Integrand size = 22, antiderivative size = 102 \[ \int \frac {\sqrt {1+x^2}}{x^4 \left (1-x^3\right )} \, dx=-\frac {\left (1+x^2\right )^{3/2}}{3 x^3}-\frac {1}{3} \arctan \left (\frac {1+x}{\sqrt {1+x^2}}\right )+\frac {\text {arctanh}\left (\frac {1-x}{\sqrt {3} \sqrt {1+x^2}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {1+x}{\sqrt {2} \sqrt {1+x^2}}\right )-\text {arctanh}\left (\sqrt {1+x^2}\right ) \] Output:
-1/3*(x^2+1)^(3/2)/x^3-1/3*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))-arctanh((x^2+1)^(1/2))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.29 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.74 \[ \int \frac {\sqrt {1+x^2}}{x^4 \left (1-x^3\right )} \, dx=\frac {1}{3} \left (-\frac {\left (1+x^2\right )^{3/2}}{x^3}+6 \text {arctanh}\left (x-\sqrt {1+x^2}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {1-x+\sqrt {1+x^2}}{\sqrt {2}}\right )-\text {RootSum}\left [1+2 \text {$\#$1}+2 \text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (-x+\sqrt {1+x^2}-\text {$\#$1}\right )-4 \log \left (-x+\sqrt {1+x^2}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-x+\sqrt {1+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{1+2 \text {$\#$1}-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ]\right ) \] Input:
Integrate[Sqrt[1 + x^2]/(x^4*(1 - x^3)),x]
Output:
(-((1 + x^2)^(3/2)/x^3) + 6*ArcTanh[x - Sqrt[1 + x^2]] + 2*Sqrt[2]*ArcTanh [(1 - x + Sqrt[1 + x^2])/Sqrt[2]] - 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) & ])/3
Time = 0.78 (sec) , antiderivative size = 102, 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.
| \(\displaystyle \int \frac {\sqrt {x^2+1}}{x^4 \left (1-x^3\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {x^2+1} (-2 x-1)}{3 \left (x^2+x+1\right )}-\frac {\sqrt {x^2+1}}{3 (x-1)}+\frac {\sqrt {x^2+1}}{x}+\frac {\sqrt {x^2+1}}{x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{3} \arctan \left (\frac {x+1}{\sqrt {x^2+1}}\right )+\frac {\text {arctanh}\left (\frac {1-x}{\sqrt {3} \sqrt {x^2+1}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {x+1}{\sqrt {2} \sqrt {x^2+1}}\right )-\text {arctanh}\left (\sqrt {x^2+1}\right )-\frac {\left (x^2+1\right )^{3/2}}{3 x^3}\) |
Input:
Int[Sqrt[1 + x^2]/(x^4*(1 - x^3)),x]
Output:
-1/3*(1 + x^2)^(3/2)/x^3 - 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]*Sq rt[1 + x^2])])/3 - ArcTanh[Sqrt[1 + x^2]]
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. \(337\) vs. \(2(81)=162\).
Time = 0.70 (sec) , antiderivative size = 338, normalized size of antiderivative = 3.31
| method | result | size |
| default | \(-\frac {\sqrt {\left (x -1\right )^{2}+2 x}}{3}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x +2\right ) \sqrt {2}}{4 \sqrt {\left (x -1\right )^{2}+2 x}}\right )}{3}-\frac {\left (x^{2}+1\right )^{\frac {3}{2}}}{3 x^{3}}+\frac {\sqrt {x^{2}+1}}{3}-\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right )-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )}{3 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}+\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )+\arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )\right )}{6 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}\) | \(338\) |
| risch | \(-\frac {x^{4}+2 x^{2}+1}{3 x^{3} \sqrt {x^{2}+1}}+\frac {\sqrt {x^{2}+1}}{3}-\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right )-\frac {\sqrt {\left (x -1\right )^{2}+2 x}}{3}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x +2\right ) \sqrt {2}}{4 \sqrt {\left (x -1\right )^{2}+2 x}}\right )}{3}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )}{3 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}+\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )+\arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )\right )}{6 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}\) | \(348\) |
| trager | \(\text {Expression too large to display}\) | \(655\) |
Input:
int((x^2+1)^(1/2)/x^4/(-x^3+1),x,method=_RETURNVERBOSE)
Output:
-1/3*((x-1)^2+2*x)^(1/2)+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)^(3/2)/x^3+1/3*(x^2+1)^(1/2)-arctanh(1/(x^2+1)^(1/2 ))-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))+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. 237 vs. \(2 (79) = 158\).
