Integrand size = 25, antiderivative size = 134 \[ \int \frac {\left (-1+x+x^4\right ) \sqrt [4]{-x^3+x^4}}{1+x} \, dx=\frac {\sqrt [4]{-x^3+x^4} \left (-32575-1060 x+10400 x^2-8064 x^3+6144 x^4\right )}{30720}-\frac {9869 \arctan \left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{4096}+2 \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {9869 \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{4096}-2 \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right ) \]
1/30720*(x^4-x^3)^(1/4)*(6144*x^4-8064*x^3+10400*x^2-1060*x-32575)-9869/40 96*arctan(x/(x^4-x^3)^(1/4))+2*2^(1/4)*arctan(2^(1/4)*x/(x^4-x^3)^(1/4))+9 869/4096*arctanh(x/(x^4-x^3)^(1/4))-2*2^(1/4)*arctanh(2^(1/4)*x/(x^4-x^3)^ (1/4))
Time = 0.72 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.28 \[ \int \frac {\left (-1+x+x^4\right ) \sqrt [4]{-x^3+x^4}}{1+x} \, dx=\frac {65150 x^3-63030 x^4-22920 x^5+36928 x^6-28416 x^7+12288 x^8-148035 (-1+x)^{3/4} x^{9/4} \arctan \left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )+122880 \sqrt [4]{2} (-1+x)^{3/4} x^{9/4} \arctan \left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )+148035 (-1+x)^{3/4} x^{9/4} \text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )-122880 \sqrt [4]{2} (-1+x)^{3/4} x^{9/4} \text {arctanh}\left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )}{61440 \left ((-1+x) x^3\right )^{3/4}} \]
(65150*x^3 - 63030*x^4 - 22920*x^5 + 36928*x^6 - 28416*x^7 + 12288*x^8 - 1 48035*(-1 + x)^(3/4)*x^(9/4)*ArcTan[((-1 + x)/x)^(-1/4)] + 122880*2^(1/4)* (-1 + x)^(3/4)*x^(9/4)*ArcTan[2^(1/4)/((-1 + x)/x)^(1/4)] + 148035*(-1 + x )^(3/4)*x^(9/4)*ArcTanh[((-1 + x)/x)^(-1/4)] - 122880*2^(1/4)*(-1 + x)^(3/ 4)*x^(9/4)*ArcTanh[2^(1/4)/((-1 + x)/x)^(1/4)])/(61440*((-1 + x)*x^3)^(3/4 ))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.74 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.37, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2467, 25, 2035, 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.
| \(\displaystyle \int \frac {\left (x^4+x-1\right ) \sqrt [4]{x^4-x^3}}{x+1} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \int -\frac {\sqrt [4]{x-1} x^{3/4} \left (-x^4-x+1\right )}{x+1}dx}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} x^{3/4} \left (-x^4-x+1\right )}{x+1}dx}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} x^{3/2} \left (-x^4-x+1\right )}{x+1}d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^4-x^3} \int \left (-\sqrt [4]{x-1} x^{9/2}+\sqrt [4]{x-1} x^{7/2}-\sqrt [4]{x-1} x^{5/2}+\sqrt [4]{x-1} \sqrt {x}-\frac {\sqrt [4]{x-1} \sqrt {x}}{x+1}\right )d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^4-x^3} \left (-\frac {\sqrt [4]{x-1} x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x,-x\right )}{3 \sqrt [4]{1-x}}+\frac {1677 \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{16384}-\frac {1677 \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{16384}-\frac {1}{20} \sqrt [4]{x-1} x^{19/4}+\frac {21}{320} \sqrt [4]{x-1} x^{15/4}-\frac {65}{768} \sqrt [4]{x-1} x^{11/4}+\frac {53 \sqrt [4]{x-1} x^{7/4}}{6144}+\frac {6515 \sqrt [4]{x-1} x^{3/4}}{24576}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
(-4*(-x^3 + x^4)^(1/4)*((6515*(-1 + x)^(1/4)*x^(3/4))/24576 + (53*(-1 + x) ^(1/4)*x^(7/4))/6144 - (65*(-1 + x)^(1/4)*x^(11/4))/768 + (21*(-1 + x)^(1/ 4)*x^(15/4))/320 - ((-1 + x)^(1/4)*x^(19/4))/20 - ((-1 + x)^(1/4)*x^(3/4)* AppellF1[3/4, -1/4, 1, 7/4, x, -x])/(3*(1 - x)^(1/4)) + (1677*ArcTan[x^(1/ 4)/(-1 + x)^(1/4)])/16384 - (1677*ArcTanh[x^(1/4)/(-1 + x)^(1/4)])/16384)) /((-1 + x)^(1/4)*x^(3/4))
3.20.28.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
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]
Time = 7.34 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.49
| method | result | size |
| pseudoelliptic | \(\frac {\left (-5 \ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}-10 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}}-\frac {49345 \ln \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )}{8192}+\frac {49345 \ln \left (\frac {x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{8192}+\frac {49345 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{4096}+\left (x^{4}-\frac {21}{16} x^{3}+\frac {325}{192} x^{2}-\frac {265}{1536} x -\frac {32575}{6144}\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}\right ) x^{15}}{5 {\left (x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}\right )}^{5} {\left (-\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}+x \right )}^{5} \left (x^{2}+\sqrt {x^{3} \left (-1+x \right )}\right )^{5}}\) | \(199\) |
