Integrand size = 22, antiderivative size = 121 \[ \int \frac {x^5 \sqrt {1+x^2}}{1-x^3} \, dx=-\sqrt {1+x^2}-\frac {1}{8} x \sqrt {1+x^2}-\frac {1}{4} x^3 \sqrt {1+x^2}+\frac {\text {arcsinh}(x)}{8}-\frac {1}{3} \arctan \left (\frac {1+x}{\sqrt {1+x^2}}\right )+\frac {\text {arctanh}\left (\frac {1-x}{\sqrt {3} \sqrt {1+x^2}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {1+x}{\sqrt {2} \sqrt {1+x^2}}\right ) \] Output:
-(x^2+1)^(1/2)-1/8*x*(x^2+1)^(1/2)-1/4*x^3*(x^2+1)^(1/2)+1/8*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 = 0.27 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.50 \[ \int \frac {x^5 \sqrt {1+x^2}}{1-x^3} \, dx=\frac {1}{24} \left (16 \sqrt {2} \text {arctanh}\left (\frac {1-x+\sqrt {1+x^2}}{\sqrt {2}}\right )-3 \left (\sqrt {1+x^2} \left (8+x+2 x^3\right )+\log \left (-x+\sqrt {1+x^2}\right )\right )-8 \text {RootSum}\left [1+2 \text {$\#$1}+2 \text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (-x+\sqrt {1+x^2}-\text {$\#$1}\right )-4 \log \left (-x+\sqrt {1+x^2}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-x+\sqrt {1+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{1+2 \text {$\#$1}-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ]\right ) \] Input:
Integrate[(x^5*Sqrt[1 + x^2])/(1 - x^3),x]
Output:
(16*Sqrt[2]*ArcTanh[(1 - x + Sqrt[1 + x^2])/Sqrt[2]] - 3*(Sqrt[1 + x^2]*(8 + x + 2*x^3) + Log[-x + Sqrt[1 + x^2]]) - 8*RootSum[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 ) & ])/24
Time = 1.05 (sec) , antiderivative size = 121, 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.
| \(\displaystyle \int \frac {x^5 \sqrt {x^2+1}}{1-x^3} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {x^2 \sqrt {x^2+1}}{1-x^3}-x^2 \sqrt {x^2+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\text {arcsinh}(x)}{8}-\frac {1}{3} \arctan \left (\frac {x+1}{\sqrt {x^2+1}}\right )+\frac {\text {arctanh}\left (\frac {1-x}{\sqrt {3} \sqrt {x^2+1}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {x+1}{\sqrt {2} \sqrt {x^2+1}}\right )-\frac {1}{8} \sqrt {x^2+1} x-\sqrt {x^2+1}-\frac {1}{4} \sqrt {x^2+1} x^3\) |
Input:
Int[(x^5*Sqrt[1 + x^2])/(1 - x^3),x]
Output:
-Sqrt[1 + x^2] - (x*Sqrt[1 + x^2])/8 - (x^3*Sqrt[1 + x^2])/4 + ArcSinh[x]/ 8 - 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. \(198\) vs. \(2(94)=188\).
