Integrand size = 22, antiderivative size = 85 \[ \int \frac {x^2 \sqrt {1+x^2}}{1-x^3} \, dx=-\sqrt {1+x^2}-\frac {1}{3} \arctan \left (\frac {1+x}{\sqrt {1+x^2}}\right )+\frac {\text {arctanh}\left (\frac {1-x}{\sqrt {3} \sqrt {1+x^2}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {1+x}{\sqrt {2} \sqrt {1+x^2}}\right ) \] Output:
-(x^2+1)^(1/2)-1/3*arctan((1+x)/(x^2+1)^(1/2))+1/3*arctanh(1/3*(1-x)*3^(1/ 2)/(x^2+1)^(1/2))*3^(1/2)+1/3*2^(1/2)*arctanh(1/2*(1+x)*2^(1/2)/(x^2+1)^(1 /2))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.17 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.86 \[ \int \frac {x^2 \sqrt {1+x^2}}{1-x^3} \, dx=\frac {1}{3} \left (-3 \sqrt {1+x^2}+2 \sqrt {2} \text {arctanh}\left (\frac {1-x+\sqrt {1+x^2}}{\sqrt {2}}\right )-\text {RootSum}\left [1+2 \text {$\#$1}+2 \text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (-x+\sqrt {1+x^2}-\text {$\#$1}\right )-4 \log \left (-x+\sqrt {1+x^2}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-x+\sqrt {1+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{1+2 \text {$\#$1}-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ]\right ) \] Input:
Integrate[(x^2*Sqrt[1 + x^2])/(1 - x^3),x]
Output:
(-3*Sqrt[1 + x^2] + 2*Sqrt[2]*ArcTanh[(1 - x + Sqrt[1 + x^2])/Sqrt[2]] - R ootSum[1 + 2*#1 + 2*#1^2 - 2*#1^3 + #1^4 & , (-Log[-x + Sqrt[1 + x^2] - #1 ] - 4*Log[-x + Sqrt[1 + x^2] - #1]*#1 + Log[-x + Sqrt[1 + x^2] - #1]*#1^2) /(1 + 2*#1 - 3*#1^2 + 2*#1^3) & ])/3
Time = 0.74 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {7276, 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.
| \(\displaystyle \int \frac {x^2 \sqrt {x^2+1}}{1-x^3} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {(-2 x-1) \sqrt {x^2+1}}{3 \left (x^2+x+1\right )}-\frac {\sqrt {x^2+1}}{3 (x-1)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{3} \arctan \left (\frac {x+1}{\sqrt {x^2+1}}\right )+\frac {\text {arctanh}\left (\frac {1-x}{\sqrt {3} \sqrt {x^2+1}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {x+1}{\sqrt {2} \sqrt {x^2+1}}\right )-\sqrt {x^2+1}\) |
Input:
Int[(x^2*Sqrt[1 + x^2])/(1 - x^3),x]
Output:
-Sqrt[1 + x^2] - ArcTan[(1 + x)/Sqrt[1 + x^2]]/3 + ArcTanh[(1 - x)/(Sqrt[3 ]*Sqrt[1 + x^2])]/Sqrt[3] + (Sqrt[2]*ArcTanh[(1 + x)/(Sqrt[2]*Sqrt[1 + x^2 ])])/3
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(186\) vs. \(2(68)=136\).
