Integrand size = 26, antiderivative size = 452 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=-\frac {1}{4} \sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x}{-\sqrt {2+\sqrt {2}} x+2^{3/4} \sqrt [4]{-x^2+x^6}}\right )-\frac {1}{4} \sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x}{\sqrt {2+\sqrt {2}} x+2^{3/4} \sqrt [4]{-x^2+x^6}}\right )-\frac {1}{4} \sqrt {1+\sqrt {2}} \arctan \left (\frac {2^{3/4} \sqrt {2+\sqrt {2}} x \sqrt [4]{-x^2+x^6}}{-2 x^2+\sqrt {2} \sqrt {-x^2+x^6}}\right )-\frac {1}{4} \sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {\frac {\sqrt [4]{2} x^2}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {-x^2+x^6}}{\sqrt [4]{2} \sqrt {2-\sqrt {2}}}}{x \sqrt [4]{-x^2+x^6}}\right )+\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (-2 x^2+2^{3/4} \sqrt {2+\sqrt {2}} x \sqrt [4]{-x^2+x^6}-\sqrt {2} \sqrt {-x^2+x^6}\right )-\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (2 \sqrt {2-\sqrt {2}} x^2+2 \sqrt [4]{2} x \sqrt [4]{-x^2+x^6}+\sqrt {4-2 \sqrt {2}} \sqrt {-x^2+x^6}\right ) \] Output:
-1/4*(2^(1/2)-1)^(1/2)*arctan((2-2^(1/2))^(1/2)*x/(-(2+2^(1/2))^(1/2)*x+2^ (3/4)*(x^6-x^2)^(1/4)))-1/4*(2^(1/2)-1)^(1/2)*arctan((2-2^(1/2))^(1/2)*x/( (2+2^(1/2))^(1/2)*x+2^(3/4)*(x^6-x^2)^(1/4)))-1/4*(1+2^(1/2))^(1/2)*arctan (2^(3/4)*(2+2^(1/2))^(1/2)*x*(x^6-x^2)^(1/4)/(-2*x^2+2^(1/2)*(x^6-x^2)^(1/ 2)))-1/4*(2^(1/2)-1)^(1/2)*arctanh((2^(1/4)*x^2/(2-2^(1/2))^(1/2)+1/2*(x^6 -x^2)^(1/2)*2^(3/4)/(2-2^(1/2))^(1/2))/x/(x^6-x^2)^(1/4))+1/8*(1+2^(1/2))^ (1/2)*ln(-2*x^2+2^(3/4)*(2+2^(1/2))^(1/2)*x*(x^6-x^2)^(1/4)-2^(1/2)*(x^6-x ^2)^(1/2))-1/8*(1+2^(1/2))^(1/2)*ln(2*(2-2^(1/2))^(1/2)*x^2+2*2^(1/4)*x*(x ^6-x^2)^(1/4)+(4-2*2^(1/2))^(1/2)*(x^6-x^2)^(1/2))
Result contains complex when optimal does not.
Time = 0.75 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.36 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=-\frac {\sqrt {x} \sqrt [4]{-1+x^4} \left (\sqrt {-1+i} \arctan \left (\frac {\sqrt {-1-i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )+\sqrt {-1-i} \arctan \left (\frac {\sqrt {-1+i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )+\sqrt {1+i} \arctan \left (\frac {\sqrt {1-i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )+\sqrt {1-i} \arctan \left (\frac {\sqrt {1+i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )\right )}{2 \sqrt {2} \sqrt [4]{x^2 \left (-1+x^4\right )}} \] Input:
Integrate[(-1 + x^4)/((1 + x^4)*(-x^2 + x^6)^(1/4)),x]
Output:
-1/2*(Sqrt[x]*(-1 + x^4)^(1/4)*(Sqrt[-1 + I]*ArcTan[(Sqrt[-1 - I]*Sqrt[x]) /(-1 + x^4)^(1/4)] + Sqrt[-1 - I]*ArcTan[(Sqrt[-1 + I]*Sqrt[x])/(-1 + x^4) ^(1/4)] + Sqrt[1 + I]*ArcTan[(Sqrt[1 - I]*Sqrt[x])/(-1 + x^4)^(1/4)] + Sqr t[1 - I]*ArcTan[(Sqrt[1 + I]*Sqrt[x])/(-1 + x^4)^(1/4)]))/(Sqrt[2]*(x^2*(- 1 + x^4))^(1/4))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.31 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.11, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2467, 966, 937, 936}
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.
