3.37 \(\int \frac{x^4 \cot ^{-1}(x)}{1+x^2} \, dx\)

Optimal. Leaf size=40 \[ \frac{x^2}{6}-\frac{2}{3} \log \left (x^2+1\right )+\frac{1}{3} x^3 \cot ^{-1}(x)-x \cot ^{-1}(x)-\frac{1}{2} \cot ^{-1}(x)^2 \]

[Out]

x^2/6 - x*ArcCot[x] + (x^3*ArcCot[x])/3 - ArcCot[x]^2/2 - (2*Log[1 + x^2])/3

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Rubi [A]  time = 0.0965447, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {4917, 4853, 266, 43, 4847, 260, 4885} \[ \frac{x^2}{6}-\frac{2}{3} \log \left (x^2+1\right )+\frac{1}{3} x^3 \cot ^{-1}(x)-x \cot ^{-1}(x)-\frac{1}{2} \cot ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]

Int[(x^4*ArcCot[x])/(1 + x^2),x]

[Out]

x^2/6 - x*ArcCot[x] + (x^3*ArcCot[x])/3 - ArcCot[x]^2/2 - (2*Log[1 + x^2])/3

Rule 4917

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 4853

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 4847

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

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 4885

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

Rubi steps

\begin{align*} \int \frac{x^4 \cot ^{-1}(x)}{1+x^2} \, dx &=\int x^2 \cot ^{-1}(x) \, dx-\int \frac{x^2 \cot ^{-1}(x)}{1+x^2} \, dx\\ &=\frac{1}{3} x^3 \cot ^{-1}(x)+\frac{1}{3} \int \frac{x^3}{1+x^2} \, dx-\int \cot ^{-1}(x) \, dx+\int \frac{\cot ^{-1}(x)}{1+x^2} \, dx\\ &=-x \cot ^{-1}(x)+\frac{1}{3} x^3 \cot ^{-1}(x)-\frac{1}{2} \cot ^{-1}(x)^2+\frac{1}{6} \operatorname{Subst}\left (\int \frac{x}{1+x} \, dx,x,x^2\right )-\int \frac{x}{1+x^2} \, dx\\ &=-x \cot ^{-1}(x)+\frac{1}{3} x^3 \cot ^{-1}(x)-\frac{1}{2} \cot ^{-1}(x)^2-\frac{1}{2} \log \left (1+x^2\right )+\frac{1}{6} \operatorname{Subst}\left (\int \left (1+\frac{1}{-1-x}\right ) \, dx,x,x^2\right )\\ &=\frac{x^2}{6}-x \cot ^{-1}(x)+\frac{1}{3} x^3 \cot ^{-1}(x)-\frac{1}{2} \cot ^{-1}(x)^2-\frac{2}{3} \log \left (1+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0274571, size = 32, normalized size = 0.8 \[ \frac{1}{6} \left (x^2-4 \log \left (x^2+1\right )+2 \left (x^2-3\right ) x \cot ^{-1}(x)-3 \cot ^{-1}(x)^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*ArcCot[x])/(1 + x^2),x]

[Out]

(x^2 + 2*x*(-3 + x^2)*ArcCot[x] - 3*ArcCot[x]^2 - 4*Log[1 + x^2])/6

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Maple [A]  time = 0.029, size = 38, normalized size = 1. \begin{align*}{\frac{{x}^{3}{\rm arccot} \left (x\right )}{3}}-x{\rm arccot} \left (x\right )+{\rm arccot} \left (x\right )\arctan \left ( x \right ) +{\frac{{x}^{2}}{6}}-{\frac{2\,\ln \left ({x}^{2}+1 \right ) }{3}}+{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arccot(x)/(x^2+1),x)

[Out]

1/3*x^3*arccot(x)-x*arccot(x)+arccot(x)*arctan(x)+1/6*x^2-2/3*ln(x^2+1)+1/2*arctan(x)^2

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Maxima [A]  time = 1.47917, size = 47, normalized size = 1.18 \begin{align*} \frac{1}{6} \, x^{2} + \frac{1}{3} \,{\left (x^{3} - 3 \, x + 3 \, \arctan \left (x\right )\right )} \operatorname{arccot}\left (x\right ) + \frac{1}{2} \, \arctan \left (x\right )^{2} - \frac{2}{3} \, \log \left (x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccot(x)/(x^2+1),x, algorithm="maxima")

[Out]

1/6*x^2 + 1/3*(x^3 - 3*x + 3*arctan(x))*arccot(x) + 1/2*arctan(x)^2 - 2/3*log(x^2 + 1)

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Fricas [A]  time = 1.99925, size = 100, normalized size = 2.5 \begin{align*} \frac{1}{6} \, x^{2} + \frac{1}{3} \,{\left (x^{3} - 3 \, x\right )} \operatorname{arccot}\left (x\right ) - \frac{1}{2} \, \operatorname{arccot}\left (x\right )^{2} - \frac{2}{3} \, \log \left (x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccot(x)/(x^2+1),x, algorithm="fricas")

[Out]

1/6*x^2 + 1/3*(x^3 - 3*x)*arccot(x) - 1/2*arccot(x)^2 - 2/3*log(x^2 + 1)

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Sympy [A]  time = 0.710818, size = 34, normalized size = 0.85 \begin{align*} \frac{x^{3} \operatorname{acot}{\left (x \right )}}{3} + \frac{x^{2}}{6} - x \operatorname{acot}{\left (x \right )} - \frac{2 \log{\left (x^{2} + 1 \right )}}{3} - \frac{\operatorname{acot}^{2}{\left (x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*acot(x)/(x**2+1),x)

[Out]

x**3*acot(x)/3 + x**2/6 - x*acot(x) - 2*log(x**2 + 1)/3 - acot(x)**2/2

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Giac [B]  time = 1.13736, size = 90, normalized size = 2.25 \begin{align*} \frac{1}{6} \, i x^{3} \log \left (-\frac{i - x}{i + x}\right ) - \frac{1}{2} \, i x \log \left (-\frac{i - x}{i + x}\right ) + \frac{1}{6} \, x^{2} + \frac{1}{8} \, \log \left (-\frac{i - x}{i + x}\right )^{2} - \frac{2}{3} \, \log \left (x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccot(x)/(x^2+1),x, algorithm="giac")

[Out]

1/6*i*x^3*log(-(i - x)/(i + x)) - 1/2*i*x*log(-(i - x)/(i + x)) + 1/6*x^2 + 1/8*log(-(i - x)/(i + x))^2 - 2/3*
log(x^2 + 1)