Time = 0.46 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.20
| method | result | size |
| risch | \(-\sqrt {x^{2}+1}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x +2\right ) \sqrt {2}}{4 \sqrt {\left (x -1\right )^{2}+2 x}}\right )}{3}+\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )-\arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )\right )}{6 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}\) | \(187\) |
| default | \(-\frac {\sqrt {\left (x -1\right )^{2}+2 x}}{3}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x +2\right ) \sqrt {2}}{4 \sqrt {\left (x -1\right )^{2}+2 x}}\right )}{3}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )}{3 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}-\frac {2 \sqrt {x^{2}+1}}{3}+\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )+\arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )\right )}{6 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}\) | \(316\) |
| trager | \(-\sqrt {x^{2}+1}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +2 \sqrt {x^{2}+1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x -1}\right )}{3}+\operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {243 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{5} x -27 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{3} x -27 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{3}+\sqrt {x^{2}+1}+3 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )}{-1+9 x \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{2}-x}\right )-9 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{3} \ln \left (\frac {27 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{3} x -3 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right ) x +\sqrt {x^{2}+1}+3 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )}{9 x \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{2}+1}\right )+\operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {27 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{3} x -3 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right ) x +\sqrt {x^{2}+1}+3 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )}{9 x \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{2}+1}\right )\) | \(364\) |
Input:
int(x^2*(x^2+1)^(1/2)/(-x^3+1),x,method=_RETURNVERBOSE)
Output:
-(x^2+1)^(1/2)+1/3*2^(1/2)*arctanh(1/4*(2*x+2)*2^(1/2)/((x-1)^2+2*x)^(1/2) )+1/6*2^(1/2)*(2*(x+1)^2/(1-x)^2+2)^(1/2)*(3^(1/2)*arctanh(1/2*(2*(x+1)^2/ (1-x)^2+2)^(1/2)*3^(1/2))-arctan(1/((x+1)^2/(1-x)^2+1)*(2*(x+1)^2/(1-x)^2+ 2)^(1/2)*(x+1)/(1-x)))/(((x+1)^2/(1-x)^2+1)/((x+1)/(1-x)+1)^2)^(1/2)/((x+1 )/(1-x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (66) = 132\).
Time = 0.08 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.08 \[ \int \frac {x^2 \sqrt {1+x^2}}{1-x^3} \, dx=-\frac {1}{6} \, \sqrt {3} \log \left (2 \, x^{2} - \sqrt {x^{2} + 1} {\left (2 \, x + \sqrt {3} + 1\right )} + \sqrt {3} {\left (x + 1\right )} + x + 3\right ) + \frac {1}{6} \, \sqrt {3} \log \left (2 \, x^{2} - \sqrt {x^{2} + 1} {\left (2 \, x - \sqrt {3} + 1\right )} - \sqrt {3} {\left (x + 1\right )} + x + 3\right ) + \frac {1}{3} \, \sqrt {2} \log \left (-\frac {\sqrt {2} {\left (x + 1\right )} + \sqrt {x^{2} + 1} {\left (\sqrt {2} + 2\right )} + x + 1}{x - 1}\right ) - \sqrt {x^{2} + 1} - \frac {1}{3} \, \arctan \left (-\sqrt {3} x + \sqrt {x^{2} + 1} {\left (\sqrt {3} + 1\right )} - x + 1\right ) + \frac {1}{3} \, \arctan \left (-\sqrt {3} x + \sqrt {x^{2} + 1} {\left (\sqrt {3} - 1\right )} + x - 1\right ) \] Input:
integrate(x^2*(x^2+1)^(1/2)/(-x^3+1),x, algorithm="fricas")
Output:
-1/6*sqrt(3)*log(2*x^2 - sqrt(x^2 + 1)*(2*x + sqrt(3) + 1) + sqrt(3)*(x + 1) + x + 3) + 1/6*sqrt(3)*log(2*x^2 - sqrt(x^2 + 1)*(2*x - sqrt(3) + 1) - sqrt(3)*(x + 1) + x + 3) + 1/3*sqrt(2)*log(-(sqrt(2)*(x + 1) + sqrt(x^2 + 1)*(sqrt(2) + 2) + x + 1)/(x - 1)) - sqrt(x^2 + 1) - 1/3*arctan(-sqrt(3)*x + sqrt(x^2 + 1)*(sqrt(3) + 1) - x + 1) + 1/3*arctan(-sqrt(3)*x + sqrt(x^2 + 1)*(sqrt(3) - 1) + x - 1)
\[ \int \frac {x^2 \sqrt {1+x^2}}{1-x^3} \, dx=- \int \frac {x^{2} \sqrt {x^{2} + 1}}{x^{3} - 1}\, dx \] Input:
integrate(x**2*(x**2+1)**(1/2)/(-x**3+1),x)
Output:
-Integral(x**2*sqrt(x**2 + 1)/(x**3 - 1), x)
\[ \int \frac {x^2 \sqrt {1+x^2}}{1-x^3} \, dx=\int { -\frac {\sqrt {x^{2} + 1} x^{2}}{x^{3} - 1} \,d x } \] Input:
integrate(x^2*(x^2+1)^(1/2)/(-x^3+1),x, algorithm="maxima")
Output:
-integrate(sqrt(x^2 + 1)*x^2/(x^3 - 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (66) = 132\).
