Integrand size = 22, antiderivative size = 92 \[ \int \frac {x^4 \sqrt {1+x^2}}{1-x^3} \, dx=-\frac {1}{3} \left (1+x^2\right )^{3/2}-\text {arcsinh}(x)-\frac {1}{3} \arctan \left (\frac {1+x}{\sqrt {1+x^2}}\right )-\frac {\text {arctanh}\left (\frac {1-x}{\sqrt {3} \sqrt {1+x^2}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {1+x}{\sqrt {2} \sqrt {1+x^2}}\right ) \] Output:
-1/3*(x^2+1)^(3/2)-arcsinh(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 = 0.21 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.07 \[ \int \frac {x^4 \sqrt {1+x^2}}{1-x^3} \, dx=\frac {1}{3} \left (-1-x^2\right ) \sqrt {1+x^2}+\frac {2}{3} \sqrt {2} \text {arctanh}\left (\frac {1}{\sqrt {2}}-\frac {x}{\sqrt {2}}+\frac {\sqrt {1+x^2}}{\sqrt {2}}\right )+\log \left (-x+\sqrt {1+x^2}\right )+\frac {2}{3} \text {RootSum}\left [1+2 \text {$\#$1}+2 \text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (-x+\sqrt {1+x^2}-\text {$\#$1}\right )-\log \left (-x+\sqrt {1+x^2}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-x+\sqrt {1+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{1+2 \text {$\#$1}-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[(x^4*Sqrt[1 + x^2])/(1 - x^3),x]
Output:
((-1 - x^2)*Sqrt[1 + x^2])/3 + (2*Sqrt[2]*ArcTanh[1/Sqrt[2] - x/Sqrt[2] + Sqrt[1 + x^2]/Sqrt[2]])/3 + Log[-x + Sqrt[1 + x^2]] + (2*RootSum[1 + 2*#1 + 2*#1^2 - 2*#1^3 + #1^4 & , (-Log[-x + Sqrt[1 + x^2] - #1] - Log[-x + Sqr t[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 = 0.85 (sec) , antiderivative size = 92, 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.
| \(\displaystyle \int \frac {x^4 \sqrt {x^2+1}}{1-x^3} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {x \sqrt {x^2+1}}{1-x^3}-x \sqrt {x^2+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\text {arcsinh}(x)-\frac {1}{3} \arctan \left (\frac {x+1}{\sqrt {x^2+1}}\right )-\frac {\text {arctanh}\left (\frac {1-x}{\sqrt {3} \sqrt {x^2+1}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {x+1}{\sqrt {2} \sqrt {x^2+1}}\right )-\frac {1}{3} \left (x^2+1\right )^{3/2}\) |
Input:
Int[(x^4*Sqrt[1 + x^2])/(1 - x^3),x]
Output:
-1/3*(1 + x^2)^(3/2) - ArcSinh[x] - ArcTan[(1 + x)/Sqrt[1 + x^2]]/3 - ArcT anh[(1 - x)/(Sqrt[3]*Sqrt[1 + x^2])]/Sqrt[3] + (Sqrt[2]*ArcTanh[(1 + x)/(S qrt[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. \(330\) vs. \(2(72)=144\).
