3.1.94 \(\int x \tan ^{-1}(x)^2 \, dx\) [94]

Optimal. Leaf size=35 \[ -x \tan ^{-1}(x)+\frac {1}{2} \tan ^{-1}(x)^2+\frac {1}{2} x^2 \tan ^{-1}(x)^2+\frac {1}{2} \log \left (1+x^2\right ) \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {4946, 5036, 4930, 266, 5004} \begin {gather*} \frac {1}{2} x^2 \text {ArcTan}(x)^2+\frac {\text {ArcTan}(x)^2}{2}-x \text {ArcTan}(x)+\frac {1}{2} \log \left (x^2+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 266

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 4930

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

Rule 4946

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^
n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTan[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 5004

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[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]

Rule 5036

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

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 26, normalized size = 0.74 \begin {gather*} \frac {1}{2} \left (-2 x \tan ^{-1}(x)+\left (1+x^2\right ) \tan ^{-1}(x)^2+\log \left (1+x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.04, size = 30, normalized size = 0.86

method	result	size
default	\(-x \arctan \left (x \right )+\frac {\arctan \left (x \right )^{2}}{2}+\frac {x^{2} \arctan \left (x \right )^{2}}{2}+\frac {\ln \left (x^{2}+1\right )}{2}\)	\(30\)
risch	\(-\frac {\left (\frac {x^{2}}{2}+\frac {1}{2}\right ) \ln \left (i x +1\right )^{2}}{4}-\frac {\left (-x^{2} \ln \left (-i x +1\right )-2 i x -\ln \left (-i x +1\right )\right ) \ln \left (i x +1\right )}{4}-\frac {x^{2} \ln \left (-i x +1\right )^{2}}{8}-\frac {\ln \left (-i x +1\right )^{2}}{8}-\frac {i x \ln \left (-i x +1\right )}{2}+\frac {\ln \left (x^{2}+1\right )}{2}\)	\(99\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arctan(x)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 1.50, size = 34, normalized size = 0.97 \begin {gather*} \frac {1}{2} \, x^{2} \arctan \left (x\right )^{2} - {\left (x - \arctan \left (x\right )\right )} \arctan \left (x\right ) - \frac {1}{2} \, \arctan \left (x\right )^{2} + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.82, size = 25, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, {\left (x^{2} + 1\right )} \arctan \left (x\right )^{2} - x \arctan \left (x\right ) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.09, size = 29, normalized size = 0.83 \begin {gather*} \frac {x^{2} \operatorname {atan}^{2}{\left (x \right )}}{2} - x \operatorname {atan}{\left (x \right )} + \frac {\log {\left (x^{2} + 1 \right )}}{2} + \frac {\operatorname {atan}^{2}{\left (x \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.46, size = 29, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, x^{2} \arctan \left (x\right )^{2} - x \arctan \left (x\right ) + \frac {1}{2} \, \arctan \left (x\right )^{2} + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.12, size = 29, normalized size = 0.83 \begin {gather*} \frac {\ln \left (x^2+1\right )}{2}+\frac {{\mathrm {atan}\left (x\right )}^2}{2}+\frac {x^2\,{\mathrm {atan}\left (x\right )}^2}{2}-x\,\mathrm {atan}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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