Integrand size = 8, antiderivative size = 117 \[ \int \sqrt {x} \arctan (x) \, dx=-\frac {4 \sqrt {x}}{3}-\frac {1}{3} \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {x}\right )+\frac {1}{3} \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {x}\right )+\frac {2}{3} x^{3/2} \arctan (x)-\frac {\log \left (1-\sqrt {2} \sqrt {x}+x\right )}{3 \sqrt {2}}+\frac {\log \left (1+\sqrt {2} \sqrt {x}+x\right )}{3 \sqrt {2}} \]
2/3*x^(3/2)*arctan(x)-1/6*ln(1+x-2^(1/2)*x^(1/2))*2^(1/2)+1/6*ln(1+x+2^(1/ 2)*x^(1/2))*2^(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)-4/3*x^(1/2)
Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.92 \[ \int \sqrt {x} \arctan (x) \, dx=\frac {1}{6} \left (-8 \sqrt {x}-2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {x}\right )+2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {x}\right )+4 x^{3/2} \arctan (x)-\sqrt {2} \log \left (1-\sqrt {2} \sqrt {x}+x\right )+\sqrt {2} \log \left (1+\sqrt {2} \sqrt {x}+x\right )\right ) \]
(-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
Time = 0.32 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.09, 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 |
| \(\displaystyle \frac {2}{3} x^{3/2} \arctan (x)-\frac {2}{3} \left (2 \sqrt {x}-2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (x+\sqrt {2} \sqrt {x}+1\right )}{2 \sqrt {2}}-\frac {\log \left (x-\sqrt {2} \sqrt {x}+1\right )}{2 \sqrt {2}}\right )\right )\right )\) |
(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
3.1.53.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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]]))
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]
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]
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]
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]
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]
Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.59
| method | result | size |
| derivativedivides | \(\frac {2 x^{\frac {3}{2}} \arctan \left (x \right )}{3}-\frac {4 \sqrt {x}}{3}+\frac {\sqrt {2}\, \left (\ln \left (\frac {1+x +\sqrt {2}\, \sqrt {x}}{1+x -\sqrt {2}\, \sqrt {x}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{6}\) | \(69\) |
| default | \(\frac {2 x^{\frac {3}{2}} \arctan \left (x \right )}{3}-\frac {4 \sqrt {x}}{3}+\frac {\sqrt {2}\, \left (\ln \left (\frac {1+x +\sqrt {2}\, \sqrt {x}}{1+x -\sqrt {2}\, \sqrt {x}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{6}\) | \(69\) |
| meijerg | \(-\frac {4 \sqrt {x}}{3}+\frac {\sqrt {x}\, \left (-\frac {\sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {1}{4}}}\right )}{3}+\frac {2 x^{\frac {5}{2}} \arctan \left (\sqrt {x^{2}}\right )}{3 \sqrt {x^{2}}}\) | \(152\) |
2/3*x^(3/2)*arctan(x)-4/3*x^(1/2)+1/6*2^(1/2)*(ln((1+x+2^(1/2)*x^(1/2))/(1 +x-2^(1/2)*x^(1/2)))+2*arctan(1+2^(1/2)*x^(1/2))+2*arctan(-1+2^(1/2)*x^(1/ 2)))
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.68 \[ \int \sqrt {x} \arctan (x) \, dx=\frac {2}{3} \, {\left (x \arctan \left (x\right ) - 2\right )} \sqrt {x} + \left (\frac {1}{6} i + \frac {1}{6}\right ) \, \sqrt {2} \log \left (\left (i + 1\right ) \, \sqrt {2} + 2 \, \sqrt {x}\right ) - \left (\frac {1}{6} i - \frac {1}{6}\right ) \, \sqrt {2} \log \left (-\left (i - 1\right ) \, \sqrt {2} + 2 \, \sqrt {x}\right ) + \left (\frac {1}{6} i - \frac {1}{6}\right ) \, \sqrt {2} \log \left (\left (i - 1\right ) \, \sqrt {2} + 2 \, \sqrt {x}\right ) - \left (\frac {1}{6} i + \frac {1}{6}\right ) \, \sqrt {2} \log \left (-\left (i + 1\right ) \, \sqrt {2} + 2 \, \sqrt {x}\right ) \]
2/3*(x*arctan(x) - 2)*sqrt(x) + (1/6*I + 1/6)*sqrt(2)*log((I + 1)*sqrt(2) + 2*sqrt(x)) - (1/6*I - 1/6)*sqrt(2)*log(-(I - 1)*sqrt(2) + 2*sqrt(x)) + ( 1/6*I - 1/6)*sqrt(2)*log((I - 1)*sqrt(2) + 2*sqrt(x)) - (1/6*I + 1/6)*sqrt (2)*log(-(I + 1)*sqrt(2) + 2*sqrt(x))
Time = 1.95 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.94 \[ \int \sqrt {x} \arctan (x) \, dx=\frac {2 x^{\frac {3}{2}} \operatorname {atan}{\left (x \right )}}{3} - \frac {4 \sqrt {x}}{3} - \frac {\sqrt {2} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{6} + \frac {\sqrt {2} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{6} + \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{3} + \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{3} \]
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
Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.74 \[ \int \sqrt {x} \arctan (x) \, dx=\frac {2}{3} \, x^{\frac {3}{2}} \arctan \left (x\right ) + \frac {1}{3} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {1}{3} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {1}{6} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {1}{6} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {4}{3} \, \sqrt {x} \]
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)
Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.74 \[ \int \sqrt {x} \arctan (x) \, dx=\frac {2}{3} \, x^{\frac {3}{2}} \arctan \left (x\right ) + \frac {1}{3} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {1}{3} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {1}{6} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {1}{6} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {4}{3} \, \sqrt {x} \]
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)
Time = 0.36 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.42 \[ \int \sqrt {x} \arctan (x) \, dx=\frac {2\,x^{3/2}\,\mathrm {atan}\left (x\right )}{3}-\frac {4\,\sqrt {x}}{3}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{3}+\frac {1}{3}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{3}-\frac {1}{3}{}\mathrm {i}\right ) \]