Integrand size = 13, antiderivative size = 129 \[ \int \frac {1}{\sqrt {x} \left (1+x^2\right )^3} \, dx=\frac {\sqrt {x}}{4 \left (1+x^2\right )^2}+\frac {7 \sqrt {x}}{16 \left (1+x^2\right )}-\frac {21 \arctan \left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {21 \arctan \left (1+\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}-\frac {21 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}+\frac {21 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}} \]
21/64*arctan(-1+2^(1/2)*x^(1/2))*2^(1/2)+21/64*arctan(1+2^(1/2)*x^(1/2))*2 ^(1/2)-21/128*ln(1+x-2^(1/2)*x^(1/2))*2^(1/2)+21/128*ln(1+x+2^(1/2)*x^(1/2 ))*2^(1/2)+1/4*x^(1/2)/(x^2+1)^2+7/16*x^(1/2)/(x^2+1)
Time = 0.10 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.56 \[ \int \frac {1}{\sqrt {x} \left (1+x^2\right )^3} \, dx=\frac {1}{64} \left (\frac {4 \sqrt {x} \left (11+7 x^2\right )}{\left (1+x^2\right )^2}+21 \sqrt {2} \arctan \left (\frac {-1+x}{\sqrt {2} \sqrt {x}}\right )+21 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x}}{1+x}\right )\right ) \]
((4*Sqrt[x]*(11 + 7*x^2))/(1 + x^2)^2 + 21*Sqrt[2]*ArcTan[(-1 + x)/(Sqrt[2 ]*Sqrt[x])] + 21*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[x])/(1 + x)])/64
Time = 0.31 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {253, 253, 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.
| \(\displaystyle \int \frac {1}{\sqrt {x} \left (x^2+1\right )^3} \, dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {7}{8} \int \frac {1}{\sqrt {x} \left (x^2+1\right )^2}dx+\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {7}{8} \left (\frac {3}{4} \int \frac {1}{\sqrt {x} \left (x^2+1\right )}dx+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )+\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {7}{8} \left (\frac {3}{2} \int \frac {1}{x^2+1}d\sqrt {x}+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )+\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {7}{8} \left (\frac {3}{2} \left (\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}+\frac {1}{2} \int \frac {x+1}{x^2+1}d\sqrt {x}\right )+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )+\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {7}{8} \left (\frac {3}{2} \left (\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}+\frac {1}{2} \int \frac {1}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}\right )\right )+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )+\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {7}{8} \left (\frac {3}{2} \left (\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}+\frac {1}{2} \left (\frac {\int \frac {1}{-x-1}d\left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-x-1}d\left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}\right )\right )+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )+\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {7}{8} \left (\frac {3}{2} \left (\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )\right )+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )+\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {7}{8} \left (\frac {3}{2} \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {x}}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {x}+1\right )}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )\right )+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )+\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {7}{8} \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {x}}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {x}+1\right )}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )\right )+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )+\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{8} \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {x}}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {x}+1}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )\right )+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )+\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {7}{8} \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (x+\sqrt {2} \sqrt {x}+1\right )}{2 \sqrt {2}}-\frac {\log \left (x-\sqrt {2} \sqrt {x}+1\right )}{2 \sqrt {2}}\right )\right )+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )+\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
