Integrand size = 24, antiderivative size = 452 \[ \int \frac {x^2}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=-\frac {1}{8} \sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x}{-\sqrt {2+\sqrt {2}} x+2^{3/4} \sqrt [4]{-x^2+x^6}}\right )-\frac {1}{8} \sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x}{\sqrt {2+\sqrt {2}} x+2^{3/4} \sqrt [4]{-x^2+x^6}}\right )+\frac {1}{8} \sqrt {-1+\sqrt {2}} \arctan \left (\frac {2^{3/4} \sqrt {2+\sqrt {2}} x \sqrt [4]{-x^2+x^6}}{-2 x^2+\sqrt {2} \sqrt {-x^2+x^6}}\right )-\frac {1}{8} \sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\frac {\sqrt [4]{2} x^2}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {-x^2+x^6}}{\sqrt [4]{2} \sqrt {2-\sqrt {2}}}}{x \sqrt [4]{-x^2+x^6}}\right )-\frac {1}{16} \sqrt {-1+\sqrt {2}} \log \left (-2 x^2+2^{3/4} \sqrt {2+\sqrt {2}} x \sqrt [4]{-x^2+x^6}-\sqrt {2} \sqrt {-x^2+x^6}\right )+\frac {1}{16} \sqrt {-1+\sqrt {2}} \log \left (2 \sqrt {2-\sqrt {2}} x^2+2 \sqrt [4]{2} x \sqrt [4]{-x^2+x^6}+\sqrt {4-2 \sqrt {2}} \sqrt {-x^2+x^6}\right ) \]
-1/8*(1+2^(1/2))^(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)))-1/8*(1+2^(1/2))^(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)))+1/8*(2^(1/2)-1)^(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)))-1/8*(1+2^(1/2))^(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))-1/16*(2^(1/2)-1) ^(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))+1/16*(2^(1/2)-1)^(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 = 0.00 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.36 \[ \int \frac {x^2}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=\frac {\sqrt {x} \sqrt [4]{-1+x^4} \left (-\sqrt {1-i} \arctan \left (\frac {\sqrt {-1-i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )-\sqrt {1+i} \arctan \left (\frac {\sqrt {-1+i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )+\sqrt {-1-i} \arctan \left (\frac {\sqrt {1-i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )+\sqrt {-1+i} \arctan \left (\frac {\sqrt {1+i} \sqrt {x}}{\sqrt [4]{-1+x^4}}\right )\right )}{4 \sqrt {2} \sqrt [4]{x^2 \left (-1+x^4\right )}} \]
(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)] + Sqrt[- 1 + I]*ArcTan[(Sqrt[1 + I]*Sqrt[x])/(-1 + x^4)^(1/4)]))/(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 = 0.23 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.11, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1948, 966, 1013, 1012}
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.
| \(\displaystyle \int \frac {x^2}{\left (x^4+1\right ) \sqrt [4]{x^6-x^2}} \, dx\) |
| \(\Big \downarrow \) 1948 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^4-1} \int \frac {x^{3/2}}{\sqrt [4]{x^4-1} \left (x^4+1\right )}dx}{\sqrt [4]{x^6-x^2}}\) |
| \(\Big \downarrow \) 966 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^4-1} \int \frac {x^2}{\sqrt [4]{x^4-1} \left (x^4+1\right )}d\sqrt {x}}{\sqrt [4]{x^6-x^2}}\) |
| \(\Big \downarrow \) 1013 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{1-x^4} \int \frac {x^2}{\sqrt [4]{1-x^4} \left (x^4+1\right )}d\sqrt {x}}{\sqrt [4]{x^6-x^2}}\) |
| \(\Big \downarrow \) 1012 |
| \(\displaystyle \frac {2 x^3 \sqrt [4]{1-x^4} \operatorname {AppellF1}\left (\frac {5}{8},\frac {1}{4},1,\frac {13}{8},x^4,-x^4\right )}{5 \sqrt [4]{x^6-x^2}}\) |
3.31.38.3.1 Defintions of rubi rules used
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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((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[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & & NeQ[m, n - 1] && !(IntegerQ[p] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Simp[e^IntPart[m]*(e*x)^FracPart[m]*( (a*x^j + b*x^(j + n))^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x ^n)^FracPart[p])) Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p, q}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && !(EqQ[n, 1] && EqQ[j, 1])
Time = 0.00 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.85
| method | result | size |
| pseudoelliptic | \(\frac {\sqrt {2}\, \left (\left (2 \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}\, x +2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {2}}}\right )+\ln \left (\frac {\sqrt {2}\, x^{2}-x \sqrt {-2+2 \sqrt {2}}\, \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )-2 \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}\, x -2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {2}}}\right )-\ln \left (\frac {x \sqrt {-2+2 \sqrt {2}}\, \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+\sqrt {2}\, x^{2}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )\right ) \sqrt {2+2 \sqrt {2}}+\sqrt {\sqrt {2}-1}\, \left (\ln \left (\frac {\sqrt {2+2 \sqrt {2}}\, x \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+\sqrt {2}\, x^{2}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )-2 \arctan \left (\frac {x \sqrt {-2+2 \sqrt {2}}+2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{\sqrt {2+2 \sqrt {2}}\, x}\right )-\ln \left (\frac {-\sqrt {2+2 \sqrt {2}}\, x \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+\sqrt {2}\, x^{2}+\sqrt {x^{6}-x^{2}}}{x^{2}}\right )+2 \arctan \left (\frac {x \sqrt {-2+2 \sqrt {2}}-2 \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{\sqrt {2+2 \sqrt {2}}\, x}\right )\right ) \sqrt {2}\right )}{32}\) | \(384\) |
| trager | \(\text {Expression too large to display}\) | \(2930\) |
1/32*2^(1/2)*((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))-ln((x*(-2+2*2^(1/2))^(1/2)*(x^6-x^2)^(1/4)+2^(1/2)*x^2 +(x^6-x^2)^(1/2))/x^2))*(2+2*2^(1/2))^(1/2)+(2^(1/2)-1)^(1/2)*(ln(((2+2*2^ (1/2))^(1/2)*x*(x^6-x^2)^(1/4)+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((-(2+ 2*2^(1/2))^(1/2)*x*(x^6-x^2)^(1/4)+2^(1/2)*x^2+(x^6-x^2)^(1/2))/x^2)+2*arc tan((x*(-2+2*2^(1/2))^(1/2)-2*(x^6-x^2)^(1/4))/(2+2*2^(1/2))^(1/2)/x))*2^( 1/2))
Result contains complex when optimal does not.
