3.270 \(\int \cos (x) \csc ^{\frac {7}{3}}(x) \, dx\)

Optimal. Leaf size=10 \[ -\frac {3}{4} \csc ^{\frac {4}{3}}(x) \]

[Out]

-3/4*csc(x)^(4/3)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2621, 30} \[ -\frac {3}{4} \csc ^{\frac {4}{3}}(x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Csc[x]^(7/3),x]

[Out]

(-3*Csc[x]^(4/3))/4

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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])

Rubi steps

\begin {align*} \int \cos (x) \csc ^{\frac {7}{3}}(x) \, dx &=-\operatorname {Subst}\left (\int \sqrt [3]{x} \, dx,x,\csc (x)\right )\\ &=-\frac {3}{4} \csc ^{\frac {4}{3}}(x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 10, normalized size = 1.00 \[ -\frac {3}{4} \csc ^{\frac {4}{3}}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Csc[x]^(7/3),x]

[Out]

(-3*Csc[x]^(4/3))/4

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 6, normalized size = 0.60 \[ -\frac {3}{4 \, \sin \relax (x)^{\frac {4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*csc(x)^(7/3),x, algorithm="fricas")

[Out]

-3/4/sin(x)^(4/3)

________________________________________________________________________________________

giac [A]  time = 0.37, size = 6, normalized size = 0.60 \[ -\frac {3}{4 \, \sin \relax (x)^{\frac {4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*csc(x)^(7/3),x, algorithm="giac")

[Out]

-3/4/sin(x)^(4/3)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 7, normalized size = 0.70 \[ -\frac {3 \left (\csc ^{\frac {4}{3}}\relax (x )\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*csc(x)^(7/3),x)

[Out]

-3/4*csc(x)^(4/3)

________________________________________________________________________________________

maxima [A]  time = 1.13, size = 6, normalized size = 0.60 \[ -\frac {3}{4 \, \sin \relax (x)^{\frac {4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*csc(x)^(7/3),x, algorithm="maxima")

[Out]

-3/4/sin(x)^(4/3)

________________________________________________________________________________________

mupad [B]  time = 0.20, size = 8, normalized size = 0.80 \[ -\frac {3\,{\left (\frac {1}{\sin \relax (x)}\right )}^{4/3}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*(1/sin(x))^(7/3),x)

[Out]

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

________________________________________________________________________________________

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(cos(x)*csc(x)**(7/3),x)

[Out]

Timed out

________________________________________________________________________________________