Integrand size = 25, antiderivative size = 239 \[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1+x^8+x^{16}\right )} \, dx=\frac {\arctan \left (\frac {-\frac {1}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}+\frac {x^4}{\sqrt {2}}}{x \sqrt {-1+x^4}}\right )}{4 \sqrt {2}}+\frac {\arctan \left (\frac {-\frac {1}{\sqrt {2} \sqrt [4]{3}}-\frac {\sqrt [4]{3} x^2}{\sqrt {2}}+\frac {x^4}{\sqrt {2} \sqrt [4]{3}}}{x \sqrt {-1+x^4}}\right )}{4 \sqrt {2} \sqrt [4]{3}}-\frac {\text {arctanh}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}+\frac {x^4}{\sqrt {2}}}{x \sqrt {-1+x^4}}\right )}{4 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {-\frac {1}{\sqrt {2} \sqrt [4]{3}}+\frac {\sqrt [4]{3} x^2}{\sqrt {2}}+\frac {x^4}{\sqrt {2} \sqrt [4]{3}}}{x \sqrt {-1+x^4}}\right )}{4 \sqrt {2} \sqrt [4]{3}} \]
1/8*arctan((-1/2*2^(1/2)-1/2*2^(1/2)*x^2+1/2*2^(1/2)*x^4)/x/(x^4-1)^(1/2)) *2^(1/2)+1/24*arctan((-1/6*2^(1/2)*3^(3/4)-1/2*3^(1/4)*x^2*2^(1/2)+1/6*x^4 *2^(1/2)*3^(3/4))/x/(x^4-1)^(1/2))*2^(1/2)*3^(3/4)-1/8*arctanh((-1/2*2^(1/ 2)+1/2*2^(1/2)*x^2+1/2*2^(1/2)*x^4)/x/(x^4-1)^(1/2))*2^(1/2)-1/24*arctanh( (-1/6*2^(1/2)*3^(3/4)+1/2*3^(1/4)*x^2*2^(1/2)+1/6*x^4*2^(1/2)*3^(3/4))/x/( x^4-1)^(1/2))*2^(1/2)*3^(3/4)
Result contains complex when optimal does not.
Time = 1.76 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.79 \[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1+x^8+x^{16}\right )} \, dx=\frac {3 \arctan \left (\frac {-1-x^2+x^4}{\sqrt {2} x \sqrt {-1+x^4}}\right )+3^{3/4} \arctan \left (\frac {-1-\left (-2 i+\sqrt {-1+4 i \sqrt {3}}\right ) x^2+x^4}{\sqrt {2} \sqrt [4]{3} x \sqrt {-1+x^4}}\right )-3 \text {arctanh}\left (\frac {-1+x^2+x^4}{\sqrt {2} x \sqrt {-1+x^4}}\right )-3^{3/4} \text {arctanh}\left (\frac {-1+\left (2 i+\sqrt {-1-4 i \sqrt {3}}\right ) x^2+x^4}{\sqrt {2} \sqrt [4]{3} x \sqrt {-1+x^4}}\right )}{12 \sqrt {2}} \]
(3*ArcTan[(-1 - x^2 + x^4)/(Sqrt[2]*x*Sqrt[-1 + x^4])] + 3^(3/4)*ArcTan[(- 1 - (-2*I + Sqrt[-1 + (4*I)*Sqrt[3]])*x^2 + x^4)/(Sqrt[2]*3^(1/4)*x*Sqrt[- 1 + x^4])] - 3*ArcTanh[(-1 + x^2 + x^4)/(Sqrt[2]*x*Sqrt[-1 + x^4])] - 3^(3 /4)*ArcTanh[(-1 + (2*I + Sqrt[-1 - (4*I)*Sqrt[3]])*x^2 + x^4)/(Sqrt[2]*3^( 1/4)*x*Sqrt[-1 + x^4])])/(12*Sqrt[2])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 4.03 (sec) , antiderivative size = 1867, normalized size of antiderivative = 7.81, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2019, 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.