| \(\displaystyle \int \frac {x^4-1}{\left (x^4+1\right ) \sqrt [4]{x^6-x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^4-1} \int \frac {\left (x^4-1\right )^{3/4}}{\sqrt {x} \left (x^4+1\right )}dx}{\sqrt [4]{x^6-x^2}}\) |
| \(\Big \downarrow \) 966 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^4-1} \int \frac {\left (x^4-1\right )^{3/4}}{x^4+1}d\sqrt {x}}{\sqrt [4]{x^6-x^2}}\) |
| \(\Big \downarrow \) 937 |
| \(\displaystyle \frac {2 \sqrt {x} \left (x^4-1\right ) \int \frac {\left (1-x^4\right )^{3/4}}{x^4+1}d\sqrt {x}}{\left (1-x^4\right )^{3/4} \sqrt [4]{x^6-x^2}}\) |
| \(\Big \downarrow \) 936 |
| \(\displaystyle \frac {2 x \left (x^4-1\right ) \operatorname {AppellF1}\left (\frac {1}{8},-\frac {3}{4},1,\frac {9}{8},x^4,-x^4\right )}{\left (1-x^4\right )^{3/4} \sqrt [4]{x^6-x^2}}\) |
Input:
Int[(-1 + x^4)/((1 + x^4)*(-x^2 + x^6)^(1/4)),x]
Output:
(2*x*(-1 + x^4)*AppellF1[1/8, -3/4, 1, 9/8, x^4, -x^4])/((1 - x^4)^(3/4)*( -x^2 + x^6)^(1/4))
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) ], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q }, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && !(IntegerQ[p] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) )^(q_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*( m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*x)^( 1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[ n, 0] && FractionQ[m] && IntegerQ[p]
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]
Time = 63.01 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.86
| method | result | size |
| pseudoelliptic | \(-\frac {\left (2+\sqrt {2}\right ) \left (\ln \left (\frac {\left (x^{6}-x^{2}\right )^{\frac {1}{4}} \sqrt {2+2 \sqrt {2}}\, x +\sqrt {2}\, x^{2}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )-2 \arctan \left (\frac {x \sqrt {-2+2 \sqrt {2}}+2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{\sqrt {2+2 \sqrt {2}}\, x}\right )-\ln \left (\frac {-\left (x^{6}-x^{2}\right )^{\frac {1}{4}} \sqrt {2+2 \sqrt {2}}\, x +\sqrt {2}\, x^{2}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )+2 \arctan \left (\frac {x \sqrt {-2+2 \sqrt {2}}-2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{\sqrt {2+2 \sqrt {2}}\, x}\right )\right ) \sqrt {-2+2 \sqrt {2}}}{16}+\frac {\sqrt {2+2 \sqrt {2}}\, \left (-2+\sqrt {2}\right ) \left (\ln \left (\frac {\sqrt {2}\, x^{2}+x \sqrt {-2+2 \sqrt {2}}\, \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )-2 \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}\, x +2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {2}}}\right )-\ln \left (\frac {\sqrt {2}\, x^{2}-x \sqrt {-2+2 \sqrt {2}}\, \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )+2 \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}\, x -2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {2}}}\right )\right )}{16}\) | \(390\) |
