3.760 \(\int x \cot (x^2) \, dx\)

Optimal. Leaf size=9 \[ \frac{1}{2} \log \left (\sin \left (x^2\right )\right ) \]

[Out]

Log[Sin[x^2]]/2

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Rubi [A]  time = 0.0066126, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3748, 3475} \[ \frac{1}{2} \log \left (\sin \left (x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[x*Cot[x^2],x]

[Out]

Log[Sin[x^2]]/2

Rule 3748

Int[((a_.) + Cot[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Cot[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 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 x \cot \left (x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \cot (x) \, dx,x,x^2\right )\\ &=\frac{1}{2} \log \left (\sin \left (x^2\right )\right )\\ \end{align*}

Mathematica [B]  time = 0.0086499, size = 19, normalized size = 2.11 \[ \frac{1}{2} \log \left (\tan \left (x^2\right )\right )+\frac{1}{2} \log \left (\cos \left (x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cot[x^2],x]

[Out]

Log[Cos[x^2]]/2 + Log[Tan[x^2]]/2

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Maple [A]  time = 0.002, size = 8, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( \sin \left ({x}^{2} \right ) \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cot(x^2),x)

[Out]

1/2*ln(sin(x^2))

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Maxima [A]  time = 0.969597, size = 9, normalized size = 1. \begin{align*} \frac{1}{2} \, \log \left (\sin \left (x^{2}\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cot(x^2),x, algorithm="maxima")

[Out]

1/2*log(sin(x^2))

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Fricas [A]  time = 2.08113, size = 43, normalized size = 4.78 \begin{align*} \frac{1}{4} \, \log \left (-\frac{1}{2} \, \cos \left (2 \, x^{2}\right ) + \frac{1}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cot(x^2),x, algorithm="fricas")

[Out]

1/4*log(-1/2*cos(2*x^2) + 1/2)

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Sympy [B]  time = 0.151614, size = 19, normalized size = 2.11 \begin{align*} - \frac{\log{\left (\tan ^{2}{\left (x^{2} \right )} + 1 \right )}}{4} + \frac{\log{\left (\tan{\left (x^{2} \right )} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cot(x**2),x)

[Out]

-log(tan(x**2)**2 + 1)/4 + log(tan(x**2))/2

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Giac [A]  time = 1.11234, size = 16, normalized size = 1.78 \begin{align*} \frac{1}{4} \, \log \left ({\left | \cos \left (x^{2}\right )^{2} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cot(x^2),x, algorithm="giac")

[Out]

1/4*log(abs(cos(x^2)^2 - 1))