\(\int \frac {\sin ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx\) [523]

Optimal result

Integrand size = 21, antiderivative size = 102 \[ \int \frac {\sin ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=-\frac {2 d^2 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac {10 \cos (e+f x)}{21 f \sqrt {d \csc (e+f x)}}+\frac {10 \sqrt {d \csc (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),2\right ) \sqrt {\sin (e+f x)}}{21 d f} \]

[Out]

-2/7*d^2*cos(f*x+e)/f/(d*csc(f*x+e))^(5/2)-10/21*cos(f*x+e)/f/(d*csc(f*x+e))^(1/2)-10/21*(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)/d/f

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 3854, 3856, 2720} \[ \int \frac {\sin ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=-\frac {2 d^2 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac {10 \cos (e+f x)}{21 f \sqrt {d \csc (e+f x)}}+\frac {10 \sqrt {\sin (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),2\right ) \sqrt {d \csc (e+f x)}}{21 d f} \]

[In]

Int[Sin[e + f*x]^3/Sqrt[d*Csc[e + f*x]],x]

[Out]

(-2*d^2*Cos[e + f*x])/(7*f*(d*Csc[e + f*x])^(5/2)) - (10*Cos[e + f*x])/(21*f*Sqrt[d*Csc[e + f*x]]) + (10*Sqrt[
d*Csc[e + f*x]]*EllipticF[(e - Pi/2 + f*x)/2, 2]*Sqrt[Sin[e + f*x]])/(21*d*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))^{7/2}} \, dx \\ & = -\frac {2 d^2 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}+\frac {1}{7} (5 d) \int \frac {1}{(d \csc (e+f x))^{3/2}} \, dx \\ & = -\frac {2 d^2 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac {10 \cos (e+f x)}{21 f \sqrt {d \csc (e+f x)}}+\frac {5 \int \sqrt {d \csc (e+f x)} \, dx}{21 d} \\ & = -\frac {2 d^2 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac {10 \cos (e+f x)}{21 f \sqrt {d \csc (e+f x)}}+\frac {\left (5 \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {1}{\sqrt {\sin (e+f x)}} \, dx}{21 d} \\ & = -\frac {2 d^2 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac {10 \cos (e+f x)}{21 f \sqrt {d \csc (e+f x)}}+\frac {10 \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)}}{21 d f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.69 \[ \int \frac {\sin ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=-\frac {\sqrt {d \csc (e+f x)} \left (40 \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),2\right ) \sqrt {\sin (e+f x)}+26 \sin (2 (e+f x))-3 \sin (4 (e+f x))\right )}{84 d f} \]

[In]

Integrate[Sin[e + f*x]^3/Sqrt[d*Csc[e + f*x]],x]

[Out]

-1/84*(Sqrt[d*Csc[e + f*x]]*(40*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]] + 26*Sin[2*(e + f*x)] -
3*Sin[4*(e + f*x)]))/(d*f)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.15 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.41

method	result	size
default	\(\frac {\sqrt {2}\, \left (5 i \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \cot \left (f x +e \right )+5 i \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \csc \left (f x +e \right )+3 \sqrt {2}\, \left (\cos ^{3}\left (f x +e \right )\right )-8 \sqrt {2}\, \cos \left (f x +e \right )\right )}{21 f \sqrt {d \csc \left (f x +e \right )}}\)	\(246\)

[In]

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

[Out]

1/21/f*2^(1/2)/(d*csc(f*x+e))^(1/2)*(5*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)*cot(f*x+e)
+5*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)*csc(f*x+e)+3*2^(1/2)*cos(f*x+e)^3-8*2^(1/2)*co
s(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 = 98, normalized size of antiderivative = 0.96 \[ \int \frac {\sin ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\frac {2 \, {\left (3 \, \cos \left (f x + e\right )^{3} - 8 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {d}{\sin \left (f x + e\right )}} \sin \left (f x + e\right ) - 5 i \, \sqrt {2 i \, d} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 5 i \, \sqrt {-2 i \, d} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}{21 \, d f} \]

[In]

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

[Out]

1/21*(2*(3*cos(f*x + e)^3 - 8*cos(f*x + e))*sqrt(d/sin(f*x + e))*sin(f*x + e) - 5*I*sqrt(2*I*d)*weierstrassPIn
verse(4, 0, cos(f*x + e) + I*sin(f*x + e)) + 5*I*sqrt(-2*I*d)*weierstrassPInverse(4, 0, cos(f*x + e) - I*sin(f
*x + e)))/(d*f)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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