3.265 \(\int \cos ^3(x) \csc ^{\frac{9}{2}}(x) \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{3} \csc ^{\frac{3}{2}}(x)-\frac{2}{7} \csc ^{\frac{7}{2}}(x) \]

[Out]

(2*Csc[x]^(3/2))/3 - (2*Csc[x]^(7/2))/7

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Rubi [A]  time = 0.0240884, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2621, 14} \[ \frac{2}{3} \csc ^{\frac{3}{2}}(x)-\frac{2}{7} \csc ^{\frac{7}{2}}(x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^3*Csc[x]^(9/2),x]

[Out]

(2*Csc[x]^(3/2))/3 - (2*Csc[x]^(7/2))/7

Rule 2621

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \cos ^3(x) \csc ^{\frac{9}{2}}(x) \, dx &=-\operatorname{Subst}\left (\int \sqrt{x} \left (-1+x^2\right ) \, dx,x,\csc (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-\sqrt{x}+x^{5/2}\right ) \, dx,x,\csc (x)\right )\\ &=\frac{2}{3} \csc ^{\frac{3}{2}}(x)-\frac{2}{7} \csc ^{\frac{7}{2}}(x)\\ \end{align*}

Mathematica [A]  time = 0.0246286, size = 18, normalized size = 0.86 \[ \frac{2}{21} \csc ^{\frac{3}{2}}(x) \left (7-3 \csc ^2(x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^3*Csc[x]^(9/2),x]

[Out]

(2*Csc[x]^(3/2)*(7 - 3*Csc[x]^2))/21

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Maple [A]  time = 0.51, size = 14, normalized size = 0.7 \begin{align*} -{\frac{2}{7} \left ( \sin \left ( x \right ) \right ) ^{-{\frac{7}{2}}}}+{\frac{2}{3} \left ( \sin \left ( x \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^3*csc(x)^(9/2),x)

[Out]

-2/7/sin(x)^(7/2)+2/3/sin(x)^(3/2)

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Maxima [A]  time = 0.963319, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{3 \, \sin \left (x\right )^{\frac{3}{2}}} - \frac{2}{7 \, \sin \left (x\right )^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*csc(x)^(9/2),x, algorithm="maxima")

[Out]

2/3/sin(x)^(3/2) - 2/7/sin(x)^(7/2)

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Fricas [A]  time = 1.02241, size = 72, normalized size = 3.43 \begin{align*} \frac{2 \,{\left (7 \, \cos \left (x\right )^{2} - 4\right )}}{21 \,{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*csc(x)^(9/2),x, algorithm="fricas")

[Out]

2/21*(7*cos(x)^2 - 4)/((cos(x)^2 - 1)*sin(x)^(3/2))

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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(cos(x)**3*csc(x)**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.13093, size = 23, normalized size = 1.1 \begin{align*} \frac{2 \,{\left (7 \, \sin \left (x\right )^{2} - 3\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )}{21 \, \sin \left (x\right )^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*csc(x)^(9/2),x, algorithm="giac")

[Out]

2/21*(7*sin(x)^2 - 3)*sgn(sin(x))/sin(x)^(7/2)