Integrand size = 18, antiderivative size = 52 \[ \int \frac {-1+x^3}{\sqrt {x} \left (1+x^2\right )} \, dx=\frac {2 x^{3/2}}{3}+\sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {x}\right )-\sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {x}\right ) \] Output:
2/3*x^(3/2)-2^(1/2)*arctan(-1+2^(1/2)*x^(1/2))-2^(1/2)*arctan(1+2^(1/2)*x^ (1/2))
Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.62 \[ \int \frac {-1+x^3}{\sqrt {x} \left (1+x^2\right )} \, dx=\frac {2 x^{3/2}}{3}-\sqrt {2} \arctan \left (\frac {-1+x}{\sqrt {2} \sqrt {x}}\right ) \] Input:
Integrate[(-1 + x^3)/(Sqrt[x]*(1 + x^2)),x]
Output:
(2*x^(3/2))/3 - Sqrt[2]*ArcTan[(-1 + x)/(Sqrt[2]*Sqrt[x])]
Time = 0.38 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2035, 25, 2426, 2009}
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.
\(\displaystyle \int \frac {x^3-1}{\sqrt {x} \left (x^2+1\right )} \, dx\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle 2 \int -\frac {1-x^3}{x^2+1}d\sqrt {x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {1-x^3}{x^2+1}d\sqrt {x}\) |
\(\Big \downarrow \) 2426 |
\(\displaystyle -2 \int \left (\frac {x+1}{x^2+1}-x\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}-\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}+\frac {x^{3/2}}{3}\right )\) |
Input:
Int[(-1 + x^3)/(Sqrt[x]*(1 + x^2)),x]
Output:
2*(x^(3/2)/3 + ArcTan[1 - Sqrt[2]*Sqrt[x]]/Sqrt[2] - ArcTan[1 + Sqrt[2]*Sq rt[x]]/Sqrt[2])
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IntegerQ[n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.15
method | result | size |
trager | \(\frac {2 x^{\frac {3}{2}}}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}+4 x^{\frac {3}{2}}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x -4 \sqrt {x}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{x^{2}+1}\right )}{2}\) | \(60\) |
risch | \(\frac {2 x^{\frac {3}{2}}}{3}-\sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )-\sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )-\frac {\sqrt {2}\, \ln \left (\frac {x +\sqrt {2}\, \sqrt {x}+1}{x -\sqrt {2}\, \sqrt {x}+1}\right )}{4}-\frac {\sqrt {2}\, \ln \left (\frac {x -\sqrt {2}\, \sqrt {x}+1}{x +\sqrt {2}\, \sqrt {x}+1}\right )}{4}\) | \(97\) |
derivativedivides | \(\frac {2 x^{\frac {3}{2}}}{3}-\frac {\sqrt {2}\, \left (\ln \left (\frac {x +\sqrt {2}\, \sqrt {x}+1}{x -\sqrt {2}\, \sqrt {x}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{4}-\frac {\sqrt {2}\, \left (\ln \left (\frac {x -\sqrt {2}\, \sqrt {x}+1}{x +\sqrt {2}\, \sqrt {x}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{4}\) | \(117\) |
default | \(\frac {2 x^{\frac {3}{2}}}{3}-\frac {\sqrt {2}\, \left (\ln \left (\frac {x +\sqrt {2}\, \sqrt {x}+1}{x -\sqrt {2}\, \sqrt {x}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{4}-\frac {\sqrt {2}\, \left (\ln \left (\frac {x -\sqrt {2}\, \sqrt {x}+1}{x +\sqrt {2}\, \sqrt {x}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{4}\) | \(117\) |
