3.826 \(\int \frac{\csc (\frac{1}{x})}{x^2} \, dx\)

Optimal. Leaf size=5 \[ \tanh ^{-1}\left (\cos \left (\frac{1}{x}\right )\right ) \]

[Out]

ArcTanh[Cos[x^(-1)]]

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Rubi [A]  time = 0.0090749, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4205, 3770} \[ \tanh ^{-1}\left (\cos \left (\frac{1}{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[Csc[x^(-1)]/x^2,x]

[Out]

ArcTanh[Cos[x^(-1)]]

Rule 4205

Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Csc[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]

Rule 3770

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

Rubi steps

\begin{align*} \int \frac{\csc \left (\frac{1}{x}\right )}{x^2} \, dx &=-\operatorname{Subst}\left (\int \csc (x) \, dx,x,\frac{1}{x}\right )\\ &=\tanh ^{-1}\left (\cos \left (\frac{1}{x}\right )\right )\\ \end{align*}

Mathematica [B]  time = 0.0148051, size = 21, normalized size = 4.2 \[ \log \left (\cos \left (\frac{1}{2 x}\right )\right )-\log \left (\sin \left (\frac{1}{2 x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x^(-1)]/x^2,x]

[Out]

Log[Cos[1/(2*x)]] - Log[Sin[1/(2*x)]]

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Maple [A]  time = 0.003, size = 11, normalized size = 2.2 \begin{align*} \ln \left ( \csc \left ({x}^{-1} \right ) +\cot \left ({x}^{-1} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(1/x)/x^2,x)

[Out]

ln(csc(1/x)+cot(1/x))

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Maxima [A]  time = 0.96226, size = 14, normalized size = 2.8 \begin{align*} \log \left (\cot \left (\frac{1}{x}\right ) + \csc \left (\frac{1}{x}\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(1/x)/x^2,x, algorithm="maxima")

[Out]

log(cot(1/x) + csc(1/x))

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Fricas [B]  time = 2.29585, size = 81, normalized size = 16.2 \begin{align*} \frac{1}{2} \, \log \left (\frac{1}{2} \, \cos \left (\frac{1}{x}\right ) + \frac{1}{2}\right ) - \frac{1}{2} \, \log \left (-\frac{1}{2} \, \cos \left (\frac{1}{x}\right ) + \frac{1}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(1/x)/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.35122, size = 10, normalized size = 2. \begin{align*} \log{\left (\cot{\left (\frac{1}{x} \right )} + \csc{\left (\frac{1}{x} \right )} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(1/x)/x**2,x)

[Out]

log(cot(1/x) + csc(1/x))

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Giac [A]  time = 1.14418, size = 14, normalized size = 2.8 \begin{align*} -\log \left ({\left | \tan \left (\frac{1}{2 \, x}\right ) \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(1/x)/x^2,x, algorithm="giac")

[Out]

-log(abs(tan(1/2/x)))