Time = 0.15 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.15 \[ \int \frac {x^2 \sqrt {1+x^2}}{1-x^3} \, dx=-\frac {1}{6} \, \pi - \frac {1}{6} \, \sqrt {3} \log \left ({\left (x + \sqrt {3} - \sqrt {x^{2} + 1} + 1\right )}^{2} + {\left (x - \sqrt {x^{2} + 1}\right )}^{2}\right ) + \frac {1}{6} \, \sqrt {3} \log \left ({\left (x - \sqrt {3} - \sqrt {x^{2} + 1} + 1\right )}^{2} + {\left (x - \sqrt {x^{2} + 1}\right )}^{2}\right ) - \frac {1}{3} \, \sqrt {2} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} + 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} + 2 \right |}}\right ) - \sqrt {x^{2} + 1} - \frac {1}{3} \, \arctan \left (-{\left (x - \sqrt {x^{2} + 1}\right )} {\left (\sqrt {3} + 1\right )} + 1\right ) - \frac {1}{3} \, \arctan \left ({\left (x - \sqrt {x^{2} + 1}\right )} {\left (\sqrt {3} - 1\right )} + 1\right ) \] Input:
integrate(x^2*(x^2+1)^(1/2)/(-x^3+1),x, algorithm="giac")
Output:
-1/6*pi - 1/6*sqrt(3)*log((x + sqrt(3) - sqrt(x^2 + 1) + 1)^2 + (x - sqrt( x^2 + 1))^2) + 1/6*sqrt(3)*log((x - sqrt(3) - sqrt(x^2 + 1) + 1)^2 + (x - sqrt(x^2 + 1))^2) - 1/3*sqrt(2)*log(abs(-2*x - 2*sqrt(2) + 2*sqrt(x^2 + 1) + 2)/abs(-2*x + 2*sqrt(2) + 2*sqrt(x^2 + 1) + 2)) - sqrt(x^2 + 1) - 1/3*a rctan(-(x - sqrt(x^2 + 1))*(sqrt(3) + 1) + 1) - 1/3*arctan((x - sqrt(x^2 + 1))*(sqrt(3) - 1) + 1)
Time = 22.35 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.48 \[ \int \frac {x^2 \sqrt {1+x^2}}{1-x^3} \, dx=-\frac {\sqrt {2}\,\left (\ln \left (x-1\right )-\ln \left (x+\sqrt {2}\,\sqrt {x^2+1}+1\right )\right )}{3}-\sqrt {x^2+1}-\frac {\left (\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (1+\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )\,\left ({\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,\sqrt {{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}}+\frac {\left (\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (1+\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}-\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )\,\left (\frac {1}{2}-{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,\sqrt {{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}} \] Input:
int(-(x^2*(x^2 + 1)^(1/2))/(x^3 - 1),x)
Output:
((log(x + (3^(1/2)*1i)/2 + 1/2) - log((3^(1/2)/2 + 1i/2)*(x^2 + 1)^(1/2) - x/2 - (3^(1/2)*x*1i)/2 + 1))*((3^(1/2)*1i)/2 - ((3^(1/2)*1i)/2 + 1/2)^2 + 1/2))/(3*((3^(1/2)*1i)/2 + 1/2)^2*(((3^(1/2)*1i)/2 + 1/2)^2 + 1)^(1/2)) - (x^2 + 1)^(1/2) - ((log(x - (3^(1/2)*1i)/2 + 1/2) - log((3^(1/2)/2 - 1i/2 )*(x^2 + 1)^(1/2) - x/2 + (3^(1/2)*x*1i)/2 + 1))*((3^(1/2)*1i)/2 + ((3^(1/ 2)*1i)/2 - 1/2)^2 - 1/2))/(3*((3^(1/2)*1i)/2 - 1/2)^2*(((3^(1/2)*1i)/2 - 1 /2)^2 + 1)^(1/2)) - (2^(1/2)*(log(x - 1) - log(x + 2^(1/2)*(x^2 + 1)^(1/2) + 1)))/3
\[ \int \frac {x^2 \sqrt {1+x^2}}{1-x^3} \, dx=-\left (\int \frac {\sqrt {x^{2}+1}\, x^{2}}{x^{3}-1}d x \right ) \] Input:
int(x^2*(x^2+1)^(1/2)/(-x^3+1),x)
Output:
- int((sqrt(x**2 + 1)*x**2)/(x**3 - 1),x)