| trager | \(\text {Expression too large to display}\) | \(2857\) |
Input:
int((x^4-1)/(x^4+1)/(x^6-x^2)^(1/4),x,method=_RETURNVERBOSE)
Output:
-1/16*(2+2^(1/2))*(ln(((x^6-x^2)^(1/4)*(2+2*2^(1/2))^(1/2)*x+2^(1/2)*x^2+( x^6-x^2)^(1/2))/x^2)-2*arctan((x*(-2+2*2^(1/2))^(1/2)+2*(x^6-x^2)^(1/4))/( 2+2*2^(1/2))^(1/2)/x)-ln((-(x^6-x^2)^(1/4)*(2+2*2^(1/2))^(1/2)*x+2^(1/2)*x ^2+(x^6-x^2)^(1/2))/x^2)+2*arctan((x*(-2+2*2^(1/2))^(1/2)-2*(x^6-x^2)^(1/4 ))/(2+2*2^(1/2))^(1/2)/x))*(-2+2*2^(1/2))^(1/2)+1/16*(2+2*2^(1/2))^(1/2)*( -2+2^(1/2))*(ln((2^(1/2)*x^2+x*(-2+2*2^(1/2))^(1/2)*(x^6-x^2)^(1/4)+(x^6-x ^2)^(1/2))/x^2)-2*arctan(((2+2*2^(1/2))^(1/2)*x+2*(x^6-x^2)^(1/4))/x/(-2+2 *2^(1/2))^(1/2))-ln((2^(1/2)*x^2-x*(-2+2*2^(1/2))^(1/2)*(x^6-x^2)^(1/4)+(x ^6-x^2)^(1/2))/x^2)+2*arctan(((2+2*2^(1/2))^(1/2)*x-2*(x^6-x^2)^(1/4))/x/( -2+2*2^(1/2))^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 2438 vs. \(2 (343) = 686\).
Time = 65.86 (sec) , antiderivative size = 2438, normalized size of antiderivative = 5.39 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=\text {Too large to display} \] Input:
integrate((x^4-1)/(x^4+1)/(x^6-x^2)^(1/4),x, algorithm="fricas")
Output:
-1/8*sqrt(sqrt(2) + 1)*arctan((25080720567741489538420*(908850053700489014 70917248255082489399665861990883067250482488177*x^12 + 1712054177418098500 7437092181820090486539062647777542204389063075454*x^10 - 34423235010160500 089058788232793646241802054821960383818334840691023*x^8 - 4773993551046395 4040312535279248618225543636646940176631091553461252*x^6 + 344232350101605 00089058788232793646241802054821960383818334840691023*x^4 + 17120541774180 985007437092181820090486539062647777542204389063075454*x^2 + sqrt(2)*(4843 46405698747637508877160078339143722104872500987070372985661805*x^12 - 2247 9453322390940824132586598591907849408200135795904234545549946716*x^10 + 72 750781193701052137100115292373632040180126384051702658718390333645*x^8 + 3 0545722900161629369857892105309312631696201383607088936429063091848*x^6 - 72750781193701052137100115292373632040180126384051702658718390333645*x^4 - 22479453322390940824132586598591907849408200135795904234545549946716*x^2 - 484346405698747637508877160078339143722104872500987070372985661805) - 90 885005370048901470917248255082489399665861990883067250482488177)*(x^6 - x^ 2)^(3/4) + 8788970999866174794871*(908850053700489014709172482550824893996 65861990883067250482488177*x^17 + 3000277228129179330891949808999313786570 2622867583655576792310903760*x^15 - 13769294004064200035623515293117458496 7208219287841535273339362764092*x^13 + 30002772281291793308919498089993137 865702622867583655576792310903760*x^11 + 276658270156464685333063147337...
