3.3.65 \(\int \cos ^3(x) \csc ^{\frac {9}{2}}(x) \, dx\) [265]

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

[Out]

2/3*csc(x)^(3/2)-2/7*csc(x)^(7/2)

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Rubi [A]
time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, 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 = {2701, 14} \begin {gather*} \frac {2}{3} \csc ^{\frac {3}{2}}(x)-\frac {2}{7} \csc ^{\frac {7}{2}}(x) \end {gather*}

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 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]]

Rule 2701

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])

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 18, normalized size = 0.86 \begin {gather*} \frac {2}{21} \csc ^{\frac {3}{2}}(x) \left (7-3 \csc ^2(x)\right ) \end {gather*}

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 = 0.09, size = 14, normalized size = 0.67

method	result	size
default	\(\frac {2}{3 \sin \left (x \right )^{\frac {3}{2}}}-\frac {2}{7 \sin \left (x \right )^{\frac {7}{2}}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{3 \, \sin \left (x\right )^{\frac {3}{2}}} - \frac {2}{7 \, \sin \left (x\right )^{\frac {7}{2}}} \end {gather*}

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 = 0.35, size = 22, normalized size = 1.05 \begin {gather*} \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 {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.46, size = 14, normalized size = 0.67 \begin {gather*} \frac {2 \, {\left (7 \, \sin \left (x\right )^{2} - 3\right )}}{21 \, \sin \left (x\right )^{\frac {7}{2}}} \end {gather*}

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)/sin(x)^(7/2)

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Mupad [B]
time = 0.33, size = 16, normalized size = 0.76 \begin {gather*} \frac {2\,\left (7\,{\sin \left (x\right )}^2-3\right )\,{\left (\frac {1}{\sin \left (x\right )}\right )}^{7/2}}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^3*(1/sin(x))^(9/2),x)

[Out]

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

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