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

________________________________________________________________________________________

Rubi [A]  time = 0.0165183, antiderivative size = 10, normalized size of antiderivative = 1., 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 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 30

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

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 = 0.0076398, size = 10, normalized size = 1. \[ -\frac{3}{4} \csc ^{\frac{4}{3}}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.026, size = 7, normalized size = 0.7 \begin{align*} -{\frac{3}{4} \left ( \csc \left ( x \right ) \right ) ^{{\frac{4}{3}}}} \end{align*}

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 = 0.962924, size = 8, normalized size = 0.8 \begin{align*} -\frac{3}{4 \, \sin \left (x\right )^{\frac{4}{3}}} \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.04112, size = 26, normalized size = 2.6 \begin{align*} -\frac{3}{4 \, \sin \left (x\right )^{\frac{4}{3}}} \end{align*}

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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.14886, size = 8, normalized size = 0.8 \begin{align*} -\frac{3}{4 \, \sin \left (x\right )^{\frac{4}{3}}} \end{align*}

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)