\(\int \sqrt {x} \arctan (x) \, dx\) [53]

Optimal result

Integrand size = 8, antiderivative size = 94 \[ \int \sqrt {x} \arctan (x) \, dx=-\frac {4 \sqrt {x}}{3}-\frac {1}{3} \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {x}\right )+\frac {1}{3} \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {x}\right )+\frac {2}{3} x^{3/2} \arctan (x)+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x}}{1+x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.15 \[ \int \sqrt {x} \arctan (x) \, dx=\frac {1}{6} \left (-8 \sqrt {x}-2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {x}\right )+2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {x}\right )+4 x^{3/2} \arctan (x)-\sqrt {2} \log \left (1-\sqrt {2} \sqrt {x}+x\right )+\sqrt {2} \log \left (1+\sqrt {2} \sqrt {x}+x\right )\right ) \] Input:

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

Output:

(-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
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.36, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.375, Rules used = {5361, 262, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sqrt {x} \arctan (x) \, dx\)

\(\Big \downarrow \) 5361

\(\displaystyle \frac {2}{3} x^{3/2} \arctan (x)-\frac {2}{3} \int \frac {x^{3/2}}{x^2+1}dx\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {2}{3} x^{3/2} \arctan (x)-\frac {2}{3} \left (2 \sqrt {x}-\int \frac {1}{\sqrt {x} \left (x^2+1\right )}dx\right )\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {2}{3} x^{3/2} \arctan (x)-\frac {2}{3} \left (2 \sqrt {x}-2 \int \frac {1}{x^2+1}d\sqrt {x}\right )\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {2}{3} x^{3/2} \arctan (x)-\frac {2}{3} \left (2 \sqrt {x}-2 \left (\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}+\frac {1}{2} \int \frac {x+1}{x^2+1}d\sqrt {x}\right )\right )\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {2}{3} x^{3/2} \arctan (x)-\frac {2}{3} \left (2 \sqrt {x}-2 \left (\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}+\frac {1}{2} \int \frac {1}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}\right )\right )\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {2}{3} x^{3/2} \arctan (x)-\frac {2}{3} \left (2 \sqrt {x}-2 \left (\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}+\frac {1}{2} \left (\frac {\int \frac {1}{-x-1}d\left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-x-1}d\left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}\right )\right )\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {2}{3} x^{3/2} \arctan (x)-\frac {2}{3} \left (2 \sqrt {x}-2 \left (\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )\right )\right )\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {2}{3} x^{3/2} \arctan (x)-\frac {2}{3} \left (2 \sqrt {x}-2 \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {x}}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {x}+1\right )}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )\right )\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2}{3} x^{3/2} \arctan (x)-\frac {2}{3} \left (2 \sqrt {x}-2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {x}}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {x}+1\right )}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )\right )\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{3} x^{3/2} \arctan (x)-\frac {2}{3} \left (2 \sqrt {x}-2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {x}}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {x}+1}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )\right )\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {2}{3} x^{3/2} \arctan (x)-\frac {2}{3} \left (2 \sqrt {x}-2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (x+\sqrt {2} \sqrt {x}+1\right )}{2 \sqrt {2}}-\frac {\log \left (x-\sqrt {2} \sqrt {x}+1\right )}{2 \sqrt {2}}\right )\right )\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[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 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[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])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[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 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
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 1479

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

rule 5361

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.73

method	result	size
derivativedivides	\(\frac {2 x^{\frac {3}{2}} \arctan \left (x \right )}{3}-\frac {4 \sqrt {x}}{3}+\frac {\sqrt {2}\, \left (\ln \left (\frac {x +\sqrt {2}\, \sqrt {x}+1}{x -\sqrt {2}\, \sqrt {x}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{6}\)	\(69\)
default	\(\frac {2 x^{\frac {3}{2}} \arctan \left (x \right )}{3}-\frac {4 \sqrt {x}}{3}+\frac {\sqrt {2}\, \left (\ln \left (\frac {x +\sqrt {2}\, \sqrt {x}+1}{x -\sqrt {2}\, \sqrt {x}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{6}\)	\(69\)
meijerg	\(-\frac {4 \sqrt {x}}{3}+\frac {\sqrt {x}\, \left (-\frac {\sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {1}{4}}}\right )}{3}+\frac {2 x^{\frac {5}{2}} \arctan \left (\sqrt {x^{2}}\right )}{3 \sqrt {x^{2}}}\)	\(152\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.80 \[ \int \sqrt {x} \arctan (x) \, dx=\frac {2}{3} \, {\left (x \arctan \left (x\right ) - 2\right )} \sqrt {x} + \frac {1}{3} \, \sqrt {2} \arctan \left (\sqrt {2} \sqrt {x} + 1\right ) + \frac {1}{3} \, \sqrt {2} \arctan \left (\sqrt {2} \sqrt {x} - 1\right ) + \frac {1}{6} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {1}{6} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 1.98 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.17 \[ \int \sqrt {x} \arctan (x) \, dx=\frac {2 x^{\frac {3}{2}} \operatorname {atan}{\left (x \right )}}{3} - \frac {4 \sqrt {x}}{3} - \frac {\sqrt {2} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{6} + \frac {\sqrt {2} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{6} + \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{3} + \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{3} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \sqrt {x} \arctan (x) \, dx=\frac {2}{3} \, x^{\frac {3}{2}} \arctan \left (x\right ) + \frac {1}{3} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {1}{3} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {1}{6} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {1}{6} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {4}{3} \, \sqrt {x} \] Input:

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

Output:

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)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \sqrt {x} \arctan (x) \, dx=\frac {2}{3} \, x^{\frac {3}{2}} \arctan \left (x\right ) + \frac {1}{3} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {1}{3} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {1}{6} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {1}{6} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {4}{3} \, \sqrt {x} \] Input:

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

Output:

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)
 

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.52 \[ \int \sqrt {x} \arctan (x) \, dx=\frac {2\,x^{3/2}\,\mathrm {atan}\left (x\right )}{3}-\frac {4\,\sqrt {x}}{3}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{3}+\frac {1}{3}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{3}-\frac {1}{3}{}\mathrm {i}\right ) \] Input:

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

Output:

(2*x^(3/2)*atan(x))/3 + 2^(1/2)*atan(2^(1/2)*x^(1/2)*(1/2 - 1i/2))*(1/3 + 
1i/3) + 2^(1/2)*atan(2^(1/2)*x^(1/2)*(1/2 + 1i/2))*(1/3 - 1i/3) - (4*x^(1/ 
2))/3
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.68 \[ \int \sqrt {x} \arctan (x) \, dx=\frac {2 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}-\sqrt {2}}{\sqrt {2}}\right )}{3}+\frac {2 \sqrt {x}\, \mathit {atan} \left (x \right ) x}{3}-\frac {\sqrt {2}\, \mathit {atan} \left (x \right )}{3}-\frac {4 \sqrt {x}}{3}-\frac {\sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, \sqrt {2}+x +1\right )}{6}+\frac {\sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, \sqrt {2}+x +1\right )}{6} \] Input:

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

Output:

(4*sqrt(2)*atan((2*sqrt(x) - sqrt(2))/sqrt(2)) + 4*sqrt(x)*atan(x)*x - 2*s 
qrt(2)*atan(x) - 8*sqrt(x) - sqrt(2)*log( - sqrt(x)*sqrt(2) + x + 1) + sqr 
t(2)*log(sqrt(x)*sqrt(2) + x + 1))/6