3.34 \(\int \frac {\tan ^{-1}(x) \log (x)}{x} \, dx\)

Optimal. Leaf size=57 \[ -\frac {1}{2} i \operatorname {PolyLog}(3,-i x)+\frac {1}{2} i \operatorname {PolyLog}(3,i x)+\frac {1}{2} i \log (x) \operatorname {PolyLog}(2,-i x)-\frac {1}{2} i \log (x) \operatorname {PolyLog}(2,i x) \]

[Out]

1/2*I*ln(x)*polylog(2,-I*x)-1/2*I*ln(x)*polylog(2,I*x)-1/2*I*polylog(3,-I*x)+1/2*I*polylog(3,I*x)

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Rubi [A]  time = 0.08, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4848, 2391, 5005, 2374, 6589} \[ -\frac {1}{2} i \text {PolyLog}(3,-i x)+\frac {1}{2} i \text {PolyLog}(3,i x)+\frac {1}{2} i \log (x) \text {PolyLog}(2,-i x)-\frac {1}{2} i \log (x) \text {PolyLog}(2,i x) \]

Antiderivative was successfully verified.

[In]

Int[(ArcTan[x]*Log[x])/x,x]

[Out]

(I/2)*Log[x]*PolyLog[2, (-I)*x] - (I/2)*Log[x]*PolyLog[2, I*x] - (I/2)*PolyLog[3, (-I)*x] + (I/2)*PolyLog[3, I
*x]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 4848

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I*c*x
]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rule 5005

Int[(ArcTan[(c_.)*(x_)^(n_.)]*Log[(d_.)*(x_)^(m_.)])/(x_), x_Symbol] :> Dist[I/2, Int[(Log[d*x^m]*Log[1 - I*c*
x^n])/x, x], x] - Dist[I/2, Int[(Log[d*x^m]*Log[1 + I*c*x^n])/x, x], x] /; FreeQ[{c, d, m, n}, x]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {\tan ^{-1}(x) \log (x)}{x} \, dx &=\frac {1}{2} i \int \frac {\log (1-i x) \log (x)}{x} \, dx-\frac {1}{2} i \int \frac {\log (1+i x) \log (x)}{x} \, dx\\ &=\frac {1}{2} i \log (x) \text {Li}_2(-i x)-\frac {1}{2} i \log (x) \text {Li}_2(i x)-\frac {1}{2} i \int \frac {\text {Li}_2(-i x)}{x} \, dx+\frac {1}{2} i \int \frac {\text {Li}_2(i x)}{x} \, dx\\ &=\frac {1}{2} i \log (x) \text {Li}_2(-i x)-\frac {1}{2} i \log (x) \text {Li}_2(i x)-\frac {1}{2} i \text {Li}_3(-i x)+\frac {1}{2} i \text {Li}_3(i x)\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 44, normalized size = 0.77 \[ \frac {1}{2} i (-\operatorname {PolyLog}(3,-i x)+\operatorname {PolyLog}(3,i x)+\log (x) \operatorname {PolyLog}(2,-i x)-\log (x) \operatorname {PolyLog}(2,i x)) \]

Antiderivative was successfully verified.

[In]

Integrate[(ArcTan[x]*Log[x])/x,x]

[Out]

(I/2)*(Log[x]*PolyLog[2, (-I)*x] - Log[x]*PolyLog[2, I*x] - PolyLog[3, (-I)*x] + PolyLog[3, I*x])

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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arctan \relax (x) \log \relax (x)}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x)*log(x)/x,x, algorithm="fricas")

[Out]

integral(arctan(x)*log(x)/x, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \relax (x) \log \relax (x)}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x)*log(x)/x,x, algorithm="giac")

[Out]

integrate(arctan(x)*log(x)/x, x)

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maple [F]  time = 0.59, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \relax (x ) \ln \relax (x )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(x)*ln(x)/x,x)

[Out]

int(arctan(x)*ln(x)/x,x)

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maxima [A]  time = 1.15, size = 31, normalized size = 0.54 \[ -\frac {1}{2} i \, {\rm Li}_2\left (i \, x\right ) \log \relax (x) + \frac {1}{2} i \, {\rm Li}_2\left (-i \, x\right ) \log \relax (x) + \frac {1}{2} i \, {\rm Li}_{3}(i \, x) - \frac {1}{2} i \, {\rm Li}_{3}(-i \, x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x)*log(x)/x,x, algorithm="maxima")

[Out]

-1/2*I*dilog(I*x)*log(x) + 1/2*I*dilog(-I*x)*log(x) + 1/2*I*polylog(3, I*x) - 1/2*I*polylog(3, -I*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {atan}\relax (x)\,\ln \relax (x)}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((atan(x)*log(x))/x,x)

[Out]

int((atan(x)*log(x))/x, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\relax (x )} \operatorname {atan}{\relax (x )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(x)*ln(x)/x,x)

[Out]

Integral(log(x)*atan(x)/x, x)

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