∫ tan−1 (x)__ x2√ ___

Optimal. Leaf size=29 \[ -\frac {\sqrt {1+x^2} \tan ^{-1}(x)}{x}-\tanh ^{-1}\left (\sqrt {1+x^2}\right ) \]

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

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Rubi [A]
time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {5064, 272, 65, 213} \begin {gather*} -\frac {\sqrt {x^2+1} \tan ^{-1}(x)}{x}-\tanh ^{-1}\left (\sqrt {x^2+1}\right ) \end {gather*}

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

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub

st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &

& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]

, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

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]]

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp

[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Dist[b*c*(p/(f*(m + 1))), Int[(

f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e,

c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]

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

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Mathematica [A]
time = 0.02, size = 33, normalized size = 1.14 \begin {gather*} -\frac {\sqrt {1+x^2} \tan ^{-1}(x)}{x}+\log (x)-\log \left (1+\sqrt {1+x^2}\right ) \end {gather*}

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

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Mathics [A]
time = 5.78, size = 21, normalized size = 0.72 \begin {gather*} -\frac {\text {ArcTan}\left [x\right ] \sqrt {1+x^2}}{x}-\text {ArcSinh}\left [\frac {1}{x}\right ] \end {gather*}

mathics('Integrate[ArcTan[x]/(x^2*Sqrt[1 + x^2]),x]')

-ArcTan[x] Sqrt[1 + x ^ 2] / x - ArcSinh[1 / x]

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Maple [C] Result contains complex when optimal does not.
time = 0.12, size = 56, normalized size = 1.93


method	result	size

default	\(-\frac {\sqrt {\left (x -i\right ) \left (x +i\right )}\, \arctan \left (x \right )}{x}-\ln \left (1+\frac {i x +1}{\sqrt {x^{2}+1}}\right )+\ln \left (\frac {i x +1}{\sqrt {x^{2}+1}}-1\right )\)	\(56\)

int(arctan(x)/x^2/(x^2+1)^(1/2),x,method=_RETURNVERBOSE)

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

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Maxima [A]
time = 0.36, size = 22, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {x^{2} + 1} \arctan \left (x\right )}{x} - \operatorname {arsinh}\left (\frac {1}{{\left | x \right |}}\right ) \end {gather*}

integrate(arctan(x)/x^2/(x^2+1)^(1/2),x, algorithm="maxima")

-sqrt(x^2 + 1)*arctan(x)/x - arcsinh(1/abs(x))

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Fricas [A]
time = 0.36, size = 47, normalized size = 1.62 \begin {gather*} -\frac {x \log \left (-x + \sqrt {x^{2} + 1} + 1\right ) - x \log \left (-x + \sqrt {x^{2} + 1} - 1\right ) + \sqrt {x^{2} + 1} \arctan \left (x\right )}{x} \end {gather*}

integrate(arctan(x)/x^2/(x^2+1)^(1/2),x, algorithm="fricas")

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

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Sympy [A]
time = 4.53, size = 19, normalized size = 0.66 \begin {gather*} - \operatorname {asinh}{\left (\frac {1}{x} \right )} - \frac {\sqrt {x^{2} + 1} \operatorname {atan}{\left (x \right )}}{x} \end {gather*}

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (25) = 50\).
time = 0.01, size = 65, normalized size = 2.24 \begin {gather*} \arctan x-2 \left (-\frac {\ln \left |\sqrt {x^{2}+1}-x-1\right |}{2}+\frac {\ln \left |\sqrt {x^{2}+1}-x+1\right |}{2}\right )+\frac {2 \arctan x}{\left (\sqrt {x^{2}+1}-x\right )^{2}-1} \end {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\mathrm {atan}\left (x\right )}{x^2\,\sqrt {x^2+1}} \,d x \end {gather*}

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3.1.31 \(\int \frac {\tan ^{-1}(x)}{x^2 \sqrt {1+x^2}} \, dx\) [31]