| \(\displaystyle \int \frac {x^{16}-1}{\sqrt {x^4-1} \left (x^{16}+x^8+1\right )} \, dx\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle \int \frac {\sqrt {x^4-1} \left (x^{12}+x^8+x^4+1\right )}{x^{16}+x^8+1}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4-1} \left (x^4+1\right )}{2 \left (x^8-x^4+1\right )}+\frac {\sqrt {x^4-1} (1-x)}{8 \left (x^2-x+1\right )}+\frac {(x+1) \sqrt {x^4-1}}{8 \left (x^2+x+1\right )}+\frac {\sqrt {x^4-1}}{4 \left (x^4-x^2+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{32} \sqrt {\frac {1}{6} \left (3+i \sqrt {3}\right )} \left (i+\sqrt {3}\right ) \arctan \left (\frac {\left (1-i \sqrt {3}\right ) x^2+2}{\sqrt {2 \left (3+i \sqrt {3}\right )} \sqrt {x^4-1}}\right )+\frac {1}{32} \sqrt {\frac {1}{6} \left (3+i \sqrt {3}\right )} \left (i+\sqrt {3}\right ) \arctan \left (\frac {4-\left (1+i \sqrt {3}\right )^2 x^2}{2 \sqrt {2 \left (3+i \sqrt {3}\right )} \sqrt {x^4-1}}\right )-\frac {\left (i-\sqrt {3}\right ) \sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticF}(\arcsin (x),-1)}{4 \left (2+\sqrt {2-2 i \sqrt {3}}\right ) \sqrt {x^4-1}}+\frac {\left (i-\sqrt {3}\right ) \sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticF}(\arcsin (x),-1)}{4 \left (2-\sqrt {2-2 i \sqrt {3}}\right ) \sqrt {x^4-1}}+\frac {\left (i+\sqrt {3}\right ) \sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticF}(\arcsin (x),-1)}{4 \sqrt {2} \left (\sqrt {2}+\sqrt {1+i \sqrt {3}}\right ) \sqrt {x^4-1}}-\frac {\left (i+\sqrt {3}\right ) \sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticF}(\arcsin (x),-1)}{4 \sqrt {2} \left (\sqrt {2}-\sqrt {1+i \sqrt {3}}\right ) \sqrt {x^4-1}}+\frac {\left (i+\sqrt {3}\right ) \sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticF}(\arcsin (x),-1)}{4 \left (3 i+\sqrt {3}\right ) \sqrt {x^4-1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticF}(\arcsin (x),-1)}{8 \sqrt {x^4-1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticF}(\arcsin (x),-1)}{8 \sqrt {x^4-1}}+\frac {\left (i-\sqrt {3}\right ) \sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticF}(\arcsin (x),-1)}{4 \left (3 i-\sqrt {3}\right ) \sqrt {x^4-1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \left (1+i \sqrt {3}+\sqrt {2+2 i \sqrt {3}}\right ) \sqrt {x^4-1}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \left (1-i \sqrt {3}+\sqrt {2-2 i \sqrt {3}}\right ) \sqrt {x^4-1}}-\frac {\left (i+\sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \left (3 i+\sqrt {3}\right ) \sqrt {x^4-1}}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (i+\sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{\sqrt {2} \left (1-\sqrt {3}\right ) \sqrt {x^4-1}}+\frac {\left (i+\sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{8 \sqrt {6} \sqrt {x^4-1}}-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (1+i \sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{\sqrt {2} \left (1-\sqrt {3}\right ) \sqrt {x^4-1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {x^4-1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {x^4-1}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \left (3 i-\sqrt {3}\right ) \sqrt {x^4-1}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{8 \sqrt {6} \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-i-\sqrt {3}\right ),\arcsin (x),-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (-\frac {4}{\left (i-\sqrt {3}\right )^2},\arcsin (x),-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (\frac {1}{2} \left (i-\sqrt {3}\right ),\arcsin (x),-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}},\arcsin (x),-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (\frac {2}{1-i \sqrt {3}},\arcsin (x),-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}},\arcsin (x),-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (\frac {2}{1+i \sqrt {3}},\arcsin (x),-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (-\frac {4}{\left (i+\sqrt {3}\right )^2},\arcsin (x),-1\right )}{4 \sqrt {x^4-1}}\) |
-1/32*(Sqrt[(3 + I*Sqrt[3])/6]*(I + Sqrt[3])*ArcTan[(2 + (1 - I*Sqrt[3])*x ^2)/(Sqrt[2*(3 + I*Sqrt[3])]*Sqrt[-1 + x^4])]) + (Sqrt[(3 + I*Sqrt[3])/6]* (I + Sqrt[3])*ArcTan[(4 - (1 + I*Sqrt[3])^2*x^2)/(2*Sqrt[2*(3 + I*Sqrt[3]) ]*Sqrt[-1 + x^4])])/32 + ((I - Sqrt[3])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*Ellipt icF[ArcSin[x], -1])/(4*(3*I - Sqrt[3])*Sqrt[-1 + x^4]) + ((1 - I*Sqrt[3])* Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[x], -1])/(8*Sqrt[-1 + x^4]) + ((1 + I*Sqrt[3])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[x], -1])/(8 *Sqrt[-1 + x^4]) + ((I + Sqrt[3])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticF[Ar cSin[x], -1])/(4*(3*I + Sqrt[3])*Sqrt[-1 + x^4]) - ((I + Sqrt[3])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[x], -1])/(4*Sqrt[2]*(Sqrt[2] - Sqrt[1 + I*Sqrt[3]])*Sqrt[-1 + x^4]) + ((I + Sqrt[3])*Sqrt[1 - x^2]*Sqrt[1 + x^2 ]*EllipticF[ArcSin[x], -1])/(4*Sqrt[2]*(Sqrt[2] + Sqrt[1 + I*Sqrt[3]])*Sqr t[-1 + x^4]) + ((I - Sqrt[3])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin [x], -1])/(4*(2 - Sqrt[2 - (2*I)*Sqrt[3]])*Sqrt[-1 + x^4]) - ((I - Sqrt[3] )*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[x], -1])/(4*(2 + Sqrt[2 - ( 2*I)*Sqrt[3]])*Sqrt[-1 + x^4]) - ((I - Sqrt[3])*Sqrt[-1 + x^2]*Sqrt[1 + x^ 2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(8*Sqrt[6]*Sqrt[-1 + x^4]) - ((I - Sqrt[3])*Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sq rt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(4*Sqrt[2]*(3*I - Sqrt[3])*Sqrt[-1 + x^4]) + ((1 - I*Sqrt[3])*Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt...
