### 3.133 $$\int \frac{1}{x (1+\log ^2(x))} \, dx$$

Optimal. Leaf size=3 $\tan ^{-1}(\log (x))$

[Out]

ArcTan[Log[x]]

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Rubi [A]  time = 0.0200639, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {203} $\tan ^{-1}(\log (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(x*(1 + Log[x]^2)),x]

[Out]

ArcTan[Log[x]]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0116724, size = 3, normalized size = 1. $\tan ^{-1}(\log (x))$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(x*(1 + Log[x]^2)),x]

[Out]

ArcTan[Log[x]]

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Maple [A]  time = 0.004, size = 4, normalized size = 1.3 \begin{align*} \arctan \left ( \ln \left ( x \right ) \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(1+ln(x)^2),x)

[Out]

arctan(ln(x))

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Maxima [A]  time = 1.605, size = 4, normalized size = 1.33 \begin{align*} \arctan \left (\log \left (x\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(log(x))

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Fricas [A]  time = 2.01827, size = 22, normalized size = 7.33 \begin{align*} \arctan \left (\log \left (x\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(log(x))

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Sympy [B]  time = 0.136765, size = 15, normalized size = 5. \begin{align*} \operatorname{RootSum}{\left (4 z^{2} + 1, \left ( i \mapsto i \log{\left (2 i + \log{\left (x \right )} \right )} \right )\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(1+ln(x)**2),x)

[Out]

RootSum(4*_z**2 + 1, Lambda(_i, _i*log(2*_i + log(x))))

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Giac [A]  time = 1.27133, size = 4, normalized size = 1.33 \begin{align*} \arctan \left (\log \left (x\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(log(x))