Integrand size = 19, antiderivative size = 49 \[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {2 x+x^2}} \, dx=-\frac {1}{2} \arctan \left (\sqrt {2 x+x^2}\right )-\frac {\text {arctanh}\left (\frac {1+2 x}{\sqrt {3} \sqrt {2 x+x^2}}\right )}{2 \sqrt {3}} \] Output:
-1/2*arctan((x^2+2*x)^(1/2))-1/6*arctanh(1/3*(1+2*x)*3^(1/2)/(x^2+2*x)^(1/ 2))*3^(1/2)
Time = 0.11 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.59 \[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {2 x+x^2}} \, dx=\frac {\sqrt {x} \sqrt {2+x} \left (3 \arctan \left (1+x-\sqrt {x} \sqrt {2+x}\right )-\sqrt {3} \text {arctanh}\left (\frac {1-x+\sqrt {x} \sqrt {2+x}}{\sqrt {3}}\right )\right )}{3 \sqrt {x (2+x)}} \] Input:
Integrate[1/((-1 + x^2)*Sqrt[2*x + x^2]),x]
Output:
(Sqrt[x]*Sqrt[2 + x]*(3*ArcTan[1 + x - Sqrt[x]*Sqrt[2 + x]] - Sqrt[3]*ArcT anh[(1 - x + Sqrt[x]*Sqrt[2 + x])/Sqrt[3]]))/(3*Sqrt[x*(2 + x)])
Time = 0.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {1316, 25, 1112, 216, 1154, 219}
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.
\(\displaystyle \int \frac {1}{\left (x^2-1\right ) \sqrt {x^2+2 x}} \, dx\) |
\(\Big \downarrow \) 1316 |
\(\displaystyle \frac {1}{2} \int -\frac {1}{(1-x) \sqrt {x^2+2 x}}dx+\frac {1}{2} \int -\frac {1}{(x+1) \sqrt {x^2+2 x}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{(1-x) \sqrt {x^2+2 x}}dx-\frac {1}{2} \int \frac {1}{(x+1) \sqrt {x^2+2 x}}dx\) |
\(\Big \downarrow \) 1112 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{(1-x) \sqrt {x^2+2 x}}dx-2 \int \frac {1}{4 \left (x^2+2 x\right )+4}d\sqrt {x^2+2 x}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{(1-x) \sqrt {x^2+2 x}}dx-\frac {1}{2} \arctan \left (\sqrt {x^2+2 x}\right )\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \int \frac {1}{12-\frac {4 (2 x+1)^2}{x^2+2 x}}d\left (-\frac {2 (2 x+1)}{\sqrt {x^2+2 x}}\right )-\frac {1}{2} \arctan \left (\sqrt {x^2+2 x}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {1}{2} \arctan \left (\sqrt {x^2+2 x}\right )-\frac {\text {arctanh}\left (\frac {2 x+1}{\sqrt {3} \sqrt {x^2+2 x}}\right )}{2 \sqrt {3}}\) |
Input:
Int[1/((-1 + x^2)*Sqrt[2*x + x^2]),x]
Output:
-1/2*ArcTan[Sqrt[2*x + x^2]] - ArcTanh[(1 + 2*x)/(Sqrt[3]*Sqrt[2*x + x^2]) ]/(2*Sqrt[3])
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symb ol] :> Simp[4*c Subst[Int[1/(b^2*e - 4*a*c*e + 4*c*e*x^2), x], x, Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[1/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Sy mbol] :> Simp[1/2 Int[1/((a - Rt[(-a)*c, 2]*x)*Sqrt[d + e*x + f*x^2]), x] , x] + Simp[1/2 Int[1/((a + Rt[(-a)*c, 2]*x)*Sqrt[d + e*x + f*x^2]), x], x] /; FreeQ[{a, c, d, e, f}, x] && NeQ[e^2 - 4*d*f, 0] && PosQ[(-a)*c]
Time = 0.42 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(\arctan \left (\frac {\sqrt {x \left (2+x \right )}}{x}\right )-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {3}\, \sqrt {x \left (2+x \right )}}{3 x}\right )}{3}\) | \(35\) |
default | \(-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (2+4 x \right ) \sqrt {3}}{6 \sqrt {\left (-1+x \right )^{2}-1+4 x}}\right )}{6}+\frac {\arctan \left (\frac {1}{\sqrt {\left (1+x \right )^{2}-1}}\right )}{2}\) | \(42\) |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +3 \sqrt {x^{2}+2 x}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{-1+x}\right )}{6}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{2}+2 x}}{1+x}\right )}{2}\) | \(75\) |
Input:
