Integrand size = 29, antiderivative size = 230 \[ \int \frac {\left (1+x^2\right ) \sqrt [4]{-x^3+x^4}}{-1+x+2 x^2} \, dx=\frac {1}{16} (-5+4 x) \sqrt [4]{-x^3+x^4}-\frac {57}{32} \arctan \left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {4}{3} \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {5 \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-x^3+x^4}}{-x^2+\sqrt {-x^3+x^4}}\right )}{12 \sqrt {2}}+\frac {57}{32} \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {4}{3} \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {5 \text {arctanh}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-x^3+x^4}}{\sqrt {2}}}{x \sqrt [4]{-x^3+x^4}}\right )}{12 \sqrt {2}} \]
1/16*(-5+4*x)*(x^4-x^3)^(1/4)-57/32*arctan(x/(x^4-x^3)^(1/4))+4/3*2^(1/4)* arctan(2^(1/4)*x/(x^4-x^3)^(1/4))+5/24*arctan(2^(1/2)*x*(x^4-x^3)^(1/4)/(- x^2+(x^4-x^3)^(1/2)))*2^(1/2)+57/32*arctanh(x/(x^4-x^3)^(1/4))-4/3*2^(1/4) *arctanh(2^(1/4)*x/(x^4-x^3)^(1/4))-5/24*arctanh((1/2*2^(1/2)*x^2+1/2*(x^4 -x^3)^(1/2)*2^(1/2))/x/(x^4-x^3)^(1/4))*2^(1/2)
Time = 0.56 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.95 \[ \int \frac {\left (1+x^2\right ) \sqrt [4]{-x^3+x^4}}{-1+x+2 x^2} \, dx=\frac {(-1+x)^{3/4} x^{9/4} \left (-30 \sqrt [4]{-1+x} x^{3/4}+24 \sqrt [4]{-1+x} x^{7/4}-171 \arctan \left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )+128 \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )+20 \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{-1+x} \sqrt [4]{x}}{\sqrt {-1+x}-\sqrt {x}}\right )+171 \text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )-128 \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )-20 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x} \sqrt [4]{x}}{\sqrt {-1+x}+\sqrt {x}}\right )\right )}{96 \left ((-1+x) x^3\right )^{3/4}} \]
((-1 + x)^(3/4)*x^(9/4)*(-30*(-1 + x)^(1/4)*x^(3/4) + 24*(-1 + x)^(1/4)*x^ (7/4) - 171*ArcTan[((-1 + x)/x)^(-1/4)] + 128*2^(1/4)*ArcTan[2^(1/4)/((-1 + x)/x)^(1/4)] + 20*Sqrt[2]*ArcTan[(Sqrt[2]*(-1 + x)^(1/4)*x^(1/4))/(Sqrt[ -1 + x] - Sqrt[x])] + 171*ArcTanh[((-1 + x)/x)^(-1/4)] - 128*2^(1/4)*ArcTa nh[2^(1/4)/((-1 + x)/x)^(1/4)] - 20*Sqrt[2]*ArcTanh[(Sqrt[2]*(-1 + x)^(1/4 )*x^(1/4))/(Sqrt[-1 + x] + Sqrt[x])]))/(96*((-1 + x)*x^3)^(3/4))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.94 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.77, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2467, 25, 2035, 7279, 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.
