\(\int (d \csc (e+f x))^{3/2} \sin ^3(e+f x) \, dx\) [517]

Optimal result

Integrand size = 21, antiderivative size = 75 \[ \int (d \csc (e+f x))^{3/2} \sin ^3(e+f x) \, dx=-\frac {2 d^2 \cos (e+f x)}{3 f \sqrt {d \csc (e+f x)}}+\frac {2 d \sqrt {d \csc (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),2\right ) \sqrt {\sin (e+f x)}}{3 f} \]

[Out]

-2/3*d^2*cos(f*x+e)/f/(d*csc(f*x+e))^(1/2)-2/3*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 = 0.04 (sec) , antiderivative size = 75, 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, 3854, 3856, 2720} \[ \int (d \csc (e+f x))^{3/2} \sin ^3(e+f x) \, dx=\frac {2 d \sqrt {\sin (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),2\right ) \sqrt {d \csc (e+f x)}}{3 f}-\frac {2 d^2 \cos (e+f x)}{3 f \sqrt {d \csc (e+f x)}} \]

[In]

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

[Out]

(-2*d^2*Cos[e + f*x])/(3*f*Sqrt[d*Csc[e + f*x]]) + (2*d*Sqrt[d*Csc[e + f*x]]*EllipticF[(e - Pi/2 + f*x)/2, 2]*
Sqrt[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 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}& = d^3 \int \frac {1}{(d \csc (e+f x))^{3/2}} \, dx \\ & = -\frac {2 d^2 \cos (e+f x)}{3 f \sqrt {d \csc (e+f x)}}+\frac {1}{3} d \int \sqrt {d \csc (e+f x)} \, dx \\ & = -\frac {2 d^2 \cos (e+f x)}{3 f \sqrt {d \csc (e+f x)}}+\frac {1}{3} \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^2 \cos (e+f x)}{3 f \sqrt {d \csc (e+f x)}}+\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)}}{3 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.75 \[ \int (d \csc (e+f x))^{3/2} \sin ^3(e+f x) \, dx=-\frac {d \sqrt {d \csc (e+f x)} \left (2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),2\right ) \sqrt {\sin (e+f x)}+\sin (2 (e+f x))\right )}{3 f} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.89 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.12

method	result	size
default	\(\frac {\sqrt {2}\, \left (i \sin \left (f x +e \right ) \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )+\left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {2}-\sqrt {2}\, \cos \left (f x +e \right )\right ) d \csc \left (f x +e \right ) \sqrt {d \csc \left (f x +e \right )}\, \left (\cos \left (f x +e \right )+1\right )}{3 f}\)	\(159\)

[In]

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

[Out]

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

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 = 1.13 \[ \int (d \csc (e+f x))^{3/2} \sin ^3(e+f x) \, dx=-\frac {2 \, d \sqrt {\frac {d}{\sin \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + i \, \sqrt {2 i \, d} d {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) - i \, \sqrt {-2 i \, d} d {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}{3 \, f} \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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