Time = 0.62 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.64
| method | result | size |
| risch | \(-\frac {\left (2 x^{3}+x +8\right ) \sqrt {x^{2}+1}}{8}+\frac {\operatorname {arcsinh}\left (x \right )}{8}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x +2\right ) \sqrt {2}}{4 \sqrt {\left (x -1\right )^{2}+2 x}}\right )}{3}+\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )-\arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )\right )}{6 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}\) | \(199\) |
| default | \(-\frac {x \left (x^{2}+1\right )^{\frac {3}{2}}}{4}+\frac {\sqrt {x^{2}+1}\, x}{8}+\frac {\operatorname {arcsinh}\left (x \right )}{8}-\frac {\sqrt {\left (x -1\right )^{2}+2 x}}{3}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x +2\right ) \sqrt {2}}{4 \sqrt {\left (x -1\right )^{2}+2 x}}\right )}{3}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )}{3 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}-\frac {2 \sqrt {x^{2}+1}}{3}+\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )+\arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )\right )}{6 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}\) | \(340\) |
| trager | \(\left (-\frac {1}{4} x^{3}-\frac {1}{8} x -1\right ) \sqrt {x^{2}+1}+\frac {\ln \left (x +\sqrt {x^{2}+1}\right )}{8}-\frac {9 \ln \left (\frac {27 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-576 \textit {\_Z}^{2}+4096\right )^{3} x -192 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-576 \textit {\_Z}^{2}+4096\right ) x +512 \sqrt {x^{2}+1}+192 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-576 \textit {\_Z}^{2}+4096\right )}{9 x \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-576 \textit {\_Z}^{2}+4096\right )^{2}+64}\right ) \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-576 \textit {\_Z}^{2}+4096\right )^{3}}{512}+\frac {\ln \left (\frac {27 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-576 \textit {\_Z}^{2}+4096\right )^{3} x -192 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-576 \textit {\_Z}^{2}+4096\right ) x +512 \sqrt {x^{2}+1}+192 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-576 \textit {\_Z}^{2}+4096\right )}{9 x \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-576 \textit {\_Z}^{2}+4096\right )^{2}+64}\right ) \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-576 \textit {\_Z}^{2}+4096\right )}{8}+\frac {\operatorname {RootOf}\left (81 \textit {\_Z}^{4}-576 \textit {\_Z}^{2}+4096\right ) \ln \left (\frac {243 x \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-576 \textit {\_Z}^{2}+4096\right )^{5}-1728 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-576 \textit {\_Z}^{2}+4096\right )^{3} x -1728 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-576 \textit {\_Z}^{2}+4096\right )^{3}+32768 \sqrt {x^{2}+1}+12288 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-576 \textit {\_Z}^{2}+4096\right )}{9 x \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-576 \textit {\_Z}^{2}+4096\right )^{2}-64 x -64}\right )}{8}-\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}\) | \(393\) |
Input:
int(x^5*(x^2+1)^(1/2)/(-x^3+1),x,method=_RETURNVERBOSE)
Output:
-1/8*(2*x^3+x+8)*(x^2+1)^(1/2)+1/8*arcsinh(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. 199 vs. \(2 (92) = 184\).
Time = 0.08 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.64 \[ \int \frac {x^5 \sqrt {1+x^2}}{1-x^3} \, dx=-\frac {1}{8} \, {\left (2 \, x^{3} + x + 8\right )} \sqrt {x^{2} + 1} - \frac {1}{6} \, \sqrt {3} \log \left (2 \, x^{2} - \sqrt {x^{2} + 1} {\left (2 \, x + \sqrt {3} + 1\right )} + \sqrt {3} {\left (x + 1\right )} + x + 3\right ) + \frac {1}{6} \, \sqrt {3} \log \left (2 \, x^{2} - \sqrt {x^{2} + 1} {\left (2 \, x - \sqrt {3} + 1\right )} - \sqrt {3} {\left (x + 1\right )} + x + 3\right ) + \frac {1}{3} \, \sqrt {2} \log \left (-\frac {\sqrt {2} {\left (x + 1\right )} + \sqrt {x^{2} + 1} {\left (\sqrt {2} + 2\right )} + x + 1}{x - 1}\right ) - \frac {1}{3} \, \arctan \left (-\sqrt {3} x + \sqrt {x^{2} + 1} {\left (\sqrt {3} + 1\right )} - x + 1\right ) + \frac {1}{3} \, \arctan \left (-\sqrt {3} x + \sqrt {x^{2} + 1} {\left (\sqrt {3} - 1\right )} + x - 1\right ) - \frac {1}{8} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \] Input:
integrate(x^5*(x^2+1)^(1/2)/(-x^3+1),x, algorithm="fricas")
Output:
-1/8*(2*x^3 + x + 8)*sqrt(x^2 + 1) - 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(-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) - 1/8*log(-x + sqrt(x^2 + 1))
\[ \int \frac {x^5 \sqrt {1+x^2}}{1-x^3} \, dx=- \int \frac {x^{5} \sqrt {x^{2} + 1}}{x^{3} - 1}\, dx \] Input:
integrate(x**5*(x**2+1)**(1/2)/(-x**3+1),x)
Output:
-Integral(x**5*sqrt(x**2 + 1)/(x**3 - 1), x)
\[ \int \frac {x^5 \sqrt {1+x^2}}{1-x^3} \, dx=\int { -\frac {\sqrt {x^{2} + 1} x^{5}}{x^{3} - 1} \,d x } \] Input:
integrate(x^5*(x^2+1)^(1/2)/(-x^3+1),x, algorithm="maxima")
Output:
-integrate(sqrt(x^2 + 1)*x^5/(x^3 - 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (92) = 184\).
