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

Optimal. Leaf size=35 \[ \frac{1}{4} i \text{PolyLog}\left (2,\frac{i}{x+1}\right )-\frac{1}{4} i \text{PolyLog}\left (2,-\frac{i}{x+1}\right ) \]

[Out]

(-I/4)*PolyLog[2, (-I)/(1 + x)] + (I/4)*PolyLog[2, I/(1 + x)]

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Rubi [A]  time = 0.036544, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5044, 12, 4849, 2391} \[ \frac{1}{4} i \text{PolyLog}\left (2,\frac{i}{x+1}\right )-\frac{1}{4} i \text{PolyLog}\left (2,-\frac{i}{x+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[1 + x]/(2 + 2*x),x]

[Out]

(-I/4)*PolyLog[2, (-I)/(1 + x)] + (I/4)*PolyLog[2, I/(1 + x)]

Rule 5044

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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]

Rubi steps

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

Mathematica [A]  time = 0.0043733, size = 35, normalized size = 1. \[ \frac{1}{4} i \text{PolyLog}\left (2,\frac{i}{x+1}\right )-\frac{1}{4} i \text{PolyLog}\left (2,-\frac{i}{x+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[1 + x]/(2 + 2*x),x]

[Out]

(-I/4)*PolyLog[2, (-I)/(1 + x)] + (I/4)*PolyLog[2, I/(1 + x)]

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Maple [B]  time = 0.036, size = 68, normalized size = 1.9 \begin{align*}{\frac{\ln \left ( x+1 \right ){\rm arccot} \left (x+1\right )}{2}}-{\frac{i}{4}}\ln \left ( x+1 \right ) \ln \left ( 1+i \left ( x+1 \right ) \right ) +{\frac{i}{4}}\ln \left ( x+1 \right ) \ln \left ( 1-i \left ( x+1 \right ) \right ) -{\frac{i}{4}}{\it dilog} \left ( 1+i \left ( x+1 \right ) \right ) +{\frac{i}{4}}{\it dilog} \left ( 1-i \left ( x+1 \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*ln(x+1)*arccot(x+1)-1/4*I*ln(x+1)*ln(1+I*(x+1))+1/4*I*ln(x+1)*ln(1-I*(x+1))-1/4*I*dilog(1+I*(x+1))+1/4*I*d
ilog(1-I*(x+1))

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Maxima [B]  time = 1.59732, size = 86, normalized size = 2.46 \begin{align*} \frac{1}{4} \, \arctan \left (x + 1, 0\right ) \log \left (x^{2} + 2 \, x + 2\right ) + \frac{1}{2} \, \operatorname{arccot}\left (x + 1\right ) \log \left (x + 1\right ) + \frac{1}{2} \, \arctan \left (x + 1\right ) \log \left (x + 1\right ) - \frac{1}{2} \, \arctan \left (x + 1\right ) \log \left ({\left | x + 1 \right |}\right ) + \frac{1}{4} i \,{\rm Li}_2\left (i \, x + i + 1\right ) - \frac{1}{4} i \,{\rm Li}_2\left (-i \, x - i + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*arctan2(x + 1, 0)*log(x^2 + 2*x + 2) + 1/2*arccot(x + 1)*log(x + 1) + 1/2*arctan(x + 1)*log(x + 1) - 1/2*a
rctan(x + 1)*log(abs(x + 1)) + 1/4*I*dilog(I*x + I + 1) - 1/4*I*dilog(-I*x - I + 1)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arccot}\left (x + 1\right )}{2 \,{\left (x + 1\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/2*arccot(x + 1)/(x + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acot(x + 1)/(x + 1), x)/2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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