### 3.8 $$\int \csc ^2(x) \, dx$$

Optimal. Leaf size=4 $-\cot (x)$

[Out]

-Cot[x]

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Rubi [A]  time = 0.005372, antiderivative size = 4, 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 = {3767, 8} $-\cot (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Csc[x]^2,x]

[Out]

-Cot[x]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \csc ^2(x) \, dx &=-\operatorname{Subst}(\int 1 \, dx,x,\cot (x))\\ &=-\cot (x)\\ \end{align*}

Mathematica [A]  time = 0.0016446, size = 4, normalized size = 1. $-\cot (x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Csc[x]^2,x]

[Out]

-Cot[x]

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Maple [A]  time = 0.003, size = 5, normalized size = 1.3 \begin{align*} -\cot \left ( x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^2,x)

[Out]

-cot(x)

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Maxima [A]  time = 0.937823, size = 8, normalized size = 2. \begin{align*} -\frac{1}{\tan \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^2,x, algorithm="maxima")

[Out]

-1/tan(x)

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Fricas [B]  time = 1.84732, size = 22, normalized size = 5.5 \begin{align*} -\frac{\cos \left (x\right )}{\sin \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^2,x, algorithm="fricas")

[Out]

-cos(x)/sin(x)

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Sympy [B]  time = 0.05926, size = 7, normalized size = 1.75 \begin{align*} - \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)**2,x)

[Out]

-cos(x)/sin(x)

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Giac [A]  time = 1.08226, size = 8, normalized size = 2. \begin{align*} -\frac{1}{\tan \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^2,x, algorithm="giac")

[Out]

-1/tan(x)