3.358 \(\int \cot ^3(x) \csc ^4(x) \, dx\)

Optimal. Leaf size=17 \[ \frac{\csc ^4(x)}{4}-\frac{\csc ^6(x)}{6} \]

[Out]

Csc[x]^4/4 - Csc[x]^6/6

________________________________________________________________________________________

Rubi [A]  time = 0.0260194, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2606, 14} \[ \frac{\csc ^4(x)}{4}-\frac{\csc ^6(x)}{6} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^3*Csc[x]^4,x]

[Out]

Csc[x]^4/4 - Csc[x]^6/6

Rule 2606

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

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 \cot ^3(x) \csc ^4(x) \, dx &=-\operatorname{Subst}\left (\int x^3 \left (-1+x^2\right ) \, dx,x,\csc (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-x^3+x^5\right ) \, dx,x,\csc (x)\right )\\ &=\frac{\csc ^4(x)}{4}-\frac{\csc ^6(x)}{6}\\ \end{align*}

Mathematica [A]  time = 0.0079304, size = 17, normalized size = 1. \[ \frac{\csc ^4(x)}{4}-\frac{\csc ^6(x)}{6} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^3*Csc[x]^4,x]

[Out]

Csc[x]^4/4 - Csc[x]^6/6

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 22, normalized size = 1.3 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{6\, \left ( \sin \left ( x \right ) \right ) ^{6}}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{12\, \left ( \sin \left ( x \right ) \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/sin(x)^6*cos(x)^4-1/12/sin(x)^4*cos(x)^4

________________________________________________________________________________________

Maxima [A]  time = 0.929369, size = 19, normalized size = 1.12 \begin{align*} \frac{3 \, \sin \left (x\right )^{2} - 2}{12 \, \sin \left (x\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*sin(x)^2 - 2)/sin(x)^6

________________________________________________________________________________________

Fricas [B]  time = 1.78264, size = 86, normalized size = 5.06 \begin{align*} \frac{3 \, \cos \left (x\right )^{2} - 1}{12 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*cos(x)^2 - 1)/(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)

________________________________________________________________________________________

Sympy [A]  time = 0.095965, size = 14, normalized size = 0.82 \begin{align*} \frac{3 \sin ^{2}{\left (x \right )} - 2}{12 \sin ^{6}{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**3/sin(x)**7,x)

[Out]

(3*sin(x)**2 - 2)/(12*sin(x)**6)

________________________________________________________________________________________

Giac [A]  time = 1.1006, size = 24, normalized size = 1.41 \begin{align*} \frac{3 \, \cos \left (x\right )^{2} - 1}{12 \,{\left (\cos \left (x\right )^{2} - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*cos(x)^2 - 1)/(cos(x)^2 - 1)^3