3.1.28 \(\int \frac {x \tan ^{-1}(x)}{\sqrt {1+x^2}} \, dx\) [28]

Optimal. Leaf size=17 \[ -\sinh ^{-1}(x)+\sqrt {1+x^2} \tan ^{-1}(x) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5050, 221} \begin {gather*} \sqrt {x^2+1} \text {ArcTan}(x)-\sinh ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*ArcTan[x])/Sqrt[1 + x^2],x]

[Out]

-ArcSinh[x] + Sqrt[1 + x^2]*ArcTan[x]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 5050

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

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} -\sinh ^{-1}(x)+\sqrt {1+x^2} \tan ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcSinh[x] + Sqrt[1 + x^2]*ArcTan[x]

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Maple [C] Result contains complex when optimal does not.
time = 0.11, size = 54, normalized size = 3.18

method	result	size
default	\(\sqrt {\left (x -i\right ) \left (x +i\right )}\, \arctan \left (x \right )+\ln \left (\frac {i x +1}{\sqrt {x^{2}+1}}-i\right )-\ln \left (\frac {i x +1}{\sqrt {x^{2}+1}}+i\right )\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 5.55, size = 15, normalized size = 0.88 \begin {gather*} \sqrt {x^{2} + 1} \arctan \left (x\right ) - \operatorname {arsinh}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^2 + 1)*arctan(x) - arcsinh(x)

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Fricas [A]
time = 0.42, size = 23, normalized size = 1.35 \begin {gather*} \sqrt {x^{2} + 1} \arctan \left (x\right ) + \log \left (-x + \sqrt {x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (14) = 28\).
time = 0.53, size = 29, normalized size = 1.71 \begin {gather*} \frac {x^{2} \operatorname {atan}{\left (x \right )}}{\sqrt {x^{2} + 1}} - \operatorname {asinh}{\left (x \right )} + \frac {\operatorname {atan}{\left (x \right )}}{\sqrt {x^{2} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*atan(x)/(x**2+1)**(1/2),x)

[Out]

x**2*atan(x)/sqrt(x**2 + 1) - asinh(x) + atan(x)/sqrt(x**2 + 1)

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Giac [A]
time = 0.47, size = 23, normalized size = 1.35 \begin {gather*} \sqrt {x^{2} + 1} \arctan \left (x\right ) + \log \left (-x + \sqrt {x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {x\,\mathrm {atan}\left (x\right )}{\sqrt {x^2+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*atan(x))/(x^2 + 1)^(1/2),x)

[Out]

int((x*atan(x))/(x^2 + 1)^(1/2), x)

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