Integrand size = 19, antiderivative size = 47 \[ \int \frac {\sqrt {1+x^2}}{1-x^3} \, dx=\frac {2}{3} \arctan \left (\frac {1+x}{\sqrt {1+x^2}}\right )+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {1+x}{\sqrt {2} \sqrt {1+x^2}}\right ) \] Output:
2/3*arctan((1+x)/(x^2+1)^(1/2))+1/3*2^(1/2)*arctanh(1/2*(1+x)*2^(1/2)/(x^2 +1)^(1/2))
Time = 0.16 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.60 \[ \int \frac {\sqrt {1+x^2}}{1-x^3} \, dx=-\frac {2}{3} \left (\arctan \left (\frac {1+x+2 x^2-(1+2 x) \sqrt {1+x^2}}{1-x+\sqrt {1+x^2}}\right )-\sqrt {2} \text {arctanh}\left (\frac {1-x+\sqrt {1+x^2}}{\sqrt {2}}\right )\right ) \] Input:
Integrate[Sqrt[1 + x^2]/(1 - x^3),x]
Output:
(-2*(ArcTan[(1 + x + 2*x^2 - (1 + 2*x)*Sqrt[1 + x^2])/(1 - x + Sqrt[1 + x^ 2])] - Sqrt[2]*ArcTanh[(1 - x + Sqrt[1 + x^2])/Sqrt[2]]))/3
Result contains complex when optimal does not.
Time = 0.70 (sec) , antiderivative size = 197, normalized size of antiderivative = 4.19, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, 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.
\(\displaystyle \int \frac {\sqrt {x^2+1}}{1-x^3} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {\sqrt {x^2+1}}{3 (1-x)}+\frac {\sqrt {x^2+1}}{3 \left (\sqrt [3]{-1} x+1\right )}+\frac {\sqrt {x^2+1}}{3 \left (1-(-1)^{2/3} x\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{3} (-1)^{2/3} \text {arcsinh}(x)+\frac {1}{3} \sqrt [3]{-1} \text {arcsinh}(x)-\frac {\text {arcsinh}(x)}{3}+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {x+1}{\sqrt {2} \sqrt {x^2+1}}\right )+\frac {1}{3} i \text {arctanh}\left (\frac {x+(-1)^{2/3}}{\sqrt {1-\sqrt [3]{-1}} \sqrt {x^2+1}}\right )-\frac {1}{3} \sqrt [3]{-1} \sqrt {1+(-1)^{2/3}} \text {arctanh}\left (\frac {2 \left ((-1)^{2/3} x+1\right )}{\left (-\sqrt {3}+i\right ) \sqrt {x^2+1}}\right )-\frac {1}{3} (-1)^{2/3} \sqrt {x^2+1}+\frac {1}{3} \sqrt [3]{-1} \sqrt {x^2+1}-\frac {\sqrt {x^2+1}}{3}\) |
Input:
Int[Sqrt[1 + x^2]/(1 - x^3),x]
Output:
-1/3*Sqrt[1 + x^2] + ((-1)^(1/3)*Sqrt[1 + x^2])/3 - ((-1)^(2/3)*Sqrt[1 + x ^2])/3 - ArcSinh[x]/3 + ((-1)^(1/3)*ArcSinh[x])/3 - ((-1)^(2/3)*ArcSinh[x] )/3 + (Sqrt[2]*ArcTanh[(1 + x)/(Sqrt[2]*Sqrt[1 + x^2])])/3 + (I/3)*ArcTanh [((-1)^(2/3) + x)/(Sqrt[1 - (-1)^(1/3)]*Sqrt[1 + x^2])] - ((-1)^(1/3)*Sqrt [1 + (-1)^(2/3)]*ArcTanh[(2*(1 + (-1)^(2/3)*x))/((I - Sqrt[3])*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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.83
method | result | size |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+2 \sqrt {x^{2}+1}}{x -1}\right )}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\sqrt {x^{2}+1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\sqrt {x^{2}+1}}{x^{2}+x +1}\right )}{3}\) | \(86\) |
default | \(-\frac {\sqrt {\left (x -1\right )^{2}+2 x}}{3}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x +2\right ) \sqrt {2}}{4 \sqrt {\left (x -1\right )^{2}+2 x}}\right )}{3}+\frac {\sqrt {x^{2}+1}}{3}+\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )+\arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )\right )}{6 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )-\arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )\right )}{6 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}\) | \(347\) |
Input:
int((x^2+1)^(1/2)/(-x^3+1),x,method=_RETURNVERBOSE)
