Integrand size = 20, antiderivative size = 35 \[ \int \frac {1-x}{\sqrt {e x} \left (1+x^2\right )} \, dx=\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} (1+x)}\right )}{\sqrt {e}} \] Output:
2^(1/2)*arctanh(2^(1/2)*(e*x)^(1/2)/e^(1/2)/(1+x))/e^(1/2)
Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {1-x}{\sqrt {e x} \left (1+x^2\right )} \, dx=\frac {\sqrt {2} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x}}{1+x}\right )}{\sqrt {e x}} \] Input:
Integrate[(1 - x)/(Sqrt[e*x]*(1 + x^2)),x]
Output:
(Sqrt[2]*Sqrt[x]*ArcTanh[(Sqrt[2]*Sqrt[x])/(1 + x)])/Sqrt[e*x]
Leaf count is larger than twice the leaf count of optimal. \(80\) vs. \(2(35)=70\).
Time = 0.37 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.29, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {554, 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.
\(\displaystyle \int \frac {1-x}{\left (x^2+1\right ) \sqrt {e x}} \, dx\) |
\(\Big \downarrow \) 554 |
\(\displaystyle 2 \int \frac {e-e x}{x^2 e^2+e^2}d\sqrt {e x}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle 2 \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e x}}{x e+e-\sqrt {2} \sqrt {e x} \sqrt {e}}d\sqrt {e x}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e x}\right )}{x e+e+\sqrt {2} \sqrt {e x} \sqrt {e}}d\sqrt {e x}}{2 \sqrt {2} \sqrt {e}}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 2 \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e x}}{x e+e-\sqrt {2} \sqrt {e x} \sqrt {e}}d\sqrt {e x}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e x}\right )}{x e+e+\sqrt {2} \sqrt {e x} \sqrt {e}}d\sqrt {e x}}{2 \sqrt {2} \sqrt {e}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e x}}{x e+e-\sqrt {2} \sqrt {e x} \sqrt {e}}d\sqrt {e x}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e x}}{x e+e+\sqrt {2} \sqrt {e x} \sqrt {e}}d\sqrt {e x}}{2 \sqrt {e}}\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle 2 \left (\frac {\log \left (e x+\sqrt {2} \sqrt {e} \sqrt {e x}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e x-\sqrt {2} \sqrt {e} \sqrt {e x}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\) |
Input:
Int[(1 - x)/(Sqrt[e*x]*(1 + x^2)),x]
Output:
2*(-1/2*Log[e + e*x - Sqrt[2]*Sqrt[e]*Sqrt[e*x]]/(Sqrt[2]*Sqrt[e]) + Log[e + e*x + Sqrt[2]*Sqrt[e]*Sqrt[e*x]]/(2*Sqrt[2]*Sqrt[e]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_) + (d_.)*(x_))/(Sqrt[(e_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*c + d*x^2)/(a*e^2 + b*x^4), x], x, Sqrt[e*x]], x] /; FreeQ[{a, b, c, d, e}, 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*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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.56 (sec) , antiderivative size = 164, normalized size of antiderivative = 4.69
method | result | size |
pseudoelliptic | \(-\frac {\left (\ln \left (\frac {-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+e \left (x +\operatorname {csgn}\left (e \right )\right )}{\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+e \left (x +\operatorname {csgn}\left (e \right )\right )}\right )-\operatorname {csgn}\left (e \right ) \ln \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+e \left (x +\operatorname {csgn}\left (e \right )\right )}{-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+e \left (x +\operatorname {csgn}\left (e \right )\right )}\right )+2 \left (\arctan \left (\frac {-\sqrt {2}\, \sqrt {e x}+\left (e^{2}\right )^{\frac {1}{4}}}{\left (e^{2}\right )^{\frac {1}{4}}}\right )-\arctan \left (\frac {\sqrt {2}\, \sqrt {e x}+\left (e^{2}\right )^{\frac {1}{4}}}{\left (e^{2}\right )^{\frac {1}{4}}}\right )\right ) \left (\operatorname {csgn}\left (e \right )-1\right )\right ) \sqrt {2}}{4 \left (e^{2}\right )^{\frac {1}{4}}}\) | \(164\) |
