\(\int (d \csc (e+f x))^{3/2} \sin (e+f x) \, dx\) [519]

Optimal result

Integrand size = 19, antiderivative size = 44 \[ \int (d \csc (e+f x))^{3/2} \sin (e+f x) \, dx=\frac {2 d \sqrt {d \csc (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),2\right ) \sqrt {\sin (e+f x)}}{f} \]

[Out]

-2*d*(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)*EllipticF(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 = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {16, 3856, 2720} \[ \int (d \csc (e+f x))^{3/2} \sin (e+f x) \, dx=\frac {2 d \sqrt {\sin (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),2\right ) \sqrt {d \csc (e+f x)}}{f} \]

[In]

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

[Out]

(2*d*Sqrt[d*Csc[e + f*x]]*EllipticF[(e - Pi/2 + f*x)/2, 2]*Sqrt[Sin[e + f*x]])/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 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}& = d \int \sqrt {d \csc (e+f x)} \, dx \\ & = \left (d \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {1}{\sqrt {\sin (e+f x)}} \, dx \\ & = \frac {2 d \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)}}{f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int (d \csc (e+f x))^{3/2} \sin (e+f x) \, dx=-\frac {2 d \sqrt {d \csc (e+f x)} \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),2\right ) \sqrt {\sin (e+f x)}}{f} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.78 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.75

method	result	size
default	\(\frac {i d \left (\cos \left (f x +e \right )+1\right ) \sqrt {d \csc \left (f x +e \right )}\, \sqrt {2}\, \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )}{f}\)	\(121\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 0.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.30 \[ \int (d \csc (e+f x))^{3/2} \sin (e+f x) \, dx=\frac {-i \, \sqrt {2 i \, d} d {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + i \, \sqrt {-2 i \, d} d {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}{f} \]

[In]

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

[Out]

(-I*sqrt(2*I*d)*d*weierstrassPInverse(4, 0, cos(f*x + e) + I*sin(f*x + e)) + I*sqrt(-2*I*d)*d*weierstrassPInve
rse(4, 0, cos(f*x + e) - I*sin(f*x + e)))/f

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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