\(\int \frac {1}{(d \csc (e+f x))^{3/2}} \, dx\) [532]

Optimal result

Integrand size = 12, antiderivative size = 77 \[ \int \frac {1}{(d \csc (e+f x))^{3/2}} \, dx=-\frac {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}}+\frac {2 \sqrt {d \csc (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),2\right ) \sqrt {\sin (e+f x)}}{3 d^2 f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3854, 3856, 2720} \[ \int \frac {1}{(d \csc (e+f x))^{3/2}} \, dx=\frac {2 \sqrt {\sin (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),2\right ) \sqrt {d \csc (e+f x)}}{3 d^2 f}-\frac {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}} \]

[In]

Int[(d*Csc[e + f*x])^(-3/2),x]

[Out]

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

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 3854

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Csc[c + d*x])^(n + 1)/(b*d*n)), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[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 {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}}+\frac {\int \sqrt {d \csc (e+f x)} \, dx}{3 d^2} \\ & = -\frac {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}}+\frac {\left (\sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {1}{\sqrt {\sin (e+f x)}} \, dx}{3 d^2} \\ & = -\frac {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}}+\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 d^2 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.82 \[ \int \frac {1}{(d \csc (e+f x))^{3/2}} \, dx=-\frac {\csc ^2(e+f x) \left (2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),2\right ) \sqrt {\sin (e+f x)}+\sin (2 (e+f x))\right )}{3 f (d \csc (e+f x))^{3/2}} \]

[In]

Integrate[(d*Csc[e + f*x])^(-3/2),x]

[Out]

-1/3*(Csc[e + f*x]^2*(2*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]] + Sin[2*(e + f*x)]))/(f*(d*Csc[
e + f*x])^(3/2))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.06 (sec) , antiderivative size = 262, normalized size of antiderivative = 3.40

method	result	size
default	\(-\frac {\sqrt {2}\, \left (i \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+i \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )\right ) \sin \left (f x +e \right )}{3 f \sqrt {d \csc \left (f x +e \right )}\, \left (\cos \left (f x +e \right )-1\right ) d \left (\cos \left (f x +e \right )+1\right )}\)	\(262\)

[In]

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

[Out]

-1/3/f*2^(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)-cs
c(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)*cos(f*x+e)*sin(f*x+e))/(d*csc(f*x+e))^(1/2)/(cos(f*x+e)-1)/d/(c
os(f*x+e)+1)*sin(f*x+e)

Fricas [C] (verification not implemented)

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

Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(d \csc (e+f x))^{3/2}} \, dx=-\frac {2 \, \sqrt {\frac {d}{\sin \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + i \, \sqrt {2 i \, d} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) - i \, \sqrt {-2 i \, d} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}{3 \, d^{2} f} \]

[In]

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

[Out]

-1/3*(2*sqrt(d/sin(f*x + e))*cos(f*x + e)*sin(f*x + e) + I*sqrt(2*I*d)*weierstrassPInverse(4, 0, cos(f*x + e)
+ I*sin(f*x + e)) - I*sqrt(-2*I*d)*weierstrassPInverse(4, 0, cos(f*x + e) - I*sin(f*x + e)))/(d^2*f)

Sympy [F]

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

[In]

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

[Out]

Integral((d*csc(e + f*x))**(-3/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate((d*csc(f*x + e))^(-3/2), x)

Giac [F]

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

[In]

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

[Out]

integrate((d*csc(f*x + e))^(-3/2), x)

Mupad [F(-1)]

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

[In]

int(1/(d/sin(e + f*x))^(3/2),x)

[Out]

int(1/(d/sin(e + f*x))^(3/2), x)