derivativedivides | \(\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {e^{2}}}{e x -\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e x}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e x -\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {e^{2}}}{e x +\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e x}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}\) | \(207\) |
default | \(\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {e^{2}}}{e x -\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e x}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e x -\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {e^{2}}}{e x +\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e x}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}\) | \(207\) |
meijerg | \(\frac {\sqrt {x}\, \left (-\frac {\sqrt {x}\, \sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {x}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {x}\, \sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {x}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {1}{4}}}\right )}{2 \sqrt {e x}}-\frac {\sqrt {x}\, \left (\frac {x^{\frac {3}{2}} \sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {3}{4}}}+\frac {x^{\frac {3}{2}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {3}{4}}}-\frac {x^{\frac {3}{2}} \sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {3}{4}}}+\frac {x^{\frac {3}{2}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {3}{4}}}\right )}{2 \sqrt {e x}}\) | \(294\) |
Input:
int((1-x)/(e*x)^(1/2)/(x^2+1),x,method=_RETURNVERBOSE)
Output:
-1/4*(ln((-(e^2)^(1/4)*(e*x)^(1/2)*2^(1/2)+e*(x+csgn(e)))/((e^2)^(1/4)*(e* x)^(1/2)*2^(1/2)+e*(x+csgn(e))))-csgn(e)*ln(((e^2)^(1/4)*(e*x)^(1/2)*2^(1/ 2)+e*(x+csgn(e)))/(-(e^2)^(1/4)*(e*x)^(1/2)*2^(1/2)+e*(x+csgn(e))))+2*(arc tan((-2^(1/2)*(e*x)^(1/2)+(e^2)^(1/4))/(e^2)^(1/4))-arctan((2^(1/2)*(e*x)^ (1/2)+(e^2)^(1/4))/(e^2)^(1/4)))*(csgn(e)-1))/(e^2)^(1/4)*2^(1/2)
Time = 0.24 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.23 \[ \int \frac {1-x}{\sqrt {e x} \left (1+x^2\right )} \, dx=\left [\frac {\sqrt {2} \log \left (\frac {x^{2} + \frac {2 \, \sqrt {2} \sqrt {e x} {\left (x + 1\right )}}{\sqrt {e}} + 4 \, x + 1}{x^{2} + 1}\right )}{2 \, \sqrt {e}}, -\sqrt {2} \sqrt {-\frac {1}{e}} \arctan \left (\frac {\sqrt {2} \sqrt {e x} {\left (x + 1\right )} \sqrt {-\frac {1}{e}}}{2 \, x}\right )\right ] \] Input:
integrate((1-x)/(e*x)^(1/2)/(x^2+1),x, algorithm="fricas")
Output:
[1/2*sqrt(2)*log((x^2 + 2*sqrt(2)*sqrt(e*x)*(x + 1)/sqrt(e) + 4*x + 1)/(x^ 2 + 1))/sqrt(e), -sqrt(2)*sqrt(-1/e)*arctan(1/2*sqrt(2)*sqrt(e*x)*(x + 1)* sqrt(-1/e)/x)]
\[ \int \frac {1-x}{\sqrt {e x} \left (1+x^2\right )} \, dx=- \int \frac {x}{x^{2} \sqrt {e x} + \sqrt {e x}}\, dx - \int \left (- \frac {1}{x^{2} \sqrt {e x} + \sqrt {e x}}\right )\, dx \] Input:
integrate((1-x)/(e*x)**(1/2)/(x**2+1),x)
Output:
-Integral(x/(x**2*sqrt(e*x) + sqrt(e*x)), x) - Integral(-1/(x**2*sqrt(e*x) + sqrt(e*x)), x)
Exception generated. \[ \int \frac {1-x}{\sqrt {e x} \left (1+x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((1-x)/(e*x)^(1/2)/(x^2+1),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (25) = 50\).