| trager | \(\left (\frac {1}{5} x^{4}-\frac {21}{80} x^{3}+\frac {65}{192} x^{2}-\frac {53}{1536} x -\frac {6515}{6144}\right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+\frac {9869 \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 )}{8192}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}} x^{2}+4 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x +4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (1+x \right )}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}} x^{2}-4 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x +4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (1+x \right )}\right )-\frac {9869 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {2 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{2}-4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+4 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x^{2}}\right )}{16384}\) | \(483\) |
| risch | \(\text {Expression too large to display}\) | \(1040\) |
1/5*(-5*ln((-2^(1/4)*x-(x^3*(-1+x))^(1/4))/(2^(1/4)*x-(x^3*(-1+x))^(1/4))) *2^(1/4)-10*arctan(1/2*2^(3/4)/x*(x^3*(-1+x))^(1/4))*2^(1/4)-49345/8192*ln (((x^3*(-1+x))^(1/4)-x)/x)+49345/8192*ln((x+(x^3*(-1+x))^(1/4))/x)+49345/4 096*arctan((x^3*(-1+x))^(1/4)/x)+(x^4-21/16*x^3+325/192*x^2-265/1536*x-325 75/6144)*(x^3*(-1+x))^(1/4))*x^15/(x+(x^3*(-1+x))^(1/4))^5/(-(x^3*(-1+x))^ (1/4)+x)^5/(x^2+(x^3*(-1+x))^(1/2))^5
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.54 \[ \int \frac {\left (-1+x+x^4\right ) \sqrt [4]{-x^3+x^4}}{1+x} \, dx=\frac {1}{30720} \, {\left (6144 \, x^{4} - 8064 \, x^{3} + 10400 \, x^{2} - 1060 \, x - 32575\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} - 2^{\frac {1}{4}} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 2^{\frac {1}{4}} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \cdot 2^{\frac {1}{4}} \log \left (\frac {i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + i \cdot 2^{\frac {1}{4}} \log \left (\frac {-i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {9869}{4096} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {9869}{8192} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {9869}{8192} \, \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
1/30720*(6144*x^4 - 8064*x^3 + 10400*x^2 - 1060*x - 32575)*(x^4 - x^3)^(1/ 4) - 2^(1/4)*log((2^(1/4)*x + (x^4 - x^3)^(1/4))/x) + 2^(1/4)*log(-(2^(1/4 )*x - (x^4 - x^3)^(1/4))/x) - I*2^(1/4)*log((I*2^(1/4)*x + (x^4 - x^3)^(1/ 4))/x) + I*2^(1/4)*log((-I*2^(1/4)*x + (x^4 - x^3)^(1/4))/x) + 9869/4096*a rctan((x^4 - x^3)^(1/4)/x) + 9869/8192*log((x + (x^4 - x^3)^(1/4))/x) - 98 69/8192*log(-(x - (x^4 - x^3)^(1/4))/x)
\[ \int \frac {\left (-1+x+x^4\right ) \sqrt [4]{-x^3+x^4}}{1+x} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )} \left (x^{4} + x - 1\right )}{x + 1}\, dx \]
\[ \int \frac {\left (-1+x+x^4\right ) \sqrt [4]{-x^3+x^4}}{1+x} \, dx=\int { \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} + x - 1\right )}}{x + 1} \,d x } \]
Time = 0.35 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.37 \[ \int \frac {\left (-1+x+x^4\right ) \sqrt [4]{-x^3+x^4}}{1+x} \, dx=-\frac {1}{30720} \, {\left (32575 \, {\left (\frac {1}{x} - 1\right )}^{4} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 131360 \, {\left (\frac {1}{x} - 1\right )}^{3} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 188230 \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 120744 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 25155 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{5} - 2 \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {9869}{4096} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {9869}{8192} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {9869}{8192} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
-1/30720*(32575*(1/x - 1)^4*(-1/x + 1)^(1/4) + 131360*(1/x - 1)^3*(-1/x + 1)^(1/4) + 188230*(1/x - 1)^2*(-1/x + 1)^(1/4) - 120744*(-1/x + 1)^(5/4) + 25155*(-1/x + 1)^(1/4))*x^5 - 2*2^(1/4)*arctan(1/2*2^(3/4)*(-1/x + 1)^(1/ 4)) - 2^(1/4)*log(2^(1/4) + (-1/x + 1)^(1/4)) + 2^(1/4)*log(abs(-2^(1/4) + (-1/x + 1)^(1/4))) + 9869/4096*arctan((-1/x + 1)^(1/4)) + 9869/8192*log(( -1/x + 1)^(1/4) + 1) - 9869/8192*log(abs((-1/x + 1)^(1/4) - 1))
Timed out. \[ \int \frac {\left (-1+x+x^4\right ) \sqrt [4]{-x^3+x^4}}{1+x} \, dx=\int \frac {{\left (x^4-x^3\right )}^{1/4}\,\left (x^4+x-1\right )}{x+1} \,d x \]