### 3.99 $$\int \cot ^3(x) \, dx$$

Optimal. Leaf size=14 $-\frac{1}{2} \cot ^2(x)-\log (\sin (x))$

[Out]

-Cot[x]^2/2 - Log[Sin[x]]

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Rubi [A]  time = 0.0074278, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, Rules used = {3473, 3475} $-\frac{1}{2} \cot ^2(x)-\log (\sin (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[Cot[x]^3,x]

[Out]

-Cot[x]^2/2 - Log[Sin[x]]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \cot ^3(x) \, dx &=-\frac{1}{2} \cot ^2(x)-\int \cot (x) \, dx\\ &=-\frac{1}{2} \cot ^2(x)-\log (\sin (x))\\ \end{align*}

Mathematica [A]  time = 0.0030025, size = 14, normalized size = 1. $-\frac{1}{2} \csc ^2(x)-\log (\sin (x))$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cot[x]^3,x]

[Out]

-Csc[x]^2/2 - Log[Sin[x]]

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Maple [A]  time = 0.002, size = 17, normalized size = 1.2 \begin{align*} -{\frac{ \left ( \cot \left ( x \right ) \right ) ^{2}}{2}}+{\frac{\ln \left ( \left ( \cot \left ( x \right ) \right ) ^{2}+1 \right ) }{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^3,x)

[Out]

-1/2*cot(x)^2+1/2*ln(cot(x)^2+1)

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Maxima [A]  time = 0.930429, size = 19, normalized size = 1.36 \begin{align*} -\frac{1}{2 \, \sin \left (x\right )^{2}} - \frac{1}{2} \, \log \left (\sin \left (x\right )^{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^3,x, algorithm="maxima")

[Out]

-1/2/sin(x)^2 - 1/2*log(sin(x)^2)

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Fricas [B]  time = 1.97195, size = 90, normalized size = 6.43 \begin{align*} -\frac{{\left (\cos \left (2 \, x\right ) - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (2 \, x\right ) + \frac{1}{2}\right ) - 2}{2 \,{\left (\cos \left (2 \, x\right ) - 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^3,x, algorithm="fricas")

[Out]

-1/2*((cos(2*x) - 1)*log(-1/2*cos(2*x) + 1/2) - 2)/(cos(2*x) - 1)

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Sympy [A]  time = 0.087574, size = 14, normalized size = 1. \begin{align*} - \log{\left (\sin{\left (x \right )} \right )} - \frac{1}{2 \sin ^{2}{\left (x \right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)**3,x)

[Out]

-log(sin(x)) - 1/(2*sin(x)**2)

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Giac [A]  time = 1.06149, size = 30, normalized size = 2.14 \begin{align*} \frac{1}{2 \,{\left (\cos \left (x\right )^{2} - 1\right )}} - \frac{1}{2} \, \log \left (-\cos \left (x\right )^{2} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^3,x, algorithm="giac")

[Out]

1/2/(cos(x)^2 - 1) - 1/2*log(-cos(x)^2 + 1)