Integrand size = 22, antiderivative size = 89 \[ \int \frac {\sqrt {1+x^2}}{x^2 \left (1-x^3\right )} \, dx=-\frac {\sqrt {1+x^2}}{x}-\frac {1}{3} \arctan \left (\frac {1+x}{\sqrt {1+x^2}}\right )-\frac {\text {arctanh}\left (\frac {1-x}{\sqrt {3} \sqrt {1+x^2}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {1+x}{\sqrt {2} \sqrt {1+x^2}}\right ) \] Output:
-(x^2+1)^(1/2)/x-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 = 0.22 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.81 \[ \int \frac {\sqrt {1+x^2}}{x^2 \left (1-x^3\right )} \, dx=-\frac {\sqrt {1+x^2}}{x}+\frac {2}{3} \sqrt {2} \text {arctanh}\left (\frac {1-x+\sqrt {1+x^2}}{\sqrt {2}}\right )+\frac {2}{3} \text {RootSum}\left [1+2 \text {$\#$1}+2 \text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (-x+\sqrt {1+x^2}-\text {$\#$1}\right )-\log \left (-x+\sqrt {1+x^2}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-x+\sqrt {1+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{1+2 \text {$\#$1}-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[Sqrt[1 + x^2]/(x^2*(1 - x^3)),x]
Output:
-(Sqrt[1 + x^2]/x) + (2*Sqrt[2]*ArcTanh[(1 - x + Sqrt[1 + x^2])/Sqrt[2]])/ 3 + (2*RootSum[1 + 2*#1 + 2*#1^2 - 2*#1^3 + #1^4 & , (-Log[-x + Sqrt[1 + x ^2] - #1] - 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 = 89, 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.
| \(\displaystyle \int \frac {\sqrt {x^2+1}}{x^2 \left (1-x^3\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {x^2+1} (x-1)}{3 \left (x^2+x+1\right )}+\frac {\sqrt {x^2+1}}{x^2}-\frac {\sqrt {x^2+1}}{3 (x-1)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{3} \arctan \left (\frac {x+1}{\sqrt {x^2+1}}\right )-\frac {\text {arctanh}\left (\frac {1-x}{\sqrt {3} \sqrt {x^2+1}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {x+1}{\sqrt {2} \sqrt {x^2+1}}\right )-\frac {\sqrt {x^2+1}}{x}\) |
Input:
Int[Sqrt[1 + x^2]/(x^2*(1 - x^3)),x]
Output:
-(Sqrt[1 + x^2]/x) - ArcTan[(1 + x)/Sqrt[1 + x^2]]/3 - ArcTanh[(1 - x)/(Sq rt[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. \(187\) vs. \(2(71)=142\).
Time = 0.64 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.11
| method | result | size |
| risch | \(-\frac {\sqrt {x^{2}+1}}{x}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x +2\right ) \sqrt {2}}{4 \sqrt {\left (x -1\right )^{2}+2 x}}\right )}{3}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )+\arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )\right )}{6 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}\) | \(188\) |
| default | \(-\frac {\sqrt {\left (x -1\right )^{2}+2 x}}{3}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x +2\right ) \sqrt {2}}{4 \sqrt {\left (x -1\right )^{2}+2 x}}\right )}{3}-\frac {\left (x^{2}+1\right )^{\frac {3}{2}}}{x}+\sqrt {x^{2}+1}\, x +\frac {\sqrt {x^{2}+1}}{3}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )-\arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )\right )}{6 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )}{3 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}\) | \(339\) |
| trager | \(-\frac {\sqrt {x^{2}+1}}{x}+72 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{3} \ln \left (-\frac {33696 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{5} x -67392 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{5}-2664 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{3} x +2520 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{3}+288 \sqrt {x^{2}+1}\, \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{2}+44 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right ) x -22 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )-3 \sqrt {x^{2}+1}}{36 x \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{2}-72 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{2}-3 x -1}\right )-2 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {33696 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{5} x -67392 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{5}-2664 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{3} x +2520 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{3}+288 \sqrt {x^{2}+1}\, \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{2}+44 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right ) x -22 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )-3 \sqrt {x^{2}+1}}{36 x \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{2}-72 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{2}-3 x -1}\right )-2 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {5184 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{5} x -10368 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{5}+936 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{3} x -1440 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{3}-288 \sqrt {x^{2}+1}\, \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{2}+22 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right ) x +22 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )+5 \sqrt {x^{2}+1}}{36 x \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{2}-72 \operatorname {RootOf}\left (1296 \textit {\_Z}^{4}-36 \textit {\_Z}^{2}+1\right )^{2}+2 x +3}\right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+2 \sqrt {x^{2}+1}}{x -1}\right )}{3}\) | \(640\) |
Input:
int((x^2+1)^(1/2)/x^2/(-x^3+1),x,method=_RETURNVERBOSE)
Output:
-(x^2+1)^(1/2)/x+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. 189 vs. \(2 (69) = 138\).
