3.53 \(\int \sqrt{x} \tan ^{-1}(x) \, dx\)

Optimal. Leaf size=117 \[ \frac{2}{3} x^{3/2} \tan ^{-1}(x)-\frac{4 \sqrt{x}}{3}-\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{3 \sqrt{2}}+\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{3 \sqrt{2}}-\frac{1}{3} \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )+\frac{1}{3} \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right ) \]

[Out]

(-4*Sqrt[x])/3 - (Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]])/3 + (Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x]])/3 + (2*x^(3/2
)*ArcTan[x])/3 - Log[1 - Sqrt[2]*Sqrt[x] + x]/(3*Sqrt[2]) + Log[1 + Sqrt[2]*Sqrt[x] + x]/(3*Sqrt[2])

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Rubi [A]  time = 0.0674999, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.125, Rules used = {4852, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{2}{3} x^{3/2} \tan ^{-1}(x)-\frac{4 \sqrt{x}}{3}-\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{3 \sqrt{2}}+\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{3 \sqrt{2}}-\frac{1}{3} \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )+\frac{1}{3} \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*ArcTan[x],x]

[Out]

(-4*Sqrt[x])/3 - (Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]])/3 + (Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x]])/3 + (2*x^(3/2
)*ArcTan[x])/3 - Log[1 - Sqrt[2]*Sqrt[x] + x]/(3*Sqrt[2]) + Log[1 + Sqrt[2]*Sqrt[x] + x]/(3*Sqrt[2])

Rule 4852

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

Rule 321

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

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rubi steps

\begin{align*} \int \sqrt{x} \tan ^{-1}(x) \, dx &=\frac{2}{3} x^{3/2} \tan ^{-1}(x)-\frac{2}{3} \int \frac{x^{3/2}}{1+x^2} \, dx\\ &=-\frac{4 \sqrt{x}}{3}+\frac{2}{3} x^{3/2} \tan ^{-1}(x)+\frac{2}{3} \int \frac{1}{\sqrt{x} \left (1+x^2\right )} \, dx\\ &=-\frac{4 \sqrt{x}}{3}+\frac{2}{3} x^{3/2} \tan ^{-1}(x)+\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=-\frac{4 \sqrt{x}}{3}+\frac{2}{3} x^{3/2} \tan ^{-1}(x)+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{x}\right )+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=-\frac{4 \sqrt{x}}{3}+\frac{2}{3} x^{3/2} \tan ^{-1}(x)+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{x}\right )}{3 \sqrt{2}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{x}\right )}{3 \sqrt{2}}\\ &=-\frac{4 \sqrt{x}}{3}+\frac{2}{3} x^{3/2} \tan ^{-1}(x)-\frac{\log \left (1-\sqrt{2} \sqrt{x}+x\right )}{3 \sqrt{2}}+\frac{\log \left (1+\sqrt{2} \sqrt{x}+x\right )}{3 \sqrt{2}}+\frac{1}{3} \sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{x}\right )-\frac{1}{3} \sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{x}\right )\\ &=-\frac{4 \sqrt{x}}{3}-\frac{1}{3} \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )+\frac{1}{3} \sqrt{2} \tan ^{-1}\left (1+\sqrt{2} \sqrt{x}\right )+\frac{2}{3} x^{3/2} \tan ^{-1}(x)-\frac{\log \left (1-\sqrt{2} \sqrt{x}+x\right )}{3 \sqrt{2}}+\frac{\log \left (1+\sqrt{2} \sqrt{x}+x\right )}{3 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0250164, size = 108, normalized size = 0.92 \[ \frac{1}{6} \left (4 x^{3/2} \tan ^{-1}(x)-8 \sqrt{x}-\sqrt{2} \log \left (x-\sqrt{2} \sqrt{x}+1\right )+\sqrt{2} \log \left (x+\sqrt{2} \sqrt{x}+1\right )-2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )+2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*ArcTan[x],x]

[Out]

