Integrand size = 22, antiderivative size = 101 \[ \int \frac {\sqrt [4]{-x^3+x^4} \left (-1+x^8\right )}{x^4} \, dx=\frac {\sqrt [4]{-x^3+x^4} \left (327680-65536 x-262144 x^2-21945 x^3-12540 x^4-9120 x^5-7296 x^6-6144 x^7+122880 x^8\right )}{737280 x^3}+\frac {1463 \arctan \left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{32768}-\frac {1463 \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{32768} \]
1/737280*(x^4-x^3)^(1/4)*(122880*x^8-6144*x^7-7296*x^6-9120*x^5-12540*x^4- 21945*x^3-262144*x^2-65536*x+327680)/x^3+1463/32768*arctan(x/(x^4-x^3)^(1/ 4))-1463/32768*arctanh(x/(x^4-x^3)^(1/4))
Time = 0.42 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt [4]{-x^3+x^4} \left (-1+x^8\right )}{x^4} \, dx=\frac {(-1+x)^{3/4} \left (2 \sqrt [4]{-1+x} \left (327680-65536 x-262144 x^2-21945 x^3-12540 x^4-9120 x^5-7296 x^6-6144 x^7+122880 x^8\right )+65835 x^{9/4} \arctan \left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )-65835 x^{9/4} \text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )}{1474560 \left ((-1+x) x^3\right )^{3/4}} \]
((-1 + x)^(3/4)*(2*(-1 + x)^(1/4)*(327680 - 65536*x - 262144*x^2 - 21945*x ^3 - 12540*x^4 - 9120*x^5 - 7296*x^6 - 6144*x^7 + 122880*x^8) + 65835*x^(9 /4)*ArcTan[((-1 + x)/x)^(-1/4)] - 65835*x^(9/4)*ArcTanh[((-1 + x)/x)^(-1/4 )]))/(1474560*((-1 + x)*x^3)^(3/4))
Leaf count is larger than twice the leaf count of optimal. \(242\) vs. \(2(101)=202\).
Time = 0.43 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.40, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2449, 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.
| \(\displaystyle \int \frac {\sqrt [4]{x^4-x^3} \left (x^8-1\right )}{x^4} \, dx\) |
| \(\Big \downarrow \) 2449 |
| \(\displaystyle \int \left (x^4 \sqrt [4]{x^4-x^3}-\frac {\sqrt [4]{x^4-x^3}}{x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1463 (x-1)^{3/4} x^{9/4} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{32768 \left (x^4-x^3\right )^{3/4}}-\frac {1463 (x-1)^{3/4} x^{9/4} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{32768 \left (x^4-x^3\right )^{3/4}}-\frac {1}{120} \sqrt [4]{x^4-x^3} x^4-\frac {19 \sqrt [4]{x^4-x^3} x^3}{1920}-\frac {209 \sqrt [4]{x^4-x^3} x}{12288}-\frac {1463 \sqrt [4]{x^4-x^3}}{49152}-\frac {4 \left (x^4-x^3\right )^{5/4}}{9 x^6}+\frac {1}{6} \sqrt [4]{x^4-x^3} x^5-\frac {16 \left (x^4-x^3\right )^{5/4}}{45 x^5}-\frac {19 \sqrt [4]{x^4-x^3} x^2}{1536}\) |
(-1463*(-x^3 + x^4)^(1/4))/49152 - (209*x*(-x^3 + x^4)^(1/4))/12288 - (19* x^2*(-x^3 + x^4)^(1/4))/1536 - (19*x^3*(-x^3 + x^4)^(1/4))/1920 - (x^4*(-x ^3 + x^4)^(1/4))/120 + (x^5*(-x^3 + x^4)^(1/4))/6 - (4*(-x^3 + x^4)^(5/4)) /(9*x^6) - (16*(-x^3 + x^4)^(5/4))/(45*x^5) - (1463*(-1 + x)^(3/4)*x^(9/4) *ArcTan[(-1 + x)^(1/4)/x^(1/4)])/(32768*(-x^3 + x^4)^(3/4)) - (1463*(-1 + x)^(3/4)*x^(9/4)*ArcTanh[(-1 + x)^(1/4)/x^(1/4)])/(32768*(-x^3 + x^4)^(3/4 ))
3.15.27.3.1 Defintions of rubi rules used
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_S ymbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ [{a, b, c, j, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && !Integer Q[p] && NeQ[n, j]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.70 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.63
| method | result | size |
| meijerg | \(\frac {4 \operatorname {signum}\left (-1+x \right )^{\frac {1}{4}} x^{\frac {23}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {23}{4}\right ], \left [\frac {27}{4}\right ], x\right )}{23 \left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{4}}}+\frac {4 \operatorname {signum}\left (-1+x \right )^{\frac {1}{4}} \left (-\frac {4}{5} x^{2}-\frac {1}{5} x +1\right ) \left (1-x \right )^{\frac {1}{4}}}{9 \left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{4}} x^{\frac {9}{4}}}\) | \(64\) |
| pseudoelliptic | \(\frac {x^{15} \left (\frac {4389 \ln \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}-x}{x}\right ) x^{3}}{32768}-\frac {4389 \ln \left (\frac {x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right ) x^{3}}{32768}-\frac {4389 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right ) x^{3}}{16384}+\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (x^{8}-\frac {1}{20} x^{7}-\frac {19}{320} x^{6}-\frac {19}{256} x^{5}-\frac {209}{2048} x^{4}-\frac {1463}{8192} x^{3}-\frac {32}{15} x^{2}-\frac {8}{15} x +\frac {8}{3}\right )\right )}{6 {\left (x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}\right )}^{6} {\left (-\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}+x \right )}^{6} \left (x^{2}+\sqrt {x^{3} \left (-1+x \right )}\right )^{6}}\) | \(161\) |
