\(\int \frac {x^3 \cot ^{-1}(x)}{1+x^2} \, dx\) [38]

Optimal result

Integrand size = 13, antiderivative size = 67 \[ \int \frac {x^3 \cot ^{-1}(x)}{1+x^2} \, dx=\frac {x}{2}+\frac {1}{2} x^2 \cot ^{-1}(x)-\frac {1}{2} i \cot ^{-1}(x)^2-\frac {\arctan (x)}{2}+\cot ^{-1}(x) \log \left (\frac {2}{1+i x}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i x}\right ) \]

[Out]

1/2*x+1/2*x^2*arccot(x)-1/2*I*arccot(x)^2-1/2*arctan(x)+arccot(x)*ln(2/(1+I*x))-1/2*I*polylog(2,1-2/(1+I*x))

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {5037, 4947, 327, 209, 5041, 4965, 2449, 2352} \[ \int \frac {x^3 \cot ^{-1}(x)}{1+x^2} \, dx=-\frac {\arctan (x)}{2}-\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{i x+1}\right )+\frac {1}{2} x^2 \cot ^{-1}(x)+\frac {x}{2}-\frac {1}{2} i \cot ^{-1}(x)^2+\log \left (\frac {2}{1+i x}\right ) \cot ^{-1}(x) \]

[In]

Int[(x^3*ArcCot[x])/(1 + x^2),x]

[Out]

x/2 + (x^2*ArcCot[x])/2 - (I/2)*ArcCot[x]^2 - ArcTan[x]/2 + ArcCot[x]*Log[2/(1 + I*x)] - (I/2)*PolyLog[2, 1 -
2/(1 + I*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])

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2352

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

Rule 2449

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 4947

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCot[c*x^
n])^p/(m + 1)), x] + Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1]

Rule 4965

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

Rule 5037

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

Rule 5041

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

Rubi steps \begin{align*} \text {integral}& = \int x \cot ^{-1}(x) \, dx-\int \frac {x \cot ^{-1}(x)}{1+x^2} \, dx \\ & = \frac {1}{2} x^2 \cot ^{-1}(x)-\frac {1}{2} i \cot ^{-1}(x)^2+\frac {1}{2} \int \frac {x^2}{1+x^2} \, dx+\int \frac {\cot ^{-1}(x)}{i-x} \, dx \\ & = \frac {x}{2}+\frac {1}{2} x^2 \cot ^{-1}(x)-\frac {1}{2} i \cot ^{-1}(x)^2+\cot ^{-1}(x) \log \left (\frac {2}{1+i x}\right )-\frac {1}{2} \int \frac {1}{1+x^2} \, dx+\int \frac {\log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx \\ & = \frac {x}{2}+\frac {1}{2} x^2 \cot ^{-1}(x)-\frac {1}{2} i \cot ^{-1}(x)^2-\frac {\arctan (x)}{2}+\cot ^{-1}(x) \log \left (\frac {2}{1+i x}\right )-i \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i x}\right ) \\ & = \frac {x}{2}+\frac {1}{2} x^2 \cot ^{-1}(x)-\frac {1}{2} i \cot ^{-1}(x)^2-\frac {\arctan (x)}{2}+\cot ^{-1}(x) \log \left (\frac {2}{1+i x}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.76 \[ \int \frac {x^3 \cot ^{-1}(x)}{1+x^2} \, dx=\frac {1}{2} \left (x-i \cot ^{-1}(x)^2+\cot ^{-1}(x) \left (1+x^2+2 \log \left (1-e^{2 i \cot ^{-1}(x)}\right )\right )-i \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(x)}\right )\right ) \]

[In]

Integrate[(x^3*ArcCot[x])/(1 + x^2),x]

[Out]

(x - I*ArcCot[x]^2 + ArcCot[x]*(1 + x^2 + 2*Log[1 - E^((2*I)*ArcCot[x])]) - I*PolyLog[2, E^((2*I)*ArcCot[x])])
/2

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (53 ) = 106\).