| \(\displaystyle \int \frac {\left (x^2+1\right ) \sqrt [4]{x^4-x^3}}{2 x^2+x-1} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \int -\frac {\sqrt [4]{x-1} x^{3/4} \left (x^2+1\right )}{-2 x^2-x+1}dx}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} x^{3/4} \left (x^2+1\right )}{-2 x^2-x+1}dx}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} x^{3/2} \left (x^2+1\right )}{-2 x^2-x+1}d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^4-x^3} \int \left (-\frac {1}{2} \sqrt [4]{x-1} x^{3/2}+\frac {1}{4} \sqrt [4]{x-1} \sqrt {x}+\frac {\sqrt [4]{x-1} (7 x-1) \sqrt {x}}{4 \left (-2 x^2-x+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^4-x^3} \left (-\frac {2 \sqrt [4]{x-1} x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x,-x\right )}{9 \sqrt [4]{1-x}}+\frac {5 \sqrt [4]{x-1} x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x,2 x\right )}{36 \sqrt [4]{1-x}}+\frac {1}{128} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}\right )-\frac {1}{128} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}\right )-\frac {1}{16} \sqrt [4]{x-1} x^{7/4}+\frac {5}{64} \sqrt [4]{x-1} x^{3/4}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
(-4*(-x^3 + x^4)^(1/4)*((5*(-1 + x)^(1/4)*x^(3/4))/64 - ((-1 + x)^(1/4)*x^ (7/4))/16 - (2*(-1 + x)^(1/4)*x^(3/4)*AppellF1[3/4, -1/4, 1, 7/4, x, -x])/ (9*(1 - x)^(1/4)) + (5*(-1 + x)^(1/4)*x^(3/4)*AppellF1[3/4, -1/4, 1, 7/4, x, 2*x])/(36*(1 - x)^(1/4)) + ArcTan[x^(1/4)/(-1 + x)^(1/4)]/128 - ArcTanh [x^(1/4)/(-1 + x)^(1/4)]/128))/((-1 + x)^(1/4)*x^(3/4))
3.27.23.3.1 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_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 3.79 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.35
| method | result | size |
| pseudoelliptic | \(\frac {x^{6} \left (48 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} x -20 \ln \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{3} \left (-1+x \right )}}{-\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{3} \left (-1+x \right )}}\right ) \sqrt {2}-40 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right ) \sqrt {2}-40 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right ) \sqrt {2}-128 \ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}-256 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}}-60 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}+171 \ln \left (\frac {x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )-171 \ln \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )+342 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )\right )}{192 {\left (x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}\right )}^{2} {\left (\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}-x \right )}^{2} \left (x^{2}+\sqrt {x^{3} \left (-1+x \right )}\right )^{2}}\) | \(310\) |
| trager | \(\text {Expression too large to display}\) | \(859\) |
1/192*x^6*(48*(x^3*(-1+x))^(1/4)*x-20*ln(((x^3*(-1+x))^(1/4)*2^(1/2)*x+x^2 +(x^3*(-1+x))^(1/2))/(-(x^3*(-1+x))^(1/4)*2^(1/2)*x+x^2+(x^3*(-1+x))^(1/2) ))*2^(1/2)-40*arctan(((x^3*(-1+x))^(1/4)*2^(1/2)+x)/x)*2^(1/2)-40*arctan(( (x^3*(-1+x))^(1/4)*2^(1/2)-x)/x)*2^(1/2)-128*ln((-2^(1/4)*x-(x^3*(-1+x))^( 1/4))/(2^(1/4)*x-(x^3*(-1+x))^(1/4)))*2^(1/4)-256*arctan(1/2*2^(3/4)/x*(x^ 3*(-1+x))^(1/4))*2^(1/4)-60*(x^3*(-1+x))^(1/4)+171*ln((x+(x^3*(-1+x))^(1/4 ))/x)-171*ln(((x^3*(-1+x))^(1/4)-x)/x)+342*arctan((x^3*(-1+x))^(1/4)/x))/( x+(x^3*(-1+x))^(1/4))^2/((x^3*(-1+x))^(1/4)-x)^2/(x^2+(x^3*(-1+x))^(1/2))^ 2
