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

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

[Out]

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

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Rubi [A]  time = 0.0224134, antiderivative size = 16, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0322744, size = 16, normalized size = 1. \[ -\frac{1}{5} \sin ^{\frac{5}{2}}(2 x) \csc ^5(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.049, size = 508, normalized size = 31.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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))*(tan(1/2*x)*(tan(
1/2*x)-1)*(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))*(tan(1/2*x)*(tan(1/2*x)-1)*
(1+tan(1/2*x)))^(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)*(1+tan(1/2*x
)))^(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)*(1+tan(1/2*x)))^(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)*(1+tan(1/2*x)))^(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)*(1+tan(1/2*x)))^(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)*(1+tan(1/2*x)))^(1/2)*tan(1/2*x)^2-(tan(1/2*x)*(tan(1/2*x)-1)*(1+tan(1/2*x)))^(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)*(1+tan(1/2*x)))^(1
/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (2 \, x\right )^{\frac{3}{2}}}{\sin \left (x\right )^{5}}\,{d x} \end{align*}

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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Fricas [B]  time = 1.93462, size = 124, normalized size = 7.75 \begin{align*} \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 )} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (2 \, x\right )^{\frac{3}{2}}}{\sin \left (x\right )^{5}}\,{d x} \end{align*}

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)