Time = 0.74 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.88

method	result	size
default	\(\frac {x^{2} \operatorname {arccot}\left (x \right )}{2}-\frac {\operatorname {arccot}\left (x \right ) \ln \left (x^{2}+1\right )}{2}+\frac {x}{2}-\frac {\arctan \left (x \right )}{2}+\frac {i \left (\ln \left (x -i\right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (i+x \right )}{2}\right )-\ln \left (x -i\right ) \ln \left (-\frac {i \left (i+x \right )}{2}\right )\right )}{4}-\frac {i \left (\ln \left (i+x \right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (i+x \right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (x -i\right )}{2}\right )-\ln \left (i+x \right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )\right )}{4}\)	\(126\)
parts	\(\frac {x^{2} \operatorname {arccot}\left (x \right )}{2}-\frac {\operatorname {arccot}\left (x \right ) \ln \left (x^{2}+1\right )}{2}+\frac {x}{2}-\frac {\arctan \left (x \right )}{2}+\frac {i \left (\ln \left (x -i\right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (i+x \right )}{2}\right )-\ln \left (x -i\right ) \ln \left (-\frac {i \left (i+x \right )}{2}\right )\right )}{4}-\frac {i \left (\ln \left (i+x \right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (i+x \right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (x -i\right )}{2}\right )-\ln \left (i+x \right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )\right )}{4}\)	\(126\)
risch	\(\frac {\pi \,x^{2}}{4}+\frac {\pi }{4}-\frac {\pi \ln \left (x^{2}+1\right )}{4}-\frac {i \ln \left (-i x +1\right ) x^{2}}{4}+\frac {i \ln \left (\frac {1}{2}+\frac {i x}{2}\right ) \ln \left (-i x +1\right )}{4}-\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i x}{2}\right )}{4}+\frac {x}{2}+\frac {i \operatorname {dilog}\left (\frac {1}{2}+\frac {i x}{2}\right )}{4}+\frac {i \ln \left (-i x +1\right )^{2}}{8}-\frac {i \ln \left (\frac {1}{2}-\frac {i x}{2}\right ) \ln \left (i x +1\right )}{4}-\frac {i \ln \left (-i x +1\right )}{4}+\frac {i \ln \left (i x +1\right )}{4}+\frac {i \ln \left (i x +1\right ) x^{2}}{4}-\frac {i \ln \left (i x +1\right )^{2}}{8}\)	\(147\)

[In]

int(x^3*arccot(x)/(x^2+1),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {x^3 \cot ^{-1}(x)}{1+x^2} \, dx=\int { \frac {x^{3} \operatorname {arccot}\left (x\right )}{x^{2} + 1} \,d x } \]

[In]

integrate(x^3*arccot(x)/(x^2+1),x, algorithm="fricas")

[Out]

integral(x^3*arccot(x)/(x^2 + 1), x)

Sympy [F]

\[ \int \frac {x^3 \cot ^{-1}(x)}{1+x^2} \, dx=\int \frac {x^{3} \operatorname {acot}{\left (x \right )}}{x^{2} + 1}\, dx \]

[In]

integrate(x**3*acot(x)/(x**2+1),x)

[Out]

Integral(x**3*acot(x)/(x**2 + 1), x)

Maxima [F]

\[ \int \frac {x^3 \cot ^{-1}(x)}{1+x^2} \, dx=\int { \frac {x^{3} \operatorname {arccot}\left (x\right )}{x^{2} + 1} \,d x } \]

[In]

integrate(x^3*arccot(x)/(x^2+1),x, algorithm="maxima")

[Out]

integrate(x^3*arccot(x)/(x^2 + 1), x)

Giac [F]

\[ \int \frac {x^3 \cot ^{-1}(x)}{1+x^2} \, dx=\int { \frac {x^{3} \operatorname {arccot}\left (x\right )}{x^{2} + 1} \,d x } \]

[In]

integrate(x^3*arccot(x)/(x^2+1),x, algorithm="giac")

[Out]

integrate(x^3*arccot(x)/(x^2 + 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \cot ^{-1}(x)}{1+x^2} \, dx=\int \frac {x^3\,\mathrm {acot}\left (x\right )}{x^2+1} \,d x \]

[In]

int((x^3*acot(x))/(x^2 + 1),x)

[Out]

int((x^3*acot(x))/(x^2 + 1), x)