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.36 \[ \int \frac {\left (1+x^2\right ) \sqrt [4]{-x^3+x^4}}{-1+x+2 x^2} \, dx=-\left (\frac {5}{48} i + \frac {5}{48}\right ) \, \sqrt {2} \log \left (\frac {\left (i + 1\right ) \, \sqrt {2} x + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \left (\frac {5}{48} i - \frac {5}{48}\right ) \, \sqrt {2} \log \left (\frac {-\left (i - 1\right ) \, \sqrt {2} x + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \left (\frac {5}{48} i - \frac {5}{48}\right ) \, \sqrt {2} \log \left (\frac {\left (i - 1\right ) \, \sqrt {2} x + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \left (\frac {5}{48} i + \frac {5}{48}\right ) \, \sqrt {2} \log \left (\frac {-\left (i + 1\right ) \, \sqrt {2} x + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{16} \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (4 \, x - 5\right )} - \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {2}{3} i \cdot 2^{\frac {1}{4}} \log \left (\frac {i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {2}{3} i \cdot 2^{\frac {1}{4}} \log \left (\frac {-i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {57}{32} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {57}{64} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {57}{64} \, \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
-(5/48*I + 5/48)*sqrt(2)*log(((I + 1)*sqrt(2)*x + 2*(x^4 - x^3)^(1/4))/x) + (5/48*I - 5/48)*sqrt(2)*log((-(I - 1)*sqrt(2)*x + 2*(x^4 - x^3)^(1/4))/x ) - (5/48*I - 5/48)*sqrt(2)*log(((I - 1)*sqrt(2)*x + 2*(x^4 - x^3)^(1/4))/ x) + (5/48*I + 5/48)*sqrt(2)*log((-(I + 1)*sqrt(2)*x + 2*(x^4 - x^3)^(1/4) )/x) + 1/16*(x^4 - x^3)^(1/4)*(4*x - 5) - 2/3*2^(1/4)*log((2^(1/4)*x + (x^ 4 - x^3)^(1/4))/x) + 2/3*2^(1/4)*log(-(2^(1/4)*x - (x^4 - x^3)^(1/4))/x) - 2/3*I*2^(1/4)*log((I*2^(1/4)*x + (x^4 - x^3)^(1/4))/x) + 2/3*I*2^(1/4)*lo g((-I*2^(1/4)*x + (x^4 - x^3)^(1/4))/x) + 57/32*arctan((x^4 - x^3)^(1/4)/x ) + 57/64*log((x + (x^4 - x^3)^(1/4))/x) - 57/64*log(-(x - (x^4 - x^3)^(1/ 4))/x)
\[ \int \frac {\left (1+x^2\right ) \sqrt [4]{-x^3+x^4}}{-1+x+2 x^2} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )} \left (x^{2} + 1\right )}{\left (x + 1\right ) \left (2 x - 1\right )}\, dx \]
\[ \int \frac {\left (1+x^2\right ) \sqrt [4]{-x^3+x^4}}{-1+x+2 x^2} \, dx=\int { \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}}{2 \, x^{2} + x - 1} \,d x } \]
Time = 0.34 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.06 \[ \int \frac {\left (1+x^2\right ) \sqrt [4]{-x^3+x^4}}{-1+x+2 x^2} \, dx=\frac {1}{16} \, {\left (5 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} - {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{2} - \frac {1}{3} \cdot 8^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {5}{24} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {5}{24} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {5}{48} \, \sqrt {2} \log \left (\sqrt {2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) + \frac {5}{48} \, \sqrt {2} \log \left (-\sqrt {2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) - \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {57}{32} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {57}{64} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {57}{64} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
1/16*(5*(-1/x + 1)^(5/4) - (-1/x + 1)^(1/4))*x^2 - 1/3*8^(3/4)*arctan(1/2* 2^(3/4)*(-1/x + 1)^(1/4)) - 5/24*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*( -1/x + 1)^(1/4))) - 5/24*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(-1/x + 1)^(1/4))) - 5/48*sqrt(2)*log(sqrt(2)*(-1/x + 1)^(1/4) + sqrt(-1/x + 1) + 1) + 5/48*sqrt(2)*log(-sqrt(2)*(-1/x + 1)^(1/4) + sqrt(-1/x + 1) + 1) - 2/ 3*2^(1/4)*log(2^(1/4) + (-1/x + 1)^(1/4)) + 2/3*2^(1/4)*log(abs(-2^(1/4) + (-1/x + 1)^(1/4))) + 57/32*arctan((-1/x + 1)^(1/4)) + 57/64*log((-1/x + 1 )^(1/4) + 1) - 57/64*log(abs((-1/x + 1)^(1/4) - 1))
Timed out. \[ \int \frac {\left (1+x^2\right ) \sqrt [4]{-x^3+x^4}}{-1+x+2 x^2} \, dx=\int \frac {\left (x^2+1\right )\,{\left (x^4-x^3\right )}^{1/4}}{2\,x^2+x-1} \,d x \]