\[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{4} + 1\right )}\, dx \] Input:
integrate((x**4-1)/(x**4+1)/(x**6-x**2)**(1/4),x)
Output:
Integral((x - 1)*(x + 1)*(x**2 + 1)/((x**2*(x - 1)*(x + 1)*(x**2 + 1))**(1 /4)*(x**4 + 1)), x)
\[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{6} - x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}} \,d x } \] Input:
integrate((x^4-1)/(x^4+1)/(x^6-x^2)^(1/4),x, algorithm="maxima")
Output:
integrate((x^4 - 1)/((x^6 - x^2)^(1/4)*(x^4 + 1)), x)
\[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{6} - x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}} \,d x } \] Input:
integrate((x^4-1)/(x^4+1)/(x^6-x^2)^(1/4),x, algorithm="giac")
Output:
integrate((x^4 - 1)/((x^6 - x^2)^(1/4)*(x^4 + 1)), x)
Timed out. \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=\int \frac {x^4-1}{\left (x^4+1\right )\,{\left (x^6-x^2\right )}^{1/4}} \,d x \] Input:
int((x^4 - 1)/((x^4 + 1)*(x^6 - x^2)^(1/4)),x)
Output:
int((x^4 - 1)/((x^4 + 1)*(x^6 - x^2)^(1/4)), x)
\[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=\frac {\frac {8 \sqrt {x}\, \left (x^{4}-1\right )^{\frac {5}{4}}}{3}+\frac {2 \sqrt {x}\, \left (x^{4}-1\right )^{\frac {1}{4}} x^{4}}{3}-\frac {2 \sqrt {x}\, \left (x^{4}-1\right )^{\frac {1}{4}}}{3}+\frac {8 \sqrt {x^{4}-1}\, \left (\int \frac {\left (x^{4}-1\right )^{\frac {3}{4}}}{\sqrt {x}\, x^{12}-\sqrt {x}\, x^{8}-\sqrt {x}\, x^{4}+\sqrt {x}}d x \right ) x^{4}}{3}-\frac {8 \sqrt {x^{4}-1}\, \left (\int \frac {\left (x^{4}-1\right )^{\frac {3}{4}}}{\sqrt {x}\, x^{12}-\sqrt {x}\, x^{8}-\sqrt {x}\, x^{4}+\sqrt {x}}d x \right )}{3}+\frac {4 \sqrt {x^{4}-1}\, \left (\int \frac {\sqrt {x}\, \left (x^{4}-1\right )^{\frac {3}{4}} x^{7}}{x^{12}-x^{8}-x^{4}+1}d x \right ) x^{4}}{3}-\frac {4 \sqrt {x^{4}-1}\, \left (\int \frac {\sqrt {x}\, \left (x^{4}-1\right )^{\frac {3}{4}} x^{7}}{x^{12}-x^{8}-x^{4}+1}d x \right )}{3}}{\sqrt {x^{4}-1}\, \left (x^{4}-1\right )} \] Input:
int((x^4-1)/(x^4+1)/(x^6-x^2)^(1/4),x)
Output:
(2*(4*sqrt(x)*(x**4 - 1)**(5/4) + sqrt(x)*(x**4 - 1)**(1/4)*x**4 - sqrt(x) *(x**4 - 1)**(1/4) + 4*sqrt(x**4 - 1)*int((x**4 - 1)**(3/4)/(sqrt(x)*x**12 - sqrt(x)*x**8 - sqrt(x)*x**4 + sqrt(x)),x)*x**4 - 4*sqrt(x**4 - 1)*int(( x**4 - 1)**(3/4)/(sqrt(x)*x**12 - sqrt(x)*x**8 - sqrt(x)*x**4 + sqrt(x)),x ) + 2*sqrt(x**4 - 1)*int((sqrt(x)*(x**4 - 1)**(3/4)*x**7)/(x**12 - x**8 - x**4 + 1),x)*x**4 - 2*sqrt(x**4 - 1)*int((sqrt(x)*(x**4 - 1)**(3/4)*x**7)/ (x**12 - x**8 - x**4 + 1),x)))/(3*sqrt(x**4 - 1)*(x**4 - 1))