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

Optimal. Leaf size=223 \[ -\frac {1}{2} i \text {Li}_2\left (-\frac {i}{c x}\right )+\frac {1}{2} i \text {Li}_2\left (\frac {i}{c x}\right )+\frac {1}{2} i \text {Li}_2\left (1-\frac {2}{1-i c x}\right )-\frac {1}{4} i \text {Li}_2\left (1-\frac {2 i c (i-x)}{(1-c) (1-i c x)}\right )-\frac {1}{4} i \text {Li}_2\left (\frac {2 i c (x+i)}{(c+1) (1-i c x)}+1\right )+\log \left (\frac {2}{1-i c x}\right ) \cot ^{-1}(c x)-\frac {1}{2} \log \left (\frac {2 i c (-x+i)}{(1-c) (1-i c x)}\right ) \cot ^{-1}(c x)-\frac {1}{2} \log \left (-\frac {2 i c (x+i)}{(c+1) (1-i c x)}\right ) \cot ^{-1}(c x) \]

[Out]

arccot(c*x)*ln(2/(1-I*c*x))-1/2*arccot(c*x)*ln(2*I*c*(I-x)/(1-c)/(1-I*c*x))-1/2*arccot(c*x)*ln(-2*I*c*(I+x)/(1
+c)/(1-I*c*x))-1/2*I*polylog(2,-I/c/x)+1/2*I*polylog(2,I/c/x)+1/2*I*polylog(2,1-2/(1-I*c*x))-1/4*I*polylog(2,1
-2*I*c*(I-x)/(1-c)/(1-I*c*x))-1/4*I*polylog(2,1+2*I*c*(I+x)/(1+c)/(1-I*c*x))

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Rubi [A]  time = 0.25, antiderivative size = 223, normalized size of antiderivative = 1.00, 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 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 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 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 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]]

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 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 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])

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*}

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Mathematica [A]  time = 0.10, size = 379, normalized size = 1.70 \[ \frac {1}{4} i \text {Li}_2\left (\frac {i c (i-x)}{1-c}\right )-\frac {1}{4} i \text {Li}_2\left (-\frac {i c (i-x)}{c+1}\right )-\frac {1}{2} i \text {Li}_2\left (-\frac {i}{c x}\right )+\frac {1}{2} i \text {Li}_2\left (\frac {i}{c x}\right )-\frac {1}{4} i \text {Li}_2\left (\frac {i c (x+i)}{1-c}\right )+\frac {1}{4} i \text {Li}_2\left (-\frac {i c (x+i)}{c+1}\right )+\frac {1}{4} i \log (-x+i) \log \left (-\frac {i (-c x+i)}{1-c}\right )+\frac {1}{4} i \log (x+i) \log \left (-\frac {i (-c x+i)}{c+1}\right )-\frac {1}{4} i \log (-x+i) \log \left (-\frac {-c x+i}{c x}\right )-\frac {1}{4} i \log (x+i) \log \left (-\frac {-c x+i}{c x}\right )-\frac {1}{4} i \log (x+i) \log \left (-\frac {i (c x+i)}{1-c}\right )-\frac {1}{4} i \log (-x+i) \log \left (-\frac {i (c x+i)}{c+1}\right )+\frac {1}{4} i \log (-x+i) \log \left (\frac {c x+i}{c x}\right )+\frac {1}{4} i \log (x+i) \log \left (\frac {c x+i}{c x}\right ) \]

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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fricas [F]  time = 1.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccot}\left (c x\right )}{x^{3} + x}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccot}\left (c x\right )}{{\left (x^{2} + 1\right )} x}\,{d x} \]

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)

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maple [A]  time = 0.26, size = 345, normalized size = 1.55 \[ \mathrm {arccot}\left (c x \right ) \ln \left (c x \right )-\frac {\ln \left (c^{2} x^{2}+c^{2}\right ) \mathrm {arccot}\left (c x \right )}{2}+\frac {i \dilog \left (\frac {-i \left (c x +i\right )+c -1}{-1+c}\right )}{4}-\frac {i \ln \left (c x -i\right ) \ln \left (\frac {i \left (c x -i\right )-c -1}{-c -1}\right )}{4}+\frac {i \dilog \left (\frac {-i \left (c x +i\right )-c -1}{-c -1}\right )}{4}+\frac {i \ln \left (c x +i\right ) \ln \left (\frac {-i \left (c x +i\right )-c -1}{-c -1}\right )}{4}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}+\frac {i \ln \left (c x +i\right ) \ln \left (\frac {-i \left (c x +i\right )+c -1}{-1+c}\right )}{4}-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}-\frac {i \dilog \left (i c x +1\right )}{2}-\frac {i \ln \left (c x -i\right ) \ln \left (\frac {i \left (c x -i\right )+c -1}{-1+c}\right )}{4}-\frac {i \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+c^{2}\right )}{4}-\frac {i \dilog \left (\frac {i \left (c x -i\right )-c -1}{-c -1}\right )}{4}+\frac {i \dilog \left (-i c x +1\right )}{2}-\frac {i \dilog \left (\frac {i \left (c x -i\right )+c -1}{-1+c}\right )}{4}+\frac {i \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+c^{2}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccot}\left (c x\right )}{{\left (x^{2} + 1\right )} x}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {acot}\left (c\,x\right )}{x\,\left (x^2+1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(c*x)/(x*(x^2 + 1)),x)

[Out]

int(acot(c*x)/(x*(x^2 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acot}{\left (c x \right )}}{x \left (x^{2} + 1\right )}\, dx \]

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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