\(\int \csc ^5(x) \sin ^{\frac {3}{2}}(2 x) \, dx\) [408]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \csc ^5(x) \sin ^{\frac {3}{2}}(2 x) \, dx=-\frac {1}{5} \csc ^5(x) \sin ^{\frac {5}{2}}(2 x) \]

[Out]

-1/5*csc(x)^5*sin(2*x)^(5/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {4377} \[ \int \csc ^5(x) \sin ^{\frac {3}{2}}(2 x) \, dx=-\frac {1}{5} \sin ^{\frac {5}{2}}(2 x) \csc ^5(x) \]

[In]

Int[Csc[x]^5*Sin[2*x]^(3/2),x]

[Out]

-1/5*(Csc[x]^5*Sin[2*x]^(5/2))

Rule 4377

Int[((e_.)*sin[(a_.) + (b_.)*(x_)])^(m_.)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Simp[(e*Sin[a + b
*x])^m*((g*Sin[c + d*x])^(p + 1)/(b*g*m)), x] /; FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && Eq
Q[d/b, 2] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{5} \csc ^5(x) \sin ^{\frac {5}{2}}(2 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \csc ^5(x) \sin ^{\frac {3}{2}}(2 x) \, dx=-\frac {1}{5} \csc ^5(x) \sin ^{\frac {5}{2}}(2 x) \]

[In]

Integrate[Csc[x]^5*Sin[2*x]^(3/2),x]

[Out]

-1/5*(Csc[x]^5*Sin[2*x]^(5/2))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.65 (sec) , antiderivative size = 508, normalized size of antiderivative = 31.75

method	result	size
default	\(\frac {\sqrt {-\frac {\tan \left (\frac {x}{2}\right )}{\tan ^{2}\left (\frac {x}{2}\right )-1}}\, \left (96 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, E\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-48 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, F\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-\sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+40 \left (\tan ^{4}\left (\frac {x}{2}\right )\right ) \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}+\left (\tan ^{4}\left (\frac {x}{2}\right )\right ) \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}+28 \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+\sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-28 \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-\sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\right )}{5 \tan \left (\frac {x}{2}\right )^{3} \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}}\)	\(508\)

[In]

int(sin(2*x)^(3/2)/sin(x)^5,x,method=_RETURNVERBOSE)

[Out]

1/5*(-tan(1/2*x)/(tan(1/2*x)^2-1))^(1/2)/tan(1/2*x)^3*(96*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(1+tan(1/2*x))^(
1/2)*(-2*tan(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)*EllipticE((1+tan(1/2*x))^(1/2),1/2*2^(1/2))*((1+tan(1/2*x))*(
tan(1/2*x)-1)*tan(1/2*x))^(1/2)*tan(1/2*x)^2-48*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(1+tan(1/2*x))^(1/2)*(-2*t
an(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)*EllipticF((1+tan(1/2*x))^(1/2),1/2*2^(1/2))*((1+tan(1/2*x))*(tan(1/2*x)
-1)*tan(1/2*x))^(1/2)*tan(1/2*x)^2-(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*((1+tan(1/2*x))*(tan(1/2*x)-1)*tan(1/2*
x))^(1/2)*tan(1/2*x)^6+40*tan(1/2*x)^4*(tan(1/2*x)^3-tan(1/2*x))^(1/2)*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)+tan
(1/2*x)^4*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*((1+tan(1/2*x))*(tan(1/2*x)-1)*tan(1/2*x))^(1/2)+28*(tan(1/2*x)^
3-tan(1/2*x))^(1/2)*((1+tan(1/2*x))*(tan(1/2*x)-1)*tan(1/2*x))^(1/2)*tan(1/2*x)^4+(tan(1/2*x)*(tan(1/2*x)^2-1)
)^(1/2)*((1+tan(1/2*x))*(tan(1/2*x)-1)*tan(1/2*x))^(1/2)*tan(1/2*x)^2-28*(tan(1/2*x)^3-tan(1/2*x))^(1/2)*((1+t
an(1/2*x))*(tan(1/2*x)-1)*tan(1/2*x))^(1/2)*tan(1/2*x)^2-(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*((1+tan(1/2*x))*(
tan(1/2*x)-1)*tan(1/2*x))^(1/2))/(tan(1/2*x)^3-tan(1/2*x))^(1/2)/((1+tan(1/2*x))*(tan(1/2*x)-1)*tan(1/2*x))^(1
/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (12) = 24\).

Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.44 \[ \int \csc ^5(x) \sin ^{\frac {3}{2}}(2 x) \, dx=\frac {4 \, {\left (\sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )\right )}}{5 \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \]

[In]

integrate(sin(2*x)^(3/2)/sin(x)^5,x, algorithm="fricas")

[Out]

4/5*(sqrt(2)*sqrt(cos(x)*sin(x))*cos(x)^2 + (cos(x)^2 - 1)*sin(x))/((cos(x)^2 - 1)*sin(x))

Sympy [F(-1)]

Timed out. \[ \int \csc ^5(x) \sin ^{\frac {3}{2}}(2 x) \, dx=\text {Timed out} \]

[In]

integrate(sin(2*x)**(3/2)/sin(x)**5,x)

[Out]

Timed out

Maxima [F]

\[ \int \csc ^5(x) \sin ^{\frac {3}{2}}(2 x) \, dx=\int { \frac {\sin \left (2 \, x\right )^{\frac {3}{2}}}{\sin \left (x\right )^{5}} \,d x } \]

[In]

integrate(sin(2*x)^(3/2)/sin(x)^5,x, algorithm="maxima")

[Out]

integrate(sin(2*x)^(3/2)/sin(x)^5, x)

Giac [F]

\[ \int \csc ^5(x) \sin ^{\frac {3}{2}}(2 x) \, dx=\int { \frac {\sin \left (2 \, x\right )^{\frac {3}{2}}}{\sin \left (x\right )^{5}} \,d x } \]

[In]

integrate(sin(2*x)^(3/2)/sin(x)^5,x, algorithm="giac")

[Out]

integrate(sin(2*x)^(3/2)/sin(x)^5, x)

Mupad [B] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \csc ^5(x) \sin ^{\frac {3}{2}}(2 x) \, dx=\frac {4\,\sqrt {\sin \left (2\,x\right )}\,\left ({\sin \left (x\right )}^2-1\right )}{5\,{\sin \left (x\right )}^3} \]

[In]

int(sin(2*x)^(3/2)/sin(x)^5,x)

[Out]

(4*sin(2*x)^(1/2)*(sin(x)^2 - 1))/(5*sin(x)^3)