Output:
1/3*RootOf(_Z^2-2)*ln(-(RootOf(_Z^2-2)*x+RootOf(_Z^2-2)+2*(x^2+1)^(1/2))/( x-1))+1/3*RootOf(_Z^2+1)*ln(((x^2+1)^(1/2)*x-RootOf(_Z^2+1)*x+(x^2+1)^(1/2 ))/(x^2+x+1))
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt {1+x^2}}{1-x^3} \, dx=\frac {1}{3} \, \sqrt {2} \log \left (-\frac {\sqrt {2} {\left (x + 1\right )} + \sqrt {x^{2} + 1} {\left (\sqrt {2} + 2\right )} + x + 1}{x - 1}\right ) - \frac {2}{3} \, \arctan \left (-\frac {x^{2} - \sqrt {x^{2} + 1} {\left (x + 1\right )} + x + 1}{x}\right ) \] Input:
integrate((x^2+1)^(1/2)/(-x^3+1),x, algorithm="fricas")
Output:
1/3*sqrt(2)*log(-(sqrt(2)*(x + 1) + sqrt(x^2 + 1)*(sqrt(2) + 2) + x + 1)/( x - 1)) - 2/3*arctan(-(x^2 - sqrt(x^2 + 1)*(x + 1) + x + 1)/x)
\[ \int \frac {\sqrt {1+x^2}}{1-x^3} \, dx=- \int \frac {\sqrt {x^{2} + 1}}{x^{3} - 1}\, dx \] Input:
integrate((x**2+1)**(1/2)/(-x**3+1),x)
Output:
-Integral(sqrt(x**2 + 1)/(x**3 - 1), x)
\[ \int \frac {\sqrt {1+x^2}}{1-x^3} \, dx=\int { -\frac {\sqrt {x^{2} + 1}}{x^{3} - 1} \,d x } \] Input:
integrate((x^2+1)^(1/2)/(-x^3+1),x, algorithm="maxima")
Output:
-integrate(sqrt(x^2 + 1)/(x^3 - 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (36) = 72\).
Time = 0.14 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.38 \[ \int \frac {\sqrt {1+x^2}}{1-x^3} \, dx=-\frac {1}{3} \, \sqrt {2} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} + 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} + 2 \right |}}\right ) + \frac {2}{3} \, \arctan \left (-\frac {1}{2} \, {\left (x - \sqrt {x^{2} + 1}\right )}^{3} - \frac {3}{2} \, {\left (x - \sqrt {x^{2} + 1}\right )}^{2} - \frac {3}{2} \, x + \frac {3}{2} \, \sqrt {x^{2} + 1} + \frac {1}{2}\right ) - \frac {2}{3} \, \arctan \left (-x + \sqrt {x^{2} + 1} - 2\right ) \] Input:
integrate((x^2+1)^(1/2)/(-x^3+1),x, algorithm="giac")
Output:
-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)) + 2/3*arctan(-1/2*(x - sqrt(x^2 + 1))^3 - 3/2*(x - sqrt(x^2 + 1))^2 - 3/2*x + 3/2*sqrt(x^2 + 1) + 1/2) - 2/3*arctan( -x + sqrt(x^2 + 1) - 2)
Time = 23.84 (sec) , antiderivative size = 164, normalized size of antiderivative = 3.49 \[ \int \frac {\sqrt {1+x^2}}{1-x^3} \, dx=-\frac {\sqrt {2}\,\left (\ln \left (x-1\right )-\ln \left (x+\sqrt {2}\,\sqrt {x^2+1}+1\right )\right )}{3}-\frac {\left (\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (1+\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )\,\sqrt {{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}}{3\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}-\frac {\left (\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (1+\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}-\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )\,\sqrt {{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}}{3\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2} \] Input:
int(-(x^2 + 1)^(1/2)/(x^3 - 1),x)
Output:
- (2^(1/2)*(log(x - 1) - log(x + 2^(1/2)*(x^2 + 1)^(1/2) + 1)))/3 - ((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 - 1/2)^2 + 1)^(1/2))/(3*((3^(1/2) *1i)/2 - 1/2)^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 + 1/2)^2 + 1)^(1/2))/(3*((3^(1/2)*1i)/2 + 1/2)^2)
\[ \int \frac {\sqrt {1+x^2}}{1-x^3} \, dx=-\left (\int \frac {\sqrt {x^{2}+1}}{x^{3}-1}d x \right ) \] Input:
int((x^2+1)^(1/2)/(-x^3+1),x)
Output:
- int(sqrt(x**2 + 1)/(x**3 - 1),x)