Sqrt[x]/(4*(1 + x^2)^2) + (7*(Sqrt[x]/(2*(1 + x^2)) + (3*((-(ArcTan[1 - Sq rt[2]*Sqrt[x]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[x]]/Sqrt[2])/2 + (-1/2*L og[1 - Sqrt[2]*Sqrt[x] + x]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[x] + x]/(2*Sqrt [2]))/2))/2))/8
3.4.33.3.1 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*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && 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]
Time = 1.81 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.59
| method | result | size |
| risch | \(\frac {\left (7 x^{2}+11\right ) \sqrt {x}}{16 \left (x^{2}+1\right )^{2}}+\frac {21 \sqrt {2}\, \left (\ln \left (\frac {1+x +\sqrt {2}\, \sqrt {x}}{1+x -\sqrt {2}\, \sqrt {x}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{128}\) | \(76\) |
| derivativedivides | \(\frac {\sqrt {x}}{4 \left (x^{2}+1\right )^{2}}+\frac {7 \sqrt {x}}{16 \left (x^{2}+1\right )}+\frac {21 \sqrt {2}\, \left (\ln \left (\frac {1+x +\sqrt {2}\, \sqrt {x}}{1+x -\sqrt {2}\, \sqrt {x}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{128}\) | \(81\) |
| default | \(\frac {\sqrt {x}}{4 \left (x^{2}+1\right )^{2}}+\frac {7 \sqrt {x}}{16 \left (x^{2}+1\right )}+\frac {21 \sqrt {2}\, \left (\ln \left (\frac {1+x +\sqrt {2}\, \sqrt {x}}{1+x -\sqrt {2}\, \sqrt {x}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{128}\) | \(81\) |
| meijerg | \(\frac {\left (7 x^{2}+11\right ) \sqrt {x}}{16 \left (x^{2}+1\right )^{2}}-\frac {21 \sqrt {x}\, \sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{128 \left (x^{2}\right )^{\frac {1}{4}}}+\frac {21 \sqrt {x}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{64 \left (x^{2}\right )^{\frac {1}{4}}}+\frac {21 \sqrt {x}\, \sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{128 \left (x^{2}\right )^{\frac {1}{4}}}+\frac {21 \sqrt {x}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{64 \left (x^{2}\right )^{\frac {1}{4}}}\) | \(158\) |
| trager | \(\frac {\left (7 x^{2}+11\right ) \sqrt {x}}{16 \left (x^{2}+1\right )^{2}}-\frac {21 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x +4 \sqrt {x}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}+x +1}\right )}{64}+\frac {21 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x +4 \sqrt {x}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-x -1}\right )}{64}\) | \(203\) |
1/16*(7*x^2+11)/(x^2+1)^2*x^(1/2)+21/128*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 = 0.27 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\sqrt {x} \left (1+x^2\right )^3} \, dx=-\frac {21 \, \sqrt {2} {\left (-\left (i + 1\right ) \, x^{4} - \left (2 i + 2\right ) \, x^{2} - i - 1\right )} \log \left (\left (i + 1\right ) \, \sqrt {2} + 2 \, \sqrt {x}\right ) + 21 \, \sqrt {2} {\left (\left (i - 1\right ) \, x^{4} + \left (2 i - 2\right ) \, x^{2} + i - 1\right )} \log \left (-\left (i - 1\right ) \, \sqrt {2} + 2 \, \sqrt {x}\right ) + 21 \, \sqrt {2} {\left (-\left (i - 1\right ) \, x^{4} - \left (2 i - 2\right ) \, x^{2} - i + 1\right )} \log \left (\left (i - 1\right ) \, \sqrt {2} + 2 \, \sqrt {x}\right ) + 21 \, \sqrt {2} {\left (\left (i + 1\right ) \, x^{4} + \left (2 i + 2\right ) \, x^{2} + i + 1\right )} \log \left (-\left (i + 1\right ) \, \sqrt {2} + 2 \, \sqrt {x}\right ) - 8 \, {\left (7 \, x^{2} + 11\right )} \sqrt {x}}{128 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \]
-1/128*(21*sqrt(2)*(-(I + 1)*x^4 - (2*I + 2)*x^2 - I - 1)*log((I + 1)*sqrt (2) + 2*sqrt(x)) + 21*sqrt(2)*((I - 1)*x^4 + (2*I - 2)*x^2 + I - 1)*log(-( I - 1)*sqrt(2) + 2*sqrt(x)) + 21*sqrt(2)*(-(I - 1)*x^4 - (2*I - 2)*x^2 - I + 1)*log((I - 1)*sqrt(2) + 2*sqrt(x)) + 21*sqrt(2)*((I + 1)*x^4 + (2*I + 2)*x^2 + I + 1)*log(-(I + 1)*sqrt(2) + 2*sqrt(x)) - 8*(7*x^2 + 11)*sqrt(x) )/(x^4 + 2*x^2 + 1)
Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (117) = 234\).
