3.1.66 \(\int \frac {1}{x (1+\log ^2(x))} \, dx\) [66]

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

[Out]

arctan(ln(x))

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Rubi [A]
time = 0.01, antiderivative size = 3, normalized size of antiderivative = 1.00, 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 = {209} \begin {gather*} \tan ^{-1}(\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(1 + Log[x]^2)),x]

[Out]

ArcTan[Log[x]]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 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 &=\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\log (x)\right )\\ &=\tan ^{-1}(\log (x))\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 3, normalized size = 1.00 \begin {gather*} \tan ^{-1}(\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 + Log[x]^2)),x]

[Out]

ArcTan[Log[x]]

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Mathics [C] Result contains complex when optimal does not.
time = 1.87, size = 22, normalized size = 7.33 \begin {gather*} \text {RootSum}\left [1+4 \text {\#1}^2\&,\text {Log}\left [\text {Log}\left [x\right ]+2 \text {\#1}\right ] \text {\#1}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(x*(1 + Log[x]^2)),x]')

[Out]

RootSum[1 + 4 #1 ^ 2&, Log[Log[x] + 2 #1] #1&]

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Maple [A]
time = 0.00, size = 4, normalized size = 1.33

method	result	size
derivativedivides	\(\arctan \left (\ln \left (x \right )\right )\)	\(4\)
default	\(\arctan \left (\ln \left (x \right )\right )\)	\(4\)
risch	\(\frac {i \ln \left (\ln \left (x \right )+i\right )}{2}-\frac {i \ln \left (\ln \left (x \right )-i\right )}{2}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(1+ln(x)^2),x,method=_RETURNVERBOSE)

[Out]

arctan(ln(x))

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Maxima [A]
time = 0.40, size = 3, normalized size = 1.00 \begin {gather*} \arctan \left (\log \left (x\right )\right ) \end {gather*}

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 = 0.32, size = 3, normalized size = 1.00 \begin {gather*} \arctan \left (\log \left (x\right )\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (3) = 6\)
time = 0.07, size = 15, normalized size = 5.00 \begin {gather*} \operatorname {RootSum} {\left (4 z^{2} + 1, \left ( i \mapsto i \log {\left (2 i + \log {\left (x \right )} \right )} \right )\right )} \end {gather*}

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 = 0.00, size = 3, normalized size = 1.00 \begin {gather*} \arctan \left (\ln x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(1+log(x)^2),x)

[Out]

arctan(log(x))

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Mupad [B]
time = 0.00, size = 3, normalized size = 1.00 \begin {gather*} \mathrm {atan}\left (\ln \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(log(x)^2 + 1)),x)

[Out]

atan(log(x))

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