(-8*Sqrt[x] - 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] + 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x]] + 4*x^(3/2)*ArcTan
[x] - Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[x] + x] + Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[x] + x])/6

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Maple [A]  time = 0.023, size = 74, normalized size = 0.6 \begin{align*}{\frac{2\,\arctan \left ( x \right ) }{3}{x}^{{\frac{3}{2}}}}-{\frac{4}{3}\sqrt{x}}+{\frac{\sqrt{2}}{3}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) }+{\frac{\sqrt{2}}{3}\arctan \left ( -1+\sqrt{2}\sqrt{x} \right ) }+{\frac{\sqrt{2}}{6}\ln \left ({ \left ( 1+x+\sqrt{2}\sqrt{x} \right ) \left ( 1+x-\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^(3/2)*arctan(x)-4/3*x^(1/2)+1/3*arctan(1+2^(1/2)*x^(1/2))*2^(1/2)+1/3*arctan(-1+2^(1/2)*x^(1/2))*2^(1/2)
+1/6*2^(1/2)*ln((1+x+2^(1/2)*x^(1/2))/(1+x-2^(1/2)*x^(1/2)))

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Maxima [A]  time = 1.48464, size = 116, normalized size = 0.99 \begin{align*} \frac{2}{3} \, x^{\frac{3}{2}} \arctan \left (x\right ) + \frac{1}{3} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{1}{3} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) + \frac{1}{6} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{1}{6} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{4}{3} \, \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^(3/2)*arctan(x) + 1/3*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 1/3*sqrt(2)*arctan(-1/2*sqrt(2
)*(sqrt(2) - 2*sqrt(x))) + 1/6*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) - 1/6*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1
) - 4/3*sqrt(x)

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Fricas [A]  time = 2.79439, size = 404, normalized size = 3.45 \begin{align*} \frac{2}{3} \,{\left (x \arctan \left (x\right ) - 2\right )} \sqrt{x} - \frac{2}{3} \, \sqrt{2} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} \sqrt{x} + x + 1} - \sqrt{2} \sqrt{x} - 1\right ) - \frac{2}{3} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{-4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4} - \sqrt{2} \sqrt{x} + 1\right ) + \frac{1}{6} \, \sqrt{2} \log \left (4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) - \frac{1}{6} \, \sqrt{2} \log \left (-4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x*arctan(x) - 2)*sqrt(x) - 2/3*sqrt(2)*arctan(sqrt(2)*sqrt(sqrt(2)*sqrt(x) + x + 1) - sqrt(2)*sqrt(x) - 1
) - 2/3*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-4*sqrt(2)*sqrt(x) + 4*x + 4) - sqrt(2)*sqrt(x) + 1) + 1/6*sqrt(2)*log
(4*sqrt(2)*sqrt(x) + 4*x + 4) - 1/6*sqrt(2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)

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Sympy [A]  time = 8.1696, size = 104, normalized size = 0.89 \begin{align*} \frac{2 x^{\frac{3}{2}} \operatorname{atan}{\left (x \right )}}{3} - \frac{4 \sqrt{x}}{3} - \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt{x} + x + 1 \right )}}{6} + \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt{x} + x + 1 \right )}}{6} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{3} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x**(3/2)*atan(x)/3 - 4*sqrt(x)/3 - sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1)/6 + sqrt(2)*log(sqrt(2)*sqrt(x) + x
+ 1)/6 + sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/3 + sqrt(2)*atan(sqrt(2)*sqrt(x) + 1)/3

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Giac [A]  time = 1.21178, size = 116, normalized size = 0.99 \begin{align*} \frac{2}{3} \, x^{\frac{3}{2}} \arctan \left (x\right ) + \frac{1}{3} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{1}{3} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) + \frac{1}{6} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{1}{6} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{4}{3} \, \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^(3/2)*arctan(x) + 1/3*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 1/3*sqrt(2)*arctan(-1/2*sqrt(2
)*(sqrt(2) - 2*sqrt(x))) + 1/6*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) - 1/6*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1
) - 4/3*sqrt(x)