Integrand size = 21, antiderivative size = 24 \[ \int \frac {-1+x}{(1+x) \sqrt {x+x^2+x^3}} \, dx=-2 \arctan \left (\frac {\sqrt {x+x^2+x^3}}{1+x+x^2}\right ) \]
Time = 0.18 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.92 \[ \int \frac {-1+x}{(1+x) \sqrt {x+x^2+x^3}} \, dx=-\frac {2 \sqrt {x} \sqrt {1+x+x^2} \arctan \left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )}{\sqrt {x \left (1+x+x^2\right )}} \]
Time = 0.34 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2467, 25, 2035, 2212, 216}
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.
| \(\displaystyle \int \frac {x-1}{(x+1) \sqrt {x^3+x^2+x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {x^2+x+1} \int -\frac {1-x}{\sqrt {x} (x+1) \sqrt {x^2+x+1}}dx}{\sqrt {x^3+x^2+x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {x^2+x+1} \int \frac {1-x}{\sqrt {x} (x+1) \sqrt {x^2+x+1}}dx}{\sqrt {x^3+x^2+x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2+x+1} \int \frac {1-x}{(x+1) \sqrt {x^2+x+1}}d\sqrt {x}}{\sqrt {x^3+x^2+x}}\) |
| \(\Big \downarrow \) 2212 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2+x+1} \int \frac {1}{x+1}d\frac {\sqrt {x}}{\sqrt {x^2+x+1}}}{\sqrt {x^3+x^2+x}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2+x+1} \arctan \left (\frac {\sqrt {x}}{\sqrt {x^2+x+1}}\right )}{\sqrt {x^3+x^2+x}}\) |
3.3.39.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Simp[A Subst[Int[1/(d - (b*d - 2*a*e)*x^2), x], x, x/Sqrt[a + b*x^2 + c*x^4]], x] /; FreeQ[{a, b, c, d, e, A, B}, x] & & EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Time = 3.53 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75
| method | result | size |
| default | \(2 \arctan \left (\frac {\sqrt {x \left (x^{2}+x +1\right )}}{x}\right )\) | \(18\) |
| pseudoelliptic | \(2 \arctan \left (\frac {\sqrt {x \left (x^{2}+x +1\right )}}{x}\right )\) | \(18\) |
| trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-2 \sqrt {x^{3}+x^{2}+x}}{\left (1+x \right )^{2}}\right )\) | \(45\) |
| elliptic | \(\frac {2 \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}}-\frac {4 \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}\, \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}\) | \(271\) |
Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int \frac {-1+x}{(1+x) \sqrt {x+x^2+x^3}} \, dx=\arctan \left (\frac {x^{2} + 1}{2 \, \sqrt {x^{3} + x^{2} + x}}\right ) \]
\[ \int \frac {-1+x}{(1+x) \sqrt {x+x^2+x^3}} \, dx=\int \frac {x - 1}{\sqrt {x \left (x^{2} + x + 1\right )} \left (x + 1\right )}\, dx \]
\[ \int \frac {-1+x}{(1+x) \sqrt {x+x^2+x^3}} \, dx=\int { \frac {x - 1}{\sqrt {x^{3} + x^{2} + x} {\left (x + 1\right )}} \,d x } \]
\[ \int \frac {-1+x}{(1+x) \sqrt {x+x^2+x^3}} \, dx=\int { \frac {x - 1}{\sqrt {x^{3} + x^{2} + x} {\left (x + 1\right )}} \,d x } \]
Time = 0.23 (sec) , antiderivative size = 179, normalized size of antiderivative = 7.46 \[ \int \frac {-1+x}{(1+x) \sqrt {x+x^2+x^3}} \, dx=\frac {\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (\sqrt {3}+1{}\mathrm {i}\right )\,\left (\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )-2\,\Pi \left (\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )\,1{}\mathrm {i}}{\sqrt {x^3+x^2-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,x}} \]
((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1 i)/2 - 1/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/ 2)*(3^(1/2) + 1i)*(ellipticF(asin((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^ (1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)) - 2*ellipticPi(1/2 - (3^(1/2)*1 i)/2, asin((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^ (1/2)*1i)/2 + 1/2)))*1i)/(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1 i)/2 + 1/2))^(1/2)