3.1.30 \(\int \frac {x \tan ^{-1}(x)}{\sqrt {1-x^2}} \, dx\) [30]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, 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 = {5094, 399, 222, 385, 209} \begin {gather*} -\text {ArcSin}(x)-\sqrt {1-x^2} \text {ArcTan}(x)+\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2}}\right ) \end {gather*}

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 209

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

Rule 222

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 385

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 399

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

Rule 5094

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]

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 \text {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*}

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Mathematica [A]
time = 0.03, size = 45, normalized size = 1.00 \begin {gather*} -\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 {gather*}

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 = 0.07, size = 0, normalized size = 0.00 \[\int \frac {x \arctan \left (x \right )}{\sqrt {-x^{2}+1}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: sign: argument cannot be imaginary
; found %i

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Fricas [A]
time = 0.46, size = 69, normalized size = 1.53 \begin {gather*} -\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 {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]

-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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \operatorname {atan}{\left (x \right )}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (37) = 74\).
time = 0.47, size = 108, normalized size = 2.40 \begin {gather*} -\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 {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]

-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))

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Mupad [B]
time = 0.03, size = 37, normalized size = 0.82 \begin {gather*} \sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{\sqrt {1-x^2}}\right )-\mathrm {atan}\left (x\right )\,\sqrt {1-x^2}-\mathrm {asin}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*atan(x))/(1 - x^2)^(1/2),x)

[Out]

2^(1/2)*atan((2^(1/2)*x)/(1 - x^2)^(1/2)) - atan(x)*(1 - x^2)^(1/2) - asin(x)

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