Time = 0.08 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.32 \[ \int \frac {\sqrt {1+x^2}}{x^4 \left (1-x^3\right )} \, dx=-\frac {\sqrt {3} x^{3} \log \left (2 \, x^{2} - \sqrt {x^{2} + 1} {\left (2 \, x + \sqrt {3} + 1\right )} + \sqrt {3} {\left (x + 1\right )} + x + 3\right ) - \sqrt {3} x^{3} \log \left (2 \, x^{2} - \sqrt {x^{2} + 1} {\left (2 \, x - \sqrt {3} + 1\right )} - \sqrt {3} {\left (x + 1\right )} + x + 3\right ) - 2 \, \sqrt {2} x^{3} \log \left (-\frac {\sqrt {2} {\left (x + 1\right )} + \sqrt {x^{2} + 1} {\left (\sqrt {2} + 2\right )} + x + 1}{x - 1}\right ) + 2 \, x^{3} \arctan \left (-\sqrt {3} x + \sqrt {x^{2} + 1} {\left (\sqrt {3} + 1\right )} - x + 1\right ) - 2 \, x^{3} \arctan \left (-\sqrt {3} x + \sqrt {x^{2} + 1} {\left (\sqrt {3} - 1\right )} + x - 1\right ) + 6 \, x^{3} \log \left (-x + \sqrt {x^{2} + 1} + 1\right ) - 6 \, x^{3} \log \left (-x + \sqrt {x^{2} + 1} - 1\right ) + 2 \, x^{3} + 2 \, {\left (x^{2} + 1\right )}^{\frac {3}{2}}}{6 \, x^{3}} \] Input:
integrate((x^2+1)^(1/2)/x^4/(-x^3+1),x, algorithm="fricas")
Output:
-1/6*(sqrt(3)*x^3*log(2*x^2 - sqrt(x^2 + 1)*(2*x + sqrt(3) + 1) + sqrt(3)* (x + 1) + x + 3) - sqrt(3)*x^3*log(2*x^2 - sqrt(x^2 + 1)*(2*x - sqrt(3) + 1) - sqrt(3)*(x + 1) + x + 3) - 2*sqrt(2)*x^3*log(-(sqrt(2)*(x + 1) + sqrt (x^2 + 1)*(sqrt(2) + 2) + x + 1)/(x - 1)) + 2*x^3*arctan(-sqrt(3)*x + sqrt (x^2 + 1)*(sqrt(3) + 1) - x + 1) - 2*x^3*arctan(-sqrt(3)*x + sqrt(x^2 + 1) *(sqrt(3) - 1) + x - 1) + 6*x^3*log(-x + sqrt(x^2 + 1) + 1) - 6*x^3*log(-x + sqrt(x^2 + 1) - 1) + 2*x^3 + 2*(x^2 + 1)^(3/2))/x^3
\[ \int \frac {\sqrt {1+x^2}}{x^4 \left (1-x^3\right )} \, dx=- \int \frac {\sqrt {x^{2} + 1}}{x^{7} - x^{4}}\, dx \] Input:
integrate((x**2+1)**(1/2)/x**4/(-x**3+1),x)
Output:
-Integral(sqrt(x**2 + 1)/(x**7 - x**4), x)
\[ \int \frac {\sqrt {1+x^2}}{x^4 \left (1-x^3\right )} \, dx=\int { -\frac {\sqrt {x^{2} + 1}}{{\left (x^{3} - 1\right )} x^{4}} \,d x } \] Input:
integrate((x^2+1)^(1/2)/x^4/(-x^3+1),x, algorithm="maxima")
Output:
-integrate(sqrt(x^2 + 1)/((x^3 - 1)*x^4), x)
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (79) = 158\).
