\sqrt {\sin (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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.74 \[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=-\frac {2 (d \csc (e+f x))^{3/2} \left (\cos (e+f x)+\operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),2\right ) \sin ^{\frac {3}{2}}(e+f x)\right )}{3 d f} \]

[In]

Integrate[Csc[e + f*x]^2*Sqrt[d*Csc[e + f*x]],x]

[Out]

(-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)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.85 (sec) , antiderivative size = 227, normalized size of antiderivative = 3.07

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {d \csc \left (f x +e \right )}\, \left (i \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+i \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\, \cot \left (f x +e \right )\right )}{3 f}\)	\(227\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 0.12 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.30 \[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=\frac {-i \, \sqrt {2 i \, d} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + i \, \sqrt {-2 i \, d} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 2 \, \sqrt {\frac {d}{\sin \left (f x + e\right )}} \cos \left (f x + e\right )}{3 \, f \sin \left (f x + e\right )} \]

[In]

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

[Out]

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, cos(f*x + e) - I*sin(f*x + e)) - 2*sqrt(d/sin(f*x + e))*cos(f*x + e))/(f*s
in(f*x + e))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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