Time = 1.58 (sec) , antiderivative size = 481, normalized size of antiderivative = 3.73 \[ \int \frac {1}{\sqrt {x} \left (1+x^2\right )^3} \, dx=\frac {56 x^{\frac {5}{2}}}{128 x^{4} + 256 x^{2} + 128} + \frac {88 \sqrt {x}}{128 x^{4} + 256 x^{2} + 128} - \frac {21 \sqrt {2} x^{4} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {21 \sqrt {2} x^{4} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {42 \sqrt {2} x^{4} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {42 \sqrt {2} x^{4} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac {42 \sqrt {2} x^{2} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {42 \sqrt {2} x^{2} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {84 \sqrt {2} x^{2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {84 \sqrt {2} x^{2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac {21 \sqrt {2} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {21 \sqrt {2} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {42 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {42 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} \]
56*x**(5/2)/(128*x**4 + 256*x**2 + 128) + 88*sqrt(x)/(128*x**4 + 256*x**2 + 128) - 21*sqrt(2)*x**4*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256 *x**2 + 128) + 21*sqrt(2)*x**4*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) + 42*sqrt(2)*x**4*atan(sqrt(2)*sqrt(x) - 1)/(128*x**4 + 256*x**2 + 128) + 42*sqrt(2)*x**4*atan(sqrt(2)*sqrt(x) + 1)/(128*x**4 + 25 6*x**2 + 128) - 42*sqrt(2)*x**2*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x** 4 + 256*x**2 + 128) + 42*sqrt(2)*x**2*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(12 8*x**4 + 256*x**2 + 128) + 84*sqrt(2)*x**2*atan(sqrt(2)*sqrt(x) - 1)/(128* x**4 + 256*x**2 + 128) + 84*sqrt(2)*x**2*atan(sqrt(2)*sqrt(x) + 1)/(128*x* *4 + 256*x**2 + 128) - 21*sqrt(2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x **4 + 256*x**2 + 128) + 21*sqrt(2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x **4 + 256*x**2 + 128) + 42*sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/(128*x**4 + 2 56*x**2 + 128) + 42*sqrt(2)*atan(sqrt(2)*sqrt(x) + 1)/(128*x**4 + 256*x**2 + 128)
Time = 0.29 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\sqrt {x} \left (1+x^2\right )^3} \, dx=\frac {21}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {21}{64} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {21}{128} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {21}{128} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {7 \, x^{\frac {5}{2}} + 11 \, \sqrt {x}}{16 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \]
21/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 21/64*sqrt(2)*ar ctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) + 21/128*sqrt(2)*log(sqrt(2)*sqrt (x) + x + 1) - 21/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) + 1/16*(7*x^(5 /2) + 11*sqrt(x))/(x^4 + 2*x^2 + 1)
Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\sqrt {x} \left (1+x^2\right )^3} \, dx=\frac {21}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {21}{64} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {21}{128} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {21}{128} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {7 \, x^{\frac {5}{2}} + 11 \, \sqrt {x}}{16 \, {\left (x^{2} + 1\right )}^{2}} \]
21/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 21/64*sqrt(2)*ar ctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) + 21/128*sqrt(2)*log(sqrt(2)*sqrt (x) + x + 1) - 21/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) + 1/16*(7*x^(5 /2) + 11*sqrt(x))/(x^2 + 1)^2
Time = 4.92 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.47 \[ \int \frac {1}{\sqrt {x} \left (1+x^2\right )^3} \, dx=\frac {\frac {11\,\sqrt {x}}{16}+\frac {7\,x^{5/2}}{16}}{x^4+2\,x^2+1}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {21}{64}+\frac {21}{64}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {21}{64}-\frac {21}{64}{}\mathrm {i}\right ) \]