Time = 12.31 (sec) , antiderivative size = 1089, normalized size of antiderivative = 2.41 \[ \int \frac {x^2}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=\text {Too large to display} \]
1/32*sqrt(2)*sqrt(I + 1)*log(((I + 1)*x^9 - (4*I - 4)*x^7 - (6*I + 6)*x^5 + (4*I - 4)*x^3 - 4*sqrt(x^6 - x^2)*(x^5 - 2*I*x^3 - x) - 2*sqrt(I + 1)*(s qrt(2)*(x^6 - x^2)^(3/4)*(I*x^4 + 2*x^2 - I) + sqrt(2)*(x^6 - x^2)^(1/4)*( -(I + 1)*x^6 + (2*I - 2)*x^4 + (I + 1)*x^2)) + (I + 1)*x)/(x^9 + 2*x^5 + x )) - 1/32*sqrt(2)*sqrt(I + 1)*log(((I + 1)*x^9 - (4*I - 4)*x^7 - (6*I + 6) *x^5 + (4*I - 4)*x^3 - 4*sqrt(x^6 - x^2)*(x^5 - 2*I*x^3 - x) - 2*sqrt(I + 1)*(sqrt(2)*(x^6 - x^2)^(3/4)*(-I*x^4 - 2*x^2 + I) + sqrt(2)*((I + 1)*x^6 - (2*I - 2)*x^4 - (I + 1)*x^2)*(x^6 - x^2)^(1/4)) + (I + 1)*x)/(x^9 + 2*x^ 5 + x)) - 1/32*sqrt(2)*sqrt(-I + 1)*log((-(I - 1)*x^9 + (4*I + 4)*x^7 + (6 *I - 6)*x^5 - (4*I + 4)*x^3 - 2*sqrt(2)*sqrt(-I + 1)*(x^6 - x^2)^(3/4)*(I* x^4 - 2*x^2 - I) - 2*sqrt(2)*sqrt(-I + 1)*(x^6 - x^2)^(1/4)*(-(I - 1)*x^6 + (2*I + 2)*x^4 + (I - 1)*x^2) - 4*sqrt(x^6 - x^2)*(x^5 + 2*I*x^3 - x) - ( I - 1)*x)/(x^9 + 2*x^5 + x)) + 1/32*sqrt(2)*sqrt(-I + 1)*log((-(I - 1)*x^9 + (4*I + 4)*x^7 + (6*I - 6)*x^5 - (4*I + 4)*x^3 - 2*sqrt(2)*sqrt(-I + 1)* (x^6 - x^2)^(3/4)*(-I*x^4 + 2*x^2 + I) - 2*sqrt(2)*sqrt(-I + 1)*(x^6 - x^2 )^(1/4)*((I - 1)*x^6 - (2*I + 2)*x^4 - (I - 1)*x^2) - 4*sqrt(x^6 - x^2)*(x ^5 + 2*I*x^3 - x) - (I - 1)*x)/(x^9 + 2*x^5 + x)) - 1/32*sqrt(2)*sqrt(I - 1)*log(((I - 1)*x^9 - (4*I + 4)*x^7 - (6*I - 6)*x^5 + (4*I + 4)*x^3 - 4*sq rt(x^6 - x^2)*(x^5 + 2*I*x^3 - x) - 2*sqrt(I - 1)*(sqrt(2)*(x^6 - x^2)^(3/ 4)*(I*x^4 - 2*x^2 - I) + sqrt(2)*(x^6 - x^2)^(1/4)*((I - 1)*x^6 - (2*I ...
\[ \int \frac {x^2}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=\int \frac {x^{2}}{\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{4} + 1\right )}\, dx \]
\[ \int \frac {x^2}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=\int { \frac {x^{2}}{{\left (x^{6} - x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}} \,d x } \]
\[ \int \frac {x^2}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=\int { \frac {x^{2}}{{\left (x^{6} - x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx=\int \frac {x^2}{\left (x^4+1\right )\,{\left (x^6-x^2\right )}^{1/4}} \,d x \]