| trager | \(\frac {\left (x^{4}-x^{3}\right )^{\frac {1}{4}} \left (122880 x^{8}-6144 x^{7}-7296 x^{6}-9120 x^{5}-12540 x^{4}-21945 x^{3}-262144 x^{2}-65536 x +327680\right )}{737280 x^{3}}+\frac {1463 \ln \left (\frac {2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-2 \sqrt {x^{4}-x^{3}}\, x +2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}-2 x^{3}+x^{2}}{x^{2}}\right )}{65536}-\frac {1463 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}}{x^{2}}\right )}{65536}\) | \(201\) |
| risch | \(\frac {\left (122880 x^{9}-129024 x^{8}-1152 x^{7}-1824 x^{6}-3420 x^{5}-9405 x^{4}-240199 x^{3}+196608 x^{2}+393216 x -327680\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{737280 x^{3} \left (-1+x \right )}+\frac {\left (\frac {1463 \ln \left (\frac {2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, x +2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}-2 x^{3}+2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x +5 x^{2}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}-4 x +1}{\left (-1+x \right )^{2}}\right )}{65536}+\frac {1463 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (-1+x \right )^{2}}\right )}{65536}\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (x \left (-1+x \right )^{3}\right )^{\frac {1}{4}}}{x \left (-1+x \right )}\) | \(445\) |
4/23*signum(-1+x)^(1/4)/(-signum(-1+x))^(1/4)*x^(23/4)*hypergeom([-1/4,23/ 4],[27/4],x)+4/9*signum(-1+x)^(1/4)/(-signum(-1+x))^(1/4)/x^(9/4)*(-4/5*x^ 2-1/5*x+1)*(1-x)^(1/4)
Time = 0.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt [4]{-x^3+x^4} \left (-1+x^8\right )}{x^4} \, dx=-\frac {131670 \, x^{3} \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 65835 \, x^{3} \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 65835 \, x^{3} \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 4 \, {\left (122880 \, x^{8} - 6144 \, x^{7} - 7296 \, x^{6} - 9120 \, x^{5} - 12540 \, x^{4} - 21945 \, x^{3} - 262144 \, x^{2} - 65536 \, x + 327680\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{2949120 \, x^{3}} \]
-1/2949120*(131670*x^3*arctan((x^4 - x^3)^(1/4)/x) + 65835*x^3*log((x + (x ^4 - x^3)^(1/4))/x) - 65835*x^3*log(-(x - (x^4 - x^3)^(1/4))/x) - 4*(12288 0*x^8 - 6144*x^7 - 7296*x^6 - 9120*x^5 - 12540*x^4 - 21945*x^3 - 262144*x^ 2 - 65536*x + 327680)*(x^4 - x^3)^(1/4))/x^3
\[ \int \frac {\sqrt [4]{-x^3+x^4} \left (-1+x^8\right )}{x^4} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}{x^{4}}\, dx \]
\[ \int \frac {\sqrt [4]{-x^3+x^4} \left (-1+x^8\right )}{x^4} \, dx=\int { \frac {{\left (x^{8} - 1\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x^{4}} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.69 \[ \int \frac {\sqrt [4]{-x^3+x^4} \left (-1+x^8\right )}{x^4} \, dx=-\frac {1}{245760} \, {\left (7315 \, {\left (\frac {1}{x} - 1\right )}^{5} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 40755 \, {\left (\frac {1}{x} - 1\right )}^{4} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 92910 \, {\left (\frac {1}{x} - 1\right )}^{3} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 109782 \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 69327 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} - 21945 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{6} + \frac {4}{9} \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - \frac {4}{5} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} - \frac {1463}{32768} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1463}{65536} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1463}{65536} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
-1/245760*(7315*(1/x - 1)^5*(-1/x + 1)^(1/4) + 40755*(1/x - 1)^4*(-1/x + 1 )^(1/4) + 92910*(1/x - 1)^3*(-1/x + 1)^(1/4) + 109782*(1/x - 1)^2*(-1/x + 1)^(1/4) - 69327*(-1/x + 1)^(5/4) - 21945*(-1/x + 1)^(1/4))*x^6 + 4/9*(1/x - 1)^2*(-1/x + 1)^(1/4) - 4/5*(-1/x + 1)^(5/4) - 1463/32768*arctan((-1/x + 1)^(1/4)) - 1463/65536*log((-1/x + 1)^(1/4) + 1) + 1463/65536*log(abs((- 1/x + 1)^(1/4) - 1))
Timed out. \[ \int \frac {\sqrt [4]{-x^3+x^4} \left (-1+x^8\right )}{x^4} \, dx=\int \frac {\left (x^8-1\right )\,{\left (x^4-x^3\right )}^{1/4}}{x^4} \,d x \]