Time = 0.14 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.35 \[ \int \frac {\sqrt {1+x^2}}{x^4 \left (1-x^3\right )} \, dx=-\frac {1}{6} \, \pi - \frac {1}{6} \, \sqrt {3} \log \left ({\left (x + \sqrt {3} - \sqrt {x^{2} + 1} + 1\right )}^{2} + {\left (x - \sqrt {x^{2} + 1}\right )}^{2}\right ) + \frac {1}{6} \, \sqrt {3} \log \left ({\left (x - \sqrt {3} - \sqrt {x^{2} + 1} + 1\right )}^{2} + {\left (x - \sqrt {x^{2} + 1}\right )}^{2}\right ) - \frac {1}{3} \, \sqrt {2} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} + 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} + 2 \right |}}\right ) + \frac {2 \, {\left (3 \, {\left (x - \sqrt {x^{2} + 1}\right )}^{4} + 1\right )}}{3 \, {\left ({\left (x - \sqrt {x^{2} + 1}\right )}^{2} - 1\right )}^{3}} - \frac {1}{3} \, \arctan \left (-{\left (x - \sqrt {x^{2} + 1}\right )} {\left (\sqrt {3} + 1\right )} + 1\right ) - \frac {1}{3} \, \arctan \left ({\left (x - \sqrt {x^{2} + 1}\right )} {\left (\sqrt {3} - 1\right )} + 1\right ) - \log \left ({\left | -x + \sqrt {x^{2} + 1} + 1 \right |}\right ) + \log \left ({\left | -x + \sqrt {x^{2} + 1} - 1 \right |}\right ) \] Input:
integrate((x^2+1)^(1/2)/x^4/(-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)) + 2/3*(3*(x - sqrt(x^2 + 1))^4 + 1)/((x - sqrt(x^2 + 1))^2 - 1)^3 - 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(abs(-x + sqrt(x^2 + 1) + 1)) + log(abs(-x + sqrt(x^2 + 1) - 1))
Time = 22.51 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.03 \[ \int \frac {\sqrt {1+x^2}}{x^4 \left (1-x^3\right )} \, dx=\mathrm {atan}\left (\sqrt {x^2+1}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}-\frac {\sqrt {2}\,\left (2\,\ln \left (x-1\right )-2\,\ln \left (x+\sqrt {2}\,\sqrt {x^2+1}+1\right )\right )}{6}+\sqrt {x^2+1}\,\left (\frac {2}{3\,x}-\frac {1}{3\,x^3}\right )-\frac {\sqrt {x^2+1}}{x}+\frac {\left (-2+\sqrt {3}\,2{}\mathrm {i}\right )\,\left (\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (1+\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )}{12\,\sqrt {{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}}-\frac {\left (2+\sqrt {3}\,2{}\mathrm {i}\right )\,\left (\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (1+\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}-\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )}{12\,\sqrt {{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}} \] Input:
int(-(x^2 + 1)^(1/2)/(x^4*(x^3 - 1)),x)
Output:
atan((x^2 + 1)^(1/2)*1i)*1i - (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)*(2/(3*x) - 1/(3*x^3)) - (x^2 + 1 )^(1/2)/x + ((3^(1/2)*2i - 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)))/(12*(((3^(1/2) *1i)/2 - 1/2)^2 + 1)^(1/2)) - ((3^(1/2)*2i + 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 )))/(12*(((3^(1/2)*1i)/2 + 1/2)^2 + 1)^(1/2))
\[ \int \frac {\sqrt {1+x^2}}{x^4 \left (1-x^3\right )} \, dx=-\left (\int \frac {\sqrt {x^{2}+1}}{x^{7}-x^{4}}d x \right ) \] Input:
int((x^2+1)^(1/2)/x^4/(-x^3+1),x)
Output:
- int(sqrt(x**2 + 1)/(x**7 - x**4),x)