3.49 \(\int \frac{\cot ^{-1}(c x)}{x (1+x^2)} \, dx\)

Optimal. Leaf size=223 \[ -\frac{1}{2} i \text{PolyLog}\left (2,-\frac{i}{c x}\right )+\frac{1}{2} i \text{PolyLog}\left (2,\frac{i}{c x}\right )+\frac{1}{2} i \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )-\frac{1}{4} i \text{PolyLog}\left (2,1-\frac{2 i c (-x+i)}{(1-c) (1-i c x)}\right )-\frac{1}{4} i \text{PolyLog}\left (2,1+\frac{2 i c (x+i)}{(c+1) (1-i c x)}\right )+\log \left (\frac{2}{1-i c x}\right ) \cot ^{-1}(c x)-\frac{1}{2} \log \left (\frac{2 i c (-x+i)}{(1-c) (1-i c x)}\right ) \cot ^{-1}(c x)-\frac{1}{2} \log \left (-\frac{2 i c (x+i)}{(c+1) (1-i c x)}\right ) \cot ^{-1}(c x) \]

[Out]

ArcCot[c*x]*Log[2/(1 - I*c*x)] - (ArcCot[c*x]*Log[((2*I)*c*(I - x))/((1 - c)*(1 - I*c*x))])/2 - (ArcCot[c*x]*L
og[((-2*I)*c*(I + x))/((1 + c)*(1 - I*c*x))])/2 - (I/2)*PolyLog[2, (-I)/(c*x)] + (I/2)*PolyLog[2, I/(c*x)] + (
I/2)*PolyLog[2, 1 - 2/(1 - I*c*x)] - (I/4)*PolyLog[2, 1 - ((2*I)*c*(I - x))/((1 - c)*(1 - I*c*x))] - (I/4)*Pol
yLog[2, 1 + ((2*I)*c*(I + x))/((1 + c)*(1 - I*c*x))]

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Rubi [A]  time = 0.24714, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {4929, 4849, 2391, 4857, 2402, 2315, 2447} \[ -\frac{1}{2} i \text{PolyLog}\left (2,-\frac{i}{c x}\right )+\frac{1}{2} i \text{PolyLog}\left (2,\frac{i}{c x}\right )+\frac{1}{2} i \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )-\frac{1}{4} i \text{PolyLog}\left (2,1-\frac{2 i c (-x+i)}{(1-c) (1-i c x)}\right )-\frac{1}{4} i \text{PolyLog}\left (2,1+\frac{2 i c (x+i)}{(c+1) (1-i c x)}\right )+\log \left (\frac{2}{1-i c x}\right ) \cot ^{-1}(c x)-\frac{1}{2} \log \left (\frac{2 i c (-x+i)}{(1-c) (1-i c x)}\right ) \cot ^{-1}(c x)-\frac{1}{2} \log \left (-\frac{2 i c (x+i)}{(c+1) (1-i c x)}\right ) \cot ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[c*x]/(x*(1 + x^2)),x]

[Out]

ArcCot[c*x]*Log[2/(1 - I*c*x)] - (ArcCot[c*x]*Log[((2*I)*c*(I - x))/((1 - c)*(1 - I*c*x))])/2 - (ArcCot[c*x]*L
og[((-2*I)*c*(I + x))/((1 + c)*(1 - I*c*x))])/2 - (I/2)*PolyLog[2, (-I)/(c*x)] + (I/2)*PolyLog[2, I/(c*x)] + (
I/2)*PolyLog[2, 1 - 2/(1 - I*c*x)] - (I/4)*PolyLog[2, 1 - ((2*I)*c*(I - x))/((1 - c)*(1 - I*c*x))] - (I/4)*Pol
yLog[2, 1 + ((2*I)*c*(I + x))/((1 + c)*(1 - I*c*x))]

Rule 4929

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcCot[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[a,
0])

Rule 4849

Int[((a_.) + ArcCot[(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 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 4857

