3.30 \(\int \frac{x \tan ^{-1}(x)}{\sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=45 \[ -\sqrt{1-x^2} \tan ^{-1}(x)+\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )-\sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0422879, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4974, 402, 216, 377, 203} \[ -\sqrt{1-x^2} \tan ^{-1}(x)+\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )-\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(x*ArcTan[x])/Sqrt[1 - x^2],x]

[Out]

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

Rule 4974

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

Rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

Rule 216

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

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 203

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

Rubi steps

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

Mathematica [A]  time = 0.0343567, size = 45, normalized size = 1. \[ -\sqrt{1-x^2} \tan ^{-1}(x)+\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )-\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.144, size = 0, normalized size = 0. \begin{align*} \int{x\arctan \left ( x \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arctan(x)/(-x^2+1)^(1/2),x)

[Out]

int(x*arctan(x)/(-x^2+1)^(1/2),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.15205, size = 182, normalized size = 4.04 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (3 \, x^{2} - 1\right )} \sqrt{-x^{2} + 1}}{4 \,{\left (x^{3} - x\right )}}\right ) - \sqrt{-x^{2} + 1} \arctan \left (x\right ) + \arctan \left (\frac{\sqrt{-x^{2} + 1} x}{x^{2} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(2)*arctan(1/4*sqrt(2)*(3*x^2 - 1)*sqrt(-x^2 + 1)/(x^3 - x)) - sqrt(-x^2 + 1)*arctan(x) + arctan(sqrt
(-x^2 + 1)*x/(x^2 - 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \operatorname{atan}{\left (x \right )}}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*atan(x)/sqrt(-(x - 1)*(x + 1)), x)

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Giac [B]  time = 1.1345, size = 146, normalized size = 3.24 \begin{align*} -\frac{1}{2} \, \pi \mathrm{sgn}\left (x\right ) + \frac{1}{2} \, \sqrt{2}{\left (\pi \mathrm{sgn}\left (x\right ) + 2 \, \arctan \left (-\frac{\sqrt{2} x{\left (\frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{4 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right )\right )} - \sqrt{-x^{2} + 1} \arctan \left (x\right ) - \arctan \left (-\frac{x{\left (\frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*pi*sgn(x) + 1/2*sqrt(2)*(pi*sgn(x) + 2*arctan(-1/4*sqrt(2)*x*((sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2
+ 1) - 1))) - sqrt(-x^2 + 1)*arctan(x) - arctan(-1/2*x*((sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1))