Time = 0.10 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.12 \[ \int \frac {\sqrt {1+x^2}}{x^2 \left (1-x^3\right )} \, dx=\frac {\sqrt {3} x \log \left (2 \, x^{2} - \sqrt {x^{2} + 1} {\left (2 \, x + \sqrt {3} + 1\right )} + \sqrt {3} {\left (x + 1\right )} + x + 3\right ) - \sqrt {3} x \log \left (2 \, x^{2} - \sqrt {x^{2} + 1} {\left (2 \, x - \sqrt {3} + 1\right )} - \sqrt {3} {\left (x + 1\right )} + x + 3\right ) + 2 \, \sqrt {2} x \log \left (-\frac {\sqrt {2} {\left (x + 1\right )} + \sqrt {x^{2} + 1} {\left (\sqrt {2} + 2\right )} + x + 1}{x - 1}\right ) - 2 \, x \arctan \left (-\sqrt {3} x + \sqrt {x^{2} + 1} {\left (\sqrt {3} + 1\right )} - x + 1\right ) + 2 \, x \arctan \left (-\sqrt {3} x + \sqrt {x^{2} + 1} {\left (\sqrt {3} - 1\right )} + x - 1\right ) - 6 \, x - 6 \, \sqrt {x^{2} + 1}}{6 \, x} \] Input:
integrate((x^2+1)^(1/2)/x^2/(-x^3+1),x, algorithm="fricas")
Output:
1/6*(sqrt(3)*x*log(2*x^2 - sqrt(x^2 + 1)*(2*x + sqrt(3) + 1) + sqrt(3)*(x + 1) + x + 3) - sqrt(3)*x*log(2*x^2 - sqrt(x^2 + 1)*(2*x - sqrt(3) + 1) - sqrt(3)*(x + 1) + x + 3) + 2*sqrt(2)*x*log(-(sqrt(2)*(x + 1) + sqrt(x^2 + 1)*(sqrt(2) + 2) + x + 1)/(x - 1)) - 2*x*arctan(-sqrt(3)*x + sqrt(x^2 + 1) *(sqrt(3) + 1) - x + 1) + 2*x*arctan(-sqrt(3)*x + sqrt(x^2 + 1)*(sqrt(3) - 1) + x - 1) - 6*x - 6*sqrt(x^2 + 1))/x
\[ \int \frac {\sqrt {1+x^2}}{x^2 \left (1-x^3\right )} \, dx=- \int \frac {\sqrt {x^{2} + 1}}{x^{5} - x^{2}}\, dx \] Input:
integrate((x**2+1)**(1/2)/x**2/(-x**3+1),x)
Output:
-Integral(sqrt(x**2 + 1)/(x**5 - x**2), x)
\[ \int \frac {\sqrt {1+x^2}}{x^2 \left (1-x^3\right )} \, dx=\int { -\frac {\sqrt {x^{2} + 1}}{{\left (x^{3} - 1\right )} x^{2}} \,d x } \] Input:
integrate((x^2+1)^(1/2)/x^2/(-x^3+1),x, algorithm="maxima")
Output:
-integrate(sqrt(x^2 + 1)/((x^3 - 1)*x^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (69) = 138\).
