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

Optimal. Leaf size=16 \[ -\frac {1}{5} \sin ^{\frac {5}{2}}(2 x) \csc ^5(x) \]

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Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {4292} \[ -\frac {1}{5} \sin ^{\frac {5}{2}}(2 x) \csc ^5(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4292

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*} \int \csc ^5(x) \sin ^{\frac {3}{2}}(2 x) \, dx &=-\frac {1}{5} \csc ^5(x) \sin ^{\frac {5}{2}}(2 x)\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 16, normalized size = 1.00 \[ -\frac {1}{5} \sin ^{\frac {5}{2}}(2 x) \csc ^5(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc ^5(x) \sin ^{\frac {3}{2}}(2 x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [B]  time = 1.00, size = 39, normalized size = 2.44 \[ \frac {4 \, {\left (\sqrt {2} \sqrt {\cos \relax (x) \sin \relax (x)} \cos \relax (x)^{2} + {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x)\right )}}{5 \, {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (2 \, x\right )^{\frac {3}{2}}}{\sin \relax (x)^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(2*x)^(3/2)/sin(x)^5, x)

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maple [C]  time = 0.27, size = 508, normalized size = 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 {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticE \left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\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 {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\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 {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+28 \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{4}\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 \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}-28 \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\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 {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-\sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\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 {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}}\)	\(508\)

Verification of antiderivative is not currently implemented for this CAS.

[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)*(tan(1/2*x)+1)^(
1/2)*(-2*tan(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)*EllipticE((tan(1/2*x)+1)^(1/2),1/2*2^(1/2))*(tan(1/2*x)*(tan(
1/2*x)-1)*(tan(1/2*x)+1))^(1/2)*tan(1/2*x)^2-48*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)+1)^(1/2)*(-2*t
an(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)*EllipticF((tan(1/2*x)+1)^(1/2),1/2*2^(1/2))*(tan(1/2*x)*(tan(1/2*x)-1)*
(tan(1/2*x)+1))^(1/2)*tan(1/2*x)^2-(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)*(tan(1/2*x)-1)*(tan(1/2*x)+
1))^(1/2)*tan(1/2*x)^6+28*(tan(1/2*x)^3-tan(1/2*x))^(1/2)*(tan(1/2*x)*(tan(1/2*x)-1)*(tan(1/2*x)+1))^(1/2)*tan
(1/2*x)^4+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*(ta
n(1/2*x)*(tan(1/2*x)-1)*(tan(1/2*x)+1))^(1/2)*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)-28*(tan(1/2*x)^3-tan(1/2*x))
^(1/2)*(tan(1/2*x)*(tan(1/2*x)-1)*(tan(1/2*x)+1))^(1/2)*tan(1/2*x)^2+(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(tan(
1/2*x)*(tan(1/2*x)-1)*(tan(1/2*x)+1))^(1/2)*tan(1/2*x)^2-(tan(1/2*x)*(tan(1/2*x)-1)*(tan(1/2*x)+1))^(1/2)*(tan
(1/2*x)*(tan(1/2*x)^2-1))^(1/2))/(tan(1/2*x)^3-tan(1/2*x))^(1/2)/(tan(1/2*x)*(tan(1/2*x)-1)*(tan(1/2*x)+1))^(1
/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (2 \, x\right )^{\frac {3}{2}}}{\sin \relax (x)^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(2*x)^(3/2)/sin(x)^5, x)

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mupad [B]  time = 0.56, size = 18, normalized size = 1.12 \[ \frac {4\,\sqrt {\sin \left (2\,x\right )}\,\left ({\sin \relax (x)}^2-1\right )}{5\,{\sin \relax (x)}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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