Time = 0.63 (sec) , antiderivative size = 331, normalized size of antiderivative = 3.60
| method | result | size |
| default | \(-\frac {\left (x^{2}+1\right )^{\frac {3}{2}}}{3}-\frac {\sqrt {\left (x -1\right )^{2}+2 x}}{3}-\operatorname {arcsinh}\left (x \right )+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x +2\right ) \sqrt {2}}{4 \sqrt {\left (x -1\right )^{2}+2 x}}\right )}{3}+\frac {\sqrt {x^{2}+1}}{3}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )-\arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )\right )}{6 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )}{3 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}\) | \(331\) |
| risch | \(-\frac {\left (x^{2}+1\right )^{\frac {3}{2}}}{3}-\frac {\sqrt {\left (x -1\right )^{2}+2 x}}{3}-\operatorname {arcsinh}\left (x \right )+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x +2\right ) \sqrt {2}}{4 \sqrt {\left (x -1\right )^{2}+2 x}}\right )}{3}+\frac {\sqrt {x^{2}+1}}{3}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )-\arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )\right )}{6 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )}{3 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}\) | \(331\) |
| trager | \(\left (-\frac {1}{3}-\frac {x^{2}}{3}\right ) \sqrt {x^{2}+1}+\ln \left (x -\sqrt {x^{2}+1}\right )+\operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {-27 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{3} x +3 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right ) x +\sqrt {x^{2}+1}-3 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )}{9 x \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{2}+1}\right )+9 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{3} \ln \left (\frac {243 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{5} x -27 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{3} x -27 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{3}+\sqrt {x^{2}+1}+3 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )}{-1+9 x \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{2}-x}\right )-\operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {243 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{5} x -27 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{3} x -27 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{3}+\sqrt {x^{2}+1}+3 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )}{-1+9 x \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{2}-x}\right )-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +2 \sqrt {x^{2}+1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x -1}\right )}{3}\) | \(405\) |
Input:
int(x^4*(x^2+1)^(1/2)/(-x^3+1),x,method=_RETURNVERBOSE)
Output:
-1/3*(x^2+1)^(3/2)-1/3*((x-1)^2+2*x)^(1/2)-arcsinh(x)+1/3*2^(1/2)*arctanh( 1/4*(2*x+2)*2^(1/2)/((x-1)^2+2*x)^(1/2))+1/3*(x^2+1)^(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)-1/3*2^ (1/2)/(((x+1)^2/(1-x)^2+1)/((x+1)/(1-x)+1)^2)^(1/2)/((x+1)/(1-x)+1)*(2*(x+ 1)^2/(1-x)^2+2)^(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))
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (70) = 140\).
Time = 0.09 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.05 \[ \int \frac {x^4 \sqrt {1+x^2}}{1-x^3} \, dx=-\frac {1}{3} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} + \frac {1}{6} \, \sqrt {3} \log \left (2 \, x^{2} - \sqrt {x^{2} + 1} {\left (2 \, x + \sqrt {3} + 1\right )} + \sqrt {3} {\left (x + 1\right )} + x + 3\right ) - \frac {1}{6} \, \sqrt {3} \log \left (2 \, x^{2} - \sqrt {x^{2} + 1} {\left (2 \, x - \sqrt {3} + 1\right )} - \sqrt {3} {\left (x + 1\right )} + x + 3\right ) + \frac {1}{3} \, \sqrt {2} \log \left (-\frac {\sqrt {2} {\left (x + 1\right )} + \sqrt {x^{2} + 1} {\left (\sqrt {2} + 2\right )} + x + 1}{x - 1}\right ) - \frac {1}{3} \, \arctan \left (-\sqrt {3} x + \sqrt {x^{2} + 1} {\left (\sqrt {3} + 1\right )} - x + 1\right ) + \frac {1}{3} \, \arctan \left (-\sqrt {3} x + \sqrt {x^{2} + 1} {\left (\sqrt {3} - 1\right )} + x - 1\right ) + \log \left (-x + \sqrt {x^{2} + 1}\right ) \] Input:
integrate(x^4*(x^2+1)^(1/2)/(-x^3+1),x, algorithm="fricas")
Output:
-1/3*(x^2 + 1)^(3/2) + 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)) - 1/3*arctan(-sqr t(3)*x + sqrt(x^2 + 1)*(sqrt(3) + 1) - x + 1) + 1/3*arctan(-sqrt(3)*x + sq rt(x^2 + 1)*(sqrt(3) - 1) + x - 1) + log(-x + sqrt(x^2 + 1))
\[ \int \frac {x^4 \sqrt {1+x^2}}{1-x^3} \, dx=- \int \frac {x^{4} \sqrt {x^{2} + 1}}{x^{3} - 1}\, dx \] Input:
integrate(x**4*(x**2+1)**(1/2)/(-x**3+1),x)
Output:
-Integral(x**4*sqrt(x**2 + 1)/(x**3 - 1), x)
\[ \int \frac {x^4 \sqrt {1+x^2}}{1-x^3} \, dx=\int { -\frac {\sqrt {x^{2} + 1} x^{4}}{x^{3} - 1} \,d x } \] Input:
integrate(x^4*(x^2+1)^(1/2)/(-x^3+1),x, algorithm="maxima")
Output:
-integrate(sqrt(x^2 + 1)*x^4/(x^3 - 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (70) = 140\).