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcCot[c*x])*Log[2/(1 -
I*c*x)])/e, x] + (-Dist[(b*c)/e, Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] + Dist[(b*c)/e, Int[Log[(2*c*(d
+ e*x))/((c*d + I*e)*(1 - I*c*x))]/(1 + c^2*x^2), x], x] + Simp[((a + b*ArcCot[c*x])*Log[(2*c*(d + e*x))/((c*
d + I*e)*(1 - I*c*x))])/e, x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rubi steps

\begin{align*} \int \frac{\cot ^{-1}(c x)}{x \left (1+x^2\right )} \, dx &=\int \left (\frac{\cot ^{-1}(c x)}{x}-\frac{x \cot ^{-1}(c x)}{1+x^2}\right ) \, dx\\ &=\int \frac{\cot ^{-1}(c x)}{x} \, dx-\int \frac{x \cot ^{-1}(c x)}{1+x^2} \, dx\\ &=\frac{1}{2} i \int \frac{\log \left (1-\frac{i}{c x}\right )}{x} \, dx-\frac{1}{2} i \int \frac{\log \left (1+\frac{i}{c x}\right )}{x} \, dx-\int \left (-\frac{\cot ^{-1}(c x)}{2 (i-x)}+\frac{\cot ^{-1}(c x)}{2 (i+x)}\right ) \, dx\\ &=-\frac{1}{2} i \text{Li}_2\left (-\frac{i}{c x}\right )+\frac{1}{2} i \text{Li}_2\left (\frac{i}{c x}\right )+\frac{1}{2} \int \frac{\cot ^{-1}(c x)}{i-x} \, dx-\frac{1}{2} \int \frac{\cot ^{-1}(c x)}{i+x} \, dx\\ &=\cot ^{-1}(c x) \log \left (\frac{2}{1-i c x}\right )-\frac{1}{2} \cot ^{-1}(c x) \log \left (\frac{2 i c (i-x)}{(1-c) (1-i c x)}\right )-\frac{1}{2} \cot ^{-1}(c x) \log \left (-\frac{2 i c (i+x)}{(1+c) (1-i c x)}\right )-\frac{1}{2} i \text{Li}_2\left (-\frac{i}{c x}\right )+\frac{1}{2} i \text{Li}_2\left (\frac{i}{c x}\right )+2 \left (\frac{1}{2} c \int \frac{\log \left (\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx\right )-\frac{1}{2} c \int \frac{\log \left (\frac{2 c (i-x)}{(-i+i c) (1-i c x)}\right )}{1+c^2 x^2} \, dx-\frac{1}{2} c \int \frac{\log \left (\frac{2 c (i+x)}{(i+i c) (1-i c x)}\right )}{1+c^2 x^2} \, dx\\ &=\cot ^{-1}(c x) \log \left (\frac{2}{1-i c x}\right )-\frac{1}{2} \cot ^{-1}(c x) \log \left (\frac{2 i c (i-x)}{(1-c) (1-i c x)}\right )-\frac{1}{2} \cot ^{-1}(c x) \log \left (-\frac{2 i c (i+x)}{(1+c) (1-i c x)}\right )-\frac{1}{2} i \text{Li}_2\left (-\frac{i}{c x}\right )+\frac{1}{2} i \text{Li}_2\left (\frac{i}{c x}\right )-\frac{1}{4} i \text{Li}_2\left (1-\frac{2 i c (i-x)}{(1-c) (1-i c x)}\right )-\frac{1}{4} i \text{Li}_2\left (1+\frac{2 i c (i+x)}{(1+c) (1-i c x)}\right )+2 \left (\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i c x}\right )\right )\\ &=\cot ^{-1}(c x) \log \left (\frac{2}{1-i c x}\right )-\frac{1}{2} \cot ^{-1}(c x) \log \left (\frac{2 i c (i-x)}{(1-c) (1-i c x)}\right )-\frac{1}{2} \cot ^{-1}(c x) \log \left (-\frac{2 i c (i+x)}{(1+c) (1-i c x)}\right )-\frac{1}{2} i \text{Li}_2\left (-\frac{i}{c x}\right )+\frac{1}{2} i \text{Li}_2\left (\frac{i}{c x}\right )+\frac{1}{2} i \text{Li}_2\left (1-\frac{2}{1-i c x}\right )-\frac{1}{4} i \text{Li}_2\left (1-\frac{2 i c (i-x)}{(1-c) (1-i c x)}\right )-\frac{1}{4} i \text{Li}_2\left (1+\frac{2 i c (i+x)}{(1+c) (1-i c x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.0912358, size = 379, normalized size = 1.7 \[ \frac{1}{4} i \text{PolyLog}\left (2,\frac{i c (-x+i)}{1-c}\right )-\frac{1}{4} i \text{PolyLog}\left (2,-\frac{i c (-x+i)}{c+1}\right )-\frac{1}{2} i \text{PolyLog}\left (2,-\frac{i}{c x}\right )+\frac{1}{2} i \text{PolyLog}\left (2,\frac{i}{c x}\right )-\frac{1}{4} i \text{PolyLog}\left (2,\frac{i c (x+i)}{1-c}\right )+\frac{1}{4} i \text{PolyLog}\left (2,-\frac{i c (x+i)}{c+1}\right )+\frac{1}{4} i \log (-x+i) \log \left (-\frac{i (-c x+i)}{1-c}\right )+\frac{1}{4} i \log (x+i) \log \left (-\frac{i (-c x+i)}{c+1}\right )-\frac{1}{4} i \log (-x+i) \log \left (-\frac{-c x+i}{c x}\right )-\frac{1}{4} i \log (x+i) \log \left (-\frac{-c x+i}{c x}\right )-\frac{1}{4} i \log (x+i) \log \left (-\frac{i (c x+i)}{1-c}\right )-\frac{1}{4} i \log (-x+i) \log \left (-\frac{i (c x+i)}{c+1}\right )+\frac{1}{4} i \log (-x+i) \log \left (\frac{c x+i}{c x}\right )+\frac{1}{4} i \log (x+i) \log \left (\frac{c x+i}{c x}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCot[c*x]/(x*(1 + x^2)),x]