Time = 0.13 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.72 \[ \int \frac {x^5 \sqrt {1+x^2}}{1-x^3} \, dx=-\frac {1}{6} \, \pi - \frac {1}{8} \, {\left ({\left (2 \, x^{2} + 1\right )} x + 8\right )} \sqrt {x^{2} + 1} - \frac {1}{6} \, \sqrt {3} \log \left ({\left (x + \sqrt {3} - \sqrt {x^{2} + 1} + 1\right )}^{2} + {\left (x - \sqrt {x^{2} + 1}\right )}^{2}\right ) + \frac {1}{6} \, \sqrt {3} \log \left ({\left (x - \sqrt {3} - \sqrt {x^{2} + 1} + 1\right )}^{2} + {\left (x - \sqrt {x^{2} + 1}\right )}^{2}\right ) - \frac {1}{3} \, \sqrt {2} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} + 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} + 2 \right |}}\right ) - \frac {1}{3} \, \arctan \left (-{\left (x - \sqrt {x^{2} + 1}\right )} {\left (\sqrt {3} + 1\right )} + 1\right ) - \frac {1}{3} \, \arctan \left ({\left (x - \sqrt {x^{2} + 1}\right )} {\left (\sqrt {3} - 1\right )} + 1\right ) - \frac {1}{8} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \] Input:
integrate(x^5*(x^2+1)^(1/2)/(-x^3+1),x, algorithm="giac")
Output:
-1/6*pi - 1/8*((2*x^2 + 1)*x + 8)*sqrt(x^2 + 1) - 1/6*sqrt(3)*log((x + sqr t(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)*lo g(abs(-2*x - 2*sqrt(2) + 2*sqrt(x^2 + 1) + 2)/abs(-2*x + 2*sqrt(2) + 2*sqr t(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) - 1/8*log(-x + sqrt(x^2 + 1))
Time = 0.42 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.86 \[ \int \frac {x^5 \sqrt {1+x^2}}{1-x^3} \, dx=\frac {\mathrm {asinh}\left (x\right )}{8}-\frac {\sqrt {2}\,\left (\ln \left (x-1\right )-\ln \left (x+\sqrt {2}\,\sqrt {x^2+1}+1\right )\right )}{3}-\sqrt {x^2+1}\,\left (\frac {x^3}{4}+\frac {x}{8}+1\right )-\frac {\left (\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (1+\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )\,\left ({\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,\sqrt {{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}}+\frac {\left (\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (1+\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}-\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )\,\left (\frac {1}{2}-{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,\sqrt {{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}} \] Input:
int(-(x^5*(x^2 + 1)^(1/2))/(x^3 - 1),x)
Output:
asinh(x)/8 - (2^(1/2)*(log(x - 1) - log(x + 2^(1/2)*(x^2 + 1)^(1/2) + 1))) /3 - (x^2 + 1)^(1/2)*(x/8 + x^3/4 + 1) - ((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)) + ((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))
\[ \int \frac {x^5 \sqrt {1+x^2}}{1-x^3} \, dx=-\left (\int \frac {\sqrt {x^{2}+1}\, x^{5}}{x^{3}-1}d x \right ) \] Input:
int(x^5*(x^2+1)^(1/2)/(-x^3+1),x)
Output:
- int((sqrt(x**2 + 1)*x**5)/(x**3 - 1),x)