3.289 \(\int x^2 \tan ^{-1}(x) \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{3} x^3 \tan ^{-1}(x)-\frac {x^2}{6}+\frac {1}{6} \log \left (x^2+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4852, 266, 43} \[ -\frac {x^2}{6}+\frac {1}{6} \log \left (x^2+1\right )+\frac {1}{3} x^3 \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[x^2*ArcTan[x],x]

[Out]

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

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 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 4852

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
n[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTan[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]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.85 \[ \frac {1}{6} \left (2 x^3 \tan ^{-1}(x)-x^2+\log \left (x^2+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcTan[x],x]

[Out]

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

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fricas [A]  time = 0.43, size = 21, normalized size = 0.78 \[ \frac {1}{3} \, x^{3} \arctan \relax (x) - \frac {1}{6} \, x^{2} + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arctan(x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.90, size = 21, normalized size = 0.78 \[ \frac {1}{3} \, x^{3} \arctan \relax (x) - \frac {1}{6} \, x^{2} + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arctan(x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 22, normalized size = 0.81 \[ \frac {x^{3} \arctan \relax (x )}{3}-\frac {x^{2}}{6}+\frac {\ln \left (x^{2}+1\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arctan(x),x)

[Out]

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

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maxima [A]  time = 0.49, size = 21, normalized size = 0.78 \[ \frac {1}{3} \, x^{3} \arctan \relax (x) - \frac {1}{6} \, x^{2} + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arctan(x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.17, size = 21, normalized size = 0.78 \[ \frac {\ln \left (x^2+1\right )}{6}+\frac {x^3\,\mathrm {atan}\relax (x)}{3}-\frac {x^2}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*atan(x),x)

[Out]

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

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sympy [A]  time = 0.35, size = 20, normalized size = 0.74 \[ \frac {x^{3} \operatorname {atan}{\relax (x )}}{3} - \frac {x^{2}}{6} + \frac {\log {\left (x^{2} + 1 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*atan(x),x)

[Out]

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

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