[Out]

(I/4)*Log[I - x]*Log[((-I)*(I - c*x))/(1 - c)] + (I/4)*Log[I + x]*Log[((-I)*(I - c*x))/(1 + c)] - (I/4)*Log[I
- x]*Log[-((I - c*x)/(c*x))] - (I/4)*Log[I + x]*Log[-((I - c*x)/(c*x))] - (I/4)*Log[I + x]*Log[((-I)*(I + c*x)
)/(1 - c)] - (I/4)*Log[I - x]*Log[((-I)*(I + c*x))/(1 + c)] + (I/4)*Log[I - x]*Log[(I + c*x)/(c*x)] + (I/4)*Lo
g[I + x]*Log[(I + c*x)/(c*x)] + (I/4)*PolyLog[2, (I*c*(I - x))/(1 - c)] - (I/4)*PolyLog[2, ((-I)*c*(I - x))/(1
+ c)] - (I/2)*PolyLog[2, (-I)/(c*x)] + (I/2)*PolyLog[2, I/(c*x)] - (I/4)*PolyLog[2, (I*c*(I + x))/(1 - c)] +
(I/4)*PolyLog[2, ((-I)*c*(I + x))/(1 + c)]

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Maple [A]  time = 0.149, size = 345, normalized size = 1.6 \begin{align*} -{\frac{\ln \left ({c}^{2}{x}^{2}+{c}^{2} \right ){\rm arccot} \left (cx\right )}{2}}+{\rm arccot} \left (cx\right )\ln \left ( cx \right ) +{\frac{i}{4}}\ln \left ( cx+i \right ) \ln \left ({\frac{-i \left ( cx+i \right ) -c-1}{-c-1}} \right ) +{\frac{i}{4}}{\it dilog} \left ({\frac{-i \left ( cx+i \right ) +c-1}{-1+c}} \right ) +{\frac{i}{2}}{\it dilog} \left ( 1-icx \right ) -{\frac{i}{2}}{\it dilog} \left ( 1+icx \right ) +{\frac{i}{2}}\ln \left ( cx \right ) \ln \left ( 1-icx \right ) -{\frac{i}{4}}{\it dilog} \left ({\frac{i \left ( cx-i \right ) -c-1}{-c-1}} \right ) -{\frac{i}{4}}\ln \left ( cx+i \right ) \ln \left ({c}^{2}{x}^{2}+{c}^{2} \right ) -{\frac{i}{4}}\ln \left ( cx-i \right ) \ln \left ({\frac{i \left ( cx-i \right ) +c-1}{-1+c}} \right ) -{\frac{i}{4}}{\it dilog} \left ({\frac{i \left ( cx-i \right ) +c-1}{-1+c}} \right ) +{\frac{i}{4}}\ln \left ( cx-i \right ) \ln \left ({c}^{2}{x}^{2}+{c}^{2} \right ) -{\frac{i}{2}}\ln \left ( cx \right ) \ln \left ( 1+icx \right ) -{\frac{i}{4}}\ln \left ( cx-i \right ) \ln \left ({\frac{i \left ( cx-i \right ) -c-1}{-c-1}} \right ) +{\frac{i}{4}}\ln \left ( cx+i \right ) \ln \left ({\frac{-i \left ( cx+i \right ) +c-1}{-1+c}} \right ) +{\frac{i}{4}}{\it dilog} \left ({\frac{-i \left ( cx+i \right ) -c-1}{-c-1}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(c*x)/x/(x^2+1),x)

[Out]

-1/2*ln(c^2*x^2+c^2)*arccot(c*x)+arccot(c*x)*ln(c*x)+1/4*I*ln(c*x+I)*ln((-I*(c*x+I)-c-1)/(-c-1))+1/4*I*dilog((
-I*(c*x+I)+c-1)/(-1+c))+1/2*I*dilog(1-I*c*x)-1/2*I*dilog(1+I*c*x)+1/2*I*ln(c*x)*ln(1-I*c*x)-1/4*I*dilog((I*(c*
x-I)-c-1)/(-c-1))-1/4*I*ln(c*x+I)*ln(c^2*x^2+c^2)-1/4*I*ln(c*x-I)*ln((I*(c*x-I)+c-1)/(-1+c))-1/4*I*dilog((I*(c
*x-I)+c-1)/(-1+c))+1/4*I*ln(c*x-I)*ln(c^2*x^2+c^2)-1/2*I*ln(c*x)*ln(1+I*c*x)-1/4*I*ln(c*x-I)*ln((I*(c*x-I)-c-1
)/(-c-1))+1/4*I*ln(c*x+I)*ln((-I*(c*x+I)+c-1)/(-1+c))+1/4*I*dilog((-I*(c*x+I)-c-1)/(-c-1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\left (c x\right )}{{\left (x^{2} + 1\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(c*x)/x/(x^2+1),x, algorithm="maxima")

[Out]

integrate(arccot(c*x)/((x^2 + 1)*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arccot}\left (c x\right )}{x^{3} + x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(c*x)/x/(x^2+1),x, algorithm="fricas")

[Out]

integral(arccot(c*x)/(x^3 + x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acot}{\left (c x \right )}}{x \left (x^{2} + 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(c*x)/x/(x**2+1),x)

[Out]

Integral(acot(c*x)/(x*(x**2 + 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\left (c x\right )}{{\left (x^{2} + 1\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(c*x)/x/(x^2+1),x, algorithm="giac")

[Out]

integrate(arccot(c*x)/((x^2 + 1)*x), x)