meijerg | \(\frac {2 x^{\frac {3}{2}}}{3}-\frac {x^{\frac {3}{2}} \left (\frac {\sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {3}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {3}{4}}}-\frac {\sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {3}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {3}{4}}}\right )}{2}+\frac {\sqrt {x}\, \sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{4 \left (x^{2}\right )^{\frac {1}{4}}}-\frac {\sqrt {x}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{2 \left (x^{2}\right )^{\frac {1}{4}}}-\frac {\sqrt {x}\, \sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{4 \left (x^{2}\right )^{\frac {1}{4}}}-\frac {\sqrt {x}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{2 \left (x^{2}\right )^{\frac {1}{4}}}\) | \(273\) |
Input:
int((x^3-1)/x^(1/2)/(x^2+1),x,method=_RETURNVERBOSE)
Output:
2/3*x^(3/2)+1/2*RootOf(_Z^2+2)*ln((RootOf(_Z^2+2)*x^2+4*x^(3/2)-4*RootOf(_ Z^2+2)*x-4*x^(1/2)+RootOf(_Z^2+2))/(x^2+1))
Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.44 \[ \int \frac {-1+x^3}{\sqrt {x} \left (1+x^2\right )} \, dx=\frac {2}{3} \, x^{\frac {3}{2}} - \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x - 1\right )}}{2 \, \sqrt {x}}\right ) \] Input:
integrate((x^3-1)/x^(1/2)/(x^2+1),x, algorithm="fricas")
Output:
2/3*x^(3/2) - sqrt(2)*arctan(1/2*sqrt(2)*(x - 1)/sqrt(x))
Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.85 \[ \int \frac {-1+x^3}{\sqrt {x} \left (1+x^2\right )} \, dx=\frac {2 x^{\frac {3}{2}}}{3} - \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )} - \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )} \] Input:
integrate((x**3-1)/x**(1/2)/(x**2+1),x)
Output:
2*x**(3/2)/3 - sqrt(2)*atan(sqrt(2)*sqrt(x) - 1) - sqrt(2)*atan(sqrt(2)*sq rt(x) + 1)
Time = 0.11 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.88 \[ \int \frac {-1+x^3}{\sqrt {x} \left (1+x^2\right )} \, dx=\frac {2}{3} \, x^{\frac {3}{2}} - \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) - \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) \] Input:
integrate((x^3-1)/x^(1/2)/(x^2+1),x, algorithm="maxima")
Output:
2/3*x^(3/2) - sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) - sqrt(2)* arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x)))
Time = 0.13 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.88 \[ \int \frac {-1+x^3}{\sqrt {x} \left (1+x^2\right )} \, dx=\frac {2}{3} \, x^{\frac {3}{2}} - \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) - \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) \] Input:
integrate((x^3-1)/x^(1/2)/(x^2+1),x, algorithm="giac")
Output:
2/3*x^(3/2) - sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) - sqrt(2)* arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x)))
Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.83 \[ \int \frac {-1+x^3}{\sqrt {x} \left (1+x^2\right )} \, dx=\frac {2\,x^{3/2}}{3}-\frac {\sqrt {2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {x}}{2}+\frac {\sqrt {2}\,x^{3/2}}{2}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {x}}{2}\right )\right )}{2} \] Input:
int((x^3 - 1)/(x^(1/2)*(x^2 + 1)),x)
Output:
(2*x^(3/2))/3 - (2^(1/2)*(2*atan((2^(1/2)*x^(1/2))/2 + (2^(1/2)*x^(3/2))/2 ) + 2*atan((2^(1/2)*x^(1/2))/2)))/2
Time = 0.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.81 \[ \int \frac {-1+x^3}{\sqrt {x} \left (1+x^2\right )} \, dx=-\sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}-\sqrt {2}}{\sqrt {2}}\right )-\sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}+\sqrt {2}}{\sqrt {2}}\right )+\frac {2 \sqrt {x}\, x}{3} \] Input:
int((x^3-1)/x^(1/2)/(x^2+1),x)
Output:
( - 3*sqrt(2)*atan((2*sqrt(x) - sqrt(2))/sqrt(2)) - 3*sqrt(2)*atan((2*sqrt (x) + sqrt(2))/sqrt(2)) + 2*sqrt(x)*x)/3