int(1/(x^2-1)/(x^2+2*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
arctan((x*(2+x))^(1/2)/x)-1/3*3^(1/2)*arctanh(1/3*3^(1/2)*(x*(2+x))^(1/2)/ x)
Time = 0.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.27 \[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {2 x+x^2}} \, dx=\frac {1}{6} \, \sqrt {3} \log \left (-\frac {\sqrt {3} {\left (2 \, x + 1\right )} + \sqrt {x^{2} + 2 \, x} {\left (2 \, \sqrt {3} - 3\right )} - 4 \, x - 2}{x - 1}\right ) - \arctan \left (-x + \sqrt {x^{2} + 2 \, x} - 1\right ) \] Input:
integrate(1/(x^2-1)/(x^2+2*x)^(1/2),x, algorithm="fricas")
Output:
1/6*sqrt(3)*log(-(sqrt(3)*(2*x + 1) + sqrt(x^2 + 2*x)*(2*sqrt(3) - 3) - 4* x - 2)/(x - 1)) - arctan(-x + sqrt(x^2 + 2*x) - 1)
\[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {2 x+x^2}} \, dx=\int \frac {1}{\sqrt {x \left (x + 2\right )} \left (x - 1\right ) \left (x + 1\right )}\, dx \] Input:
integrate(1/(x**2-1)/(x**2+2*x)**(1/2),x)
Output:
Integral(1/(sqrt(x*(x + 2))*(x - 1)*(x + 1)), x)
Time = 0.10 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {2 x+x^2}} \, dx=-\frac {1}{6} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{2} + 2 \, x}}{{\left | 2 \, x - 2 \right |}} + \frac {6}{{\left | 2 \, x - 2 \right |}} + 2\right ) + \frac {1}{2} \, \arcsin \left (\frac {2}{{\left | 2 \, x + 2 \right |}}\right ) \] Input:
integrate(1/(x^2-1)/(x^2+2*x)^(1/2),x, algorithm="maxima")
Output:
-1/6*sqrt(3)*log(2*sqrt(3)*sqrt(x^2 + 2*x)/abs(2*x - 2) + 6/abs(2*x - 2) + 2) + 1/2*arcsin(2/abs(2*x + 2))
Time = 0.14 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.45 \[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {2 x+x^2}} \, dx=\frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {3} + 2 \, \sqrt {x^{2} + 2 \, x} + 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {3} + 2 \, \sqrt {x^{2} + 2 \, x} + 2 \right |}}\right ) - \arctan \left (-x + \sqrt {x^{2} + 2 \, x} - 1\right ) \] Input:
integrate(1/(x^2-1)/(x^2+2*x)^(1/2),x, algorithm="giac")
Output:
1/6*sqrt(3)*log(abs(-2*x - 2*sqrt(3) + 2*sqrt(x^2 + 2*x) + 2)/abs(-2*x + 2 *sqrt(3) + 2*sqrt(x^2 + 2*x) + 2)) - arctan(-x + sqrt(x^2 + 2*x) - 1)
Timed out. \[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {2 x+x^2}} \, dx=\int \frac {1}{\sqrt {x^2+2\,x}\,\left (x^2-1\right )} \,d x \] Input:
int(1/((2*x + x^2)^(1/2)*(x^2 - 1)),x)
Output:
int(1/((2*x + x^2)^(1/2)*(x^2 - 1)), x)
Time = 0.16 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.55 \[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {2 x+x^2}} \, dx=-\mathit {atan} \left (\sqrt {x +2}+\sqrt {x}-1\right )+\mathit {atan} \left (\sqrt {x +2}+\sqrt {x}+1\right )+\frac {\sqrt {3}\, \mathrm {log}\left (\frac {2 \sqrt {x +2}+2 \sqrt {x}-2 \sqrt {3}-2}{\sqrt {2}}\right )}{6}-\frac {\sqrt {3}\, \mathrm {log}\left (\frac {2 \sqrt {x +2}+2 \sqrt {x}-2 \sqrt {3}+2}{\sqrt {2}}\right )}{6}-\frac {\sqrt {3}\, \mathrm {log}\left (\frac {2 \sqrt {x +2}+2 \sqrt {x}+2 \sqrt {3}-2}{\sqrt {2}}\right )}{6}+\frac {\sqrt {3}\, \mathrm {log}\left (\frac {2 \sqrt {x +2}+2 \sqrt {x}+2 \sqrt {3}+2}{\sqrt {2}}\right )}{6} \] Input:
int(1/(x^2-1)/(x^2+2*x)^(1/2),x)
Output:
( - 6*atan(sqrt(x + 2) + sqrt(x) - 1) + 6*atan(sqrt(x + 2) + sqrt(x) + 1) + sqrt(3)*log((2*sqrt(x + 2) + 2*sqrt(x) - 2*sqrt(3) - 2)/sqrt(2)) - sqrt( 3)*log((2*sqrt(x + 2) + 2*sqrt(x) - 2*sqrt(3) + 2)/sqrt(2)) - sqrt(3)*log( (2*sqrt(x + 2) + 2*sqrt(x) + 2*sqrt(3) - 2)/sqrt(2)) + sqrt(3)*log((2*sqrt (x + 2) + 2*sqrt(x) + 2*sqrt(3) + 2)/sqrt(2)))/6