Time = 0.13 (sec) , antiderivative size = 174, normalized size of antiderivative = 4.97 \[ \int \frac {1-x}{\sqrt {e x} \left (1+x^2\right )} \, dx=\frac {\sqrt {2} {\left (e \sqrt {{\left | e \right |}} - {\left | e \right |}^{\frac {3}{2}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | e \right |}} + 2 \, \sqrt {e x}\right )}}{2 \, \sqrt {{\left | e \right |}}}\right )}{2 \, e^{2}} + \frac {\sqrt {2} {\left (e \sqrt {{\left | e \right |}} - {\left | e \right |}^{\frac {3}{2}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | e \right |}} - 2 \, \sqrt {e x}\right )}}{2 \, \sqrt {{\left | e \right |}}}\right )}{2 \, e^{2}} + \frac {\sqrt {2} {\left (e \sqrt {{\left | e \right |}} + {\left | e \right |}^{\frac {3}{2}}\right )} \log \left (e x + \sqrt {2} \sqrt {e x} \sqrt {{\left | e \right |}} + {\left | e \right |}\right )}{4 \, e^{2}} - \frac {\sqrt {2} {\left (e \sqrt {{\left | e \right |}} + {\left | e \right |}^{\frac {3}{2}}\right )} \log \left (e x - \sqrt {2} \sqrt {e x} \sqrt {{\left | e \right |}} + {\left | e \right |}\right )}{4 \, e^{2}} \] Input:
integrate((1-x)/(e*x)^(1/2)/(x^2+1),x, algorithm="giac")
Output:
1/2*sqrt(2)*(e*sqrt(abs(e)) - abs(e)^(3/2))*arctan(1/2*sqrt(2)*(sqrt(2)*sq rt(abs(e)) + 2*sqrt(e*x))/sqrt(abs(e)))/e^2 + 1/2*sqrt(2)*(e*sqrt(abs(e)) - abs(e)^(3/2))*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(e)) - 2*sqrt(e*x))/s qrt(abs(e)))/e^2 + 1/4*sqrt(2)*(e*sqrt(abs(e)) + abs(e)^(3/2))*log(e*x + s qrt(2)*sqrt(e*x)*sqrt(abs(e)) + abs(e))/e^2 - 1/4*sqrt(2)*(e*sqrt(abs(e)) + abs(e)^(3/2))*log(e*x - sqrt(2)*sqrt(e*x)*sqrt(abs(e)) + abs(e))/e^2
Time = 7.86 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {1-x}{\sqrt {e x} \left (1+x^2\right )} \, dx=\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {8\,\sqrt {2}\,\sqrt {e}\,\sqrt {e\,x}}{8\,e+8\,e\,x}\right )}{\sqrt {e}} \] Input:
int(-(x - 1)/((x^2 + 1)*(e*x)^(1/2)),x)
Output:
(2^(1/2)*atanh((8*2^(1/2)*e^(1/2)*(e*x)^(1/2))/(8*e + 8*e*x)))/e^(1/2)
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {1-x}{\sqrt {e x} \left (1+x^2\right )} \, dx=\frac {\sqrt {e}\, \sqrt {2}\, \left (-\mathrm {log}\left (-\sqrt {x}\, \sqrt {2}+x +1\right )+\mathrm {log}\left (\sqrt {x}\, \sqrt {2}+x +1\right )\right )}{2 e} \] Input:
int((1-x)/(e*x)^(1/2)/(x^2+1),x)
Output:
(sqrt(e)*sqrt(2)*( - log( - sqrt(x)*sqrt(2) + x + 1) + log(sqrt(x)*sqrt(2) + x + 1)))/(2*e)