Time = 0.18 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.17 \[ \int \frac {\sqrt {1+x^2}}{x^2 \left (1-x^3\right )} \, dx=-\frac {1}{6} \, \pi + \frac {1}{6} \, \sqrt {3} \log \left ({\left (x + \sqrt {3} - \sqrt {x^{2} + 1} + 1\right )}^{2} + {\left (x - \sqrt {x^{2} + 1}\right )}^{2}\right ) - \frac {1}{6} \, \sqrt {3} \log \left ({\left (x - \sqrt {3} - \sqrt {x^{2} + 1} + 1\right )}^{2} + {\left (x - \sqrt {x^{2} + 1}\right )}^{2}\right ) - \frac {1}{3} \, \sqrt {2} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} + 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} + 2 \right |}}\right ) + \frac {2}{{\left (x - \sqrt {x^{2} + 1}\right )}^{2} - 1} - \frac {1}{3} \, \arctan \left (-{\left (x - \sqrt {x^{2} + 1}\right )} {\left (\sqrt {3} + 1\right )} + 1\right ) - \frac {1}{3} \, \arctan \left ({\left (x - \sqrt {x^{2} + 1}\right )} {\left (\sqrt {3} - 1\right )} + 1\right ) \] Input:
integrate((x^2+1)^(1/2)/x^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)) + 2/((x - sqrt(x^2 + 1) )^2 - 1) - 1/3*arctan(-(x - sqrt(x^2 + 1))*(sqrt(3) + 1) + 1) - 1/3*arctan ((x - sqrt(x^2 + 1))*(sqrt(3) - 1) + 1)
Time = 22.56 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.89 \[ \int \frac {\sqrt {1+x^2}}{x^2 \left (1-x^3\right )} \, dx=-\frac {\sqrt {2}\,\left (2\,\ln \left (x-1\right )-2\,\ln \left (x+\sqrt {2}\,\sqrt {x^2+1}+1\right )\right )}{6}-\frac {\sqrt {x^2+1}}{x}-\frac {\left (\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,2{}\mathrm {i}-\ln \left (1+\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\,2{}\mathrm {i}\right )\,1{}\mathrm {i}}{6\,\sqrt {{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}}-\frac {\left (\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,2{}\mathrm {i}-\ln \left (1+\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}-\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\,2{}\mathrm {i}\right )\,1{}\mathrm {i}}{6\,\sqrt {{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}} \] Input:
int(-(x^2 + 1)^(1/2)/(x^2*(x^3 - 1)),x)
Output:
- ((log(x - (3^(1/2)*1i)/2 + 1/2)*2i - log((3^(1/2)/2 - 1i/2)*(x^2 + 1)^(1 /2) - x/2 + (3^(1/2)*x*1i)/2 + 1)*2i)*1i)/(6*(((3^(1/2)*1i)/2 - 1/2)^2 + 1 )^(1/2)) - ((log(x + (3^(1/2)*1i)/2 + 1/2)*2i - log((3^(1/2)/2 + 1i/2)*(x^ 2 + 1)^(1/2) - x/2 - (3^(1/2)*x*1i)/2 + 1)*2i)*1i)/(6*(((3^(1/2)*1i)/2 + 1 /2)^2 + 1)^(1/2)) - (2^(1/2)*(2*log(x - 1) - 2*log(x + 2^(1/2)*(x^2 + 1)^( 1/2) + 1)))/6 - (x^2 + 1)^(1/2)/x
\[ \int \frac {\sqrt {1+x^2}}{x^2 \left (1-x^3\right )} \, dx=\frac {2 \sqrt {x^{2}+1}\, x -4 \sqrt {x^{2}+1}+\sqrt {2}\, \mathrm {log}\left (-\sqrt {x^{2}+1}\, \sqrt {2}-x -1\right ) x -\sqrt {2}\, \mathrm {log}\left (x -1\right ) x -\left (\int \frac {\sqrt {x^{2}+1}}{x^{6}+x^{5}+2 x^{4}+x^{3}+x^{2}}d x \right ) x -\left (\int \frac {\sqrt {x^{2}+1}}{x^{5}+x^{4}+2 x^{3}+x^{2}+x}d x \right ) x -2 \left (\int \frac {\sqrt {x^{2}+1}\, x^{3}}{x^{4}+x^{3}+2 x^{2}+x +1}d x \right ) x -2 \left (\int \frac {\sqrt {x^{2}+1}\, x^{2}}{x^{4}+x^{3}+2 x^{2}+x +1}d x \right ) x}{3 x} \] Input:
int((x^2+1)^(1/2)/x^2/(-x^3+1),x)
Output:
(2*sqrt(x**2 + 1)*x - 4*sqrt(x**2 + 1) + sqrt(2)*log( - sqrt(x**2 + 1)*sqr t(2) - x - 1)*x - sqrt(2)*log(x - 1)*x - int(sqrt(x**2 + 1)/(x**6 + x**5 + 2*x**4 + x**3 + x**2),x)*x - int(sqrt(x**2 + 1)/(x**5 + x**4 + 2*x**3 + x **2 + x),x)*x - 2*int((sqrt(x**2 + 1)*x**3)/(x**4 + x**3 + 2*x**2 + x + 1) ,x)*x - 2*int((sqrt(x**2 + 1)*x**2)/(x**4 + x**3 + 2*x**2 + x + 1),x)*x)/( 3*x)