Time = 0.15 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.12 \[ \int \frac {x^4 \sqrt {1+x^2}}{1-x^3} \, dx=-\frac {1}{6} \, \pi - \frac {1}{3} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} + \frac {1}{6} \, \sqrt {3} \log \left ({\left (x + \sqrt {3} - \sqrt {x^{2} + 1} + 1\right )}^{2} + {\left (x - \sqrt {x^{2} + 1}\right )}^{2}\right ) - \frac {1}{6} \, \sqrt {3} \log \left ({\left (x - \sqrt {3} - \sqrt {x^{2} + 1} + 1\right )}^{2} + {\left (x - \sqrt {x^{2} + 1}\right )}^{2}\right ) - \frac {1}{3} \, \sqrt {2} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} + 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} + 2 \right |}}\right ) - \frac {1}{3} \, \arctan \left (-{\left (x - \sqrt {x^{2} + 1}\right )} {\left (\sqrt {3} + 1\right )} + 1\right ) - \frac {1}{3} \, \arctan \left ({\left (x - \sqrt {x^{2} + 1}\right )} {\left (\sqrt {3} - 1\right )} + 1\right ) + \log \left (-x + \sqrt {x^{2} + 1}\right ) \] Input:
integrate(x^4*(x^2+1)^(1/2)/(-x^3+1),x, algorithm="giac")
Output:
-1/6*pi - 1/3*(x^2 + 1)^(3/2) + 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*sqr t(2) + 2*sqrt(x^2 + 1) + 2)/abs(-2*x + 2*sqrt(2) + 2*sqrt(x^2 + 1) + 2)) - 1/3*arctan(-(x - sqrt(x^2 + 1))*(sqrt(3) + 1) + 1) - 1/3*arctan((x - sqrt (x^2 + 1))*(sqrt(3) - 1) + 1) + log(-x + sqrt(x^2 + 1))
Time = 22.46 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.17 \[ \int \frac {x^4 \sqrt {1+x^2}}{1-x^3} \, dx=-\mathrm {asinh}\left (x\right )-\frac {\sqrt {2}\,\left (\ln \left (x-1\right )-\ln \left (x+\sqrt {2}\,\sqrt {x^2+1}+1\right )\right )}{3}-\sqrt {x^2+1}\,\left (\frac {x^2}{3}+\frac {1}{3}\right )+\frac {\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (1+\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}-\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )}{3\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,\sqrt {{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}}-\frac {\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (1+\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )}{3\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,\sqrt {{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}} \] Input:
int(-(x^4*(x^2 + 1)^(1/2))/(x^3 - 1),x)
Output:
(((3^(1/2)*1i)/2 - 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*((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 - (x^2 + 1)^(1/2)*(x^2/3 + 1/3) - as inh(x) - (((3^(1/2)*1i)/2 + 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*((3^(1/2 )*1i)/2 - 1/2)^2*(((3^(1/2)*1i)/2 - 1/2)^2 + 1)^(1/2))
\[ \int \frac {x^4 \sqrt {1+x^2}}{1-x^3} \, dx=-\left (\int \frac {\sqrt {x^{2}+1}\, x^{4}}{x^{3}-1}d x \right ) \] Input:
int(x^4*(x^2+1)^(1/2)/(-x^3+1),x)
Output:
- int((sqrt(x**2 + 1)*x**4)/(x**3 - 1),x)