### 3.100 $$\int \cot ^4(x) \csc ^4(x) \, dx$$

Optimal. Leaf size=17 $-\frac{1}{7} \cot ^7(x)-\frac{\cot ^5(x)}{5}$

[Out]

-Cot[x]^5/5 - Cot[x]^7/7

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Rubi [A]  time = 0.0260087, 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 = {2607, 14} $-\frac{1}{7} \cot ^7(x)-\frac{\cot ^5(x)}{5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Cot[x]^5/5 - Cot[x]^7/7

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 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 ^4(x) \csc ^4(x) \, dx &=\operatorname{Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (x)\right )\\ &=\operatorname{Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (x)\right )\\ &=-\frac{1}{5} \cot ^5(x)-\frac{\cot ^7(x)}{7}\\ \end{align*}

Mathematica [B]  time = 0.0266781, size = 37, normalized size = 2.18 $-\frac{2 \cot (x)}{35}-\frac{1}{7} \cot (x) \csc ^6(x)+\frac{8}{35} \cot (x) \csc ^4(x)-\frac{1}{35} \cot (x) \csc ^2(x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*Cot[x])/35 - (Cot[x]*Csc[x]^2)/35 + (8*Cot[x]*Csc[x]^4)/35 - (Cot[x]*Csc[x]^6)/7

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Maple [A]  time = 0.013, size = 22, normalized size = 1.3 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5}}{7\, \left ( \sin \left ( x \right ) \right ) ^{7}}}-{\frac{2\, \left ( \cos \left ( x \right ) \right ) ^{5}}{35\, \left ( \sin \left ( x \right ) \right ) ^{5}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^4*csc(x)^4,x)

[Out]

-1/7/sin(x)^7*cos(x)^5-2/35/sin(x)^5*cos(x)^5

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Maxima [A]  time = 0.933675, size = 19, normalized size = 1.12 \begin{align*} -\frac{7 \, \tan \left (x\right )^{2} + 5}{35 \, \tan \left (x\right )^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^4*csc(x)^4,x, algorithm="maxima")

[Out]

-1/35*(7*tan(x)^2 + 5)/tan(x)^7

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Fricas [B]  time = 1.95796, size = 112, normalized size = 6.59 \begin{align*} -\frac{2 \, \cos \left (x\right )^{7} - 7 \, \cos \left (x\right )^{5}}{35 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^4*csc(x)^4,x, algorithm="fricas")

[Out]

-1/35*(2*cos(x)^7 - 7*cos(x)^5)/((cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)*sin(x))

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Sympy [B]  time = 0.066887, size = 41, normalized size = 2.41 \begin{align*} - \frac{2 \cos{\left (x \right )}}{35 \sin{\left (x \right )}} - \frac{\cos{\left (x \right )}}{35 \sin ^{3}{\left (x \right )}} + \frac{8 \cos{\left (x \right )}}{35 \sin ^{5}{\left (x \right )}} - \frac{\cos{\left (x \right )}}{7 \sin ^{7}{\left (x \right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)**4*csc(x)**4,x)

[Out]

-2*cos(x)/(35*sin(x)) - cos(x)/(35*sin(x)**3) + 8*cos(x)/(35*sin(x)**5) - cos(x)/(7*sin(x)**7)

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Giac [A]  time = 1.07736, size = 19, normalized size = 1.12 \begin{align*} -\frac{7 \, \tan \left (x\right )^{2} + 5}{35 \, \tan \left (x\right )^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^4*csc(x)^4,x, algorithm="giac")

[Out]

-1/35*(7*tan(x)^2 + 5)/tan(x)^7