3.27.70.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
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 = 21.34 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.95
| method | result | size |
| elliptic | \(\frac {\left (-\frac {\ln \left (\frac {x^{4}-1}{x^{2}}+\frac {\sqrt {x^{4}-1}\, \sqrt {2}}{x}+1\right )}{8}+\frac {\arctan \left (1+\frac {\sqrt {x^{4}-1}\, \sqrt {2}}{x}\right )}{4}+\frac {3^{\frac {3}{4}} \left (\ln \left (\frac {\frac {x^{4}-1}{2 x^{2}}-\frac {3^{\frac {1}{4}} \sqrt {x^{4}-1}\, \sqrt {2}}{2 x}+\frac {\sqrt {3}}{2}}{\frac {x^{4}-1}{2 x^{2}}+\frac {3^{\frac {1}{4}} \sqrt {x^{4}-1}\, \sqrt {2}}{2 x}+\frac {\sqrt {3}}{2}}\right )+2 \arctan \left (\frac {3^{\frac {3}{4}} \sqrt {x^{4}-1}\, \sqrt {2}}{3 x}+1\right )+2 \arctan \left (\frac {3^{\frac {3}{4}} \sqrt {x^{4}-1}\, \sqrt {2}}{3 x}-1\right )\right )}{24}+\frac {\ln \left (\frac {x^{4}-1}{x^{2}}-\frac {\sqrt {x^{4}-1}\, \sqrt {2}}{x}+1\right )}{8}+\frac {\arctan \left (-1+\frac {\sqrt {x^{4}-1}\, \sqrt {2}}{x}\right )}{4}\right ) \sqrt {2}}{2}\) | \(226\) |
| default | \(\frac {-\frac {\sqrt {2}\, \left (\left (\left (-1+i\right ) x^{2}-1-i\right ) \sqrt {3}+\left (-1-i\right ) x^{2}-4 i x +1-i\right ) \sqrt {9 i-3 \sqrt {3}}\, \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (2 i+\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \arctan \left (\frac {3^{\frac {3}{4}} \left (\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (-1-i\right ) x -2 i\right )}{\sqrt {\frac {i \left (-x^{4}+1\right )}{\left (2 i+\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \left (\left (-6+6 i\right ) x \sqrt {3}+12 x^{2}+\left (6+6 i\right ) x +12 i\right )}\right )}{6}+\frac {\left (\left (\left (-1+i\right ) x^{2}-1-i\right ) \sqrt {3}+\left (-1-i\right ) x^{2}+4 i x +1-i\right ) \sqrt {2}\, \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (-2 i+\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \sqrt {9 i-3 \sqrt {3}}\, \arctan \left (\frac {3^{\frac {3}{4}} \left (\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (-1-i\right ) x +2 i\right )}{\sqrt {\frac {i \left (-x^{4}+1\right )}{\left (-2 i+\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \left (\left (-6+6 i\right ) x \sqrt {3}-12 x^{2}+\left (6+6 i\right ) x -12 i\right )}\right )}{6}-\frac {2 \left (\left (\left (1+i\right ) x^{2}-i x -1+i\right ) 3^{\frac {3}{4}}+3 \,3^{\frac {1}{4}} x \right ) \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (-2 i+\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \operatorname {arctanh}\left (2 \,3^{\frac {1}{4}} \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (-2 i+\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (1+i\right ) x \right )^{2}}}\right )}{3}+\frac {2 \left (\left (\left (1+i\right ) x^{2}+i x -1+i\right ) 3^{\frac {3}{4}}-3 \,3^{\frac {1}{4}} x \right ) \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (2 i+\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \operatorname {arctanh}\left (2 \,3^{\frac {1}{4}} \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (2 i+\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (1+i\right ) x \right )^{2}}}\right )}{3}+x \sqrt {2}\, \left (\left (1-i\right ) \arctan \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {x^{4}-1}\, \sqrt {2}}{x}\right )+\left (1+i\right ) \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {x^{4}-1}\, \sqrt {2}}{x}\right )\right ) \sqrt {\frac {x^{4}-1}{x^{2}}}}{8 \sqrt {\frac {x^{4}-1}{x^{2}}}\, x}\) | \(615\) |
| pseudoelliptic | \(\frac {-\frac {\sqrt {2}\, \left (\left (\left (-1+i\right ) x^{2}-1-i\right ) \sqrt {3}+\left (-1-i\right ) x^{2}-4 i x +1-i\right ) \sqrt {9 i-3 \sqrt {3}}\, \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (2 i+\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \arctan \left (\frac {3^{\frac {3}{4}} \left (\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (-1-i\right ) x -2 i\right )}{\sqrt {\frac {i \left (-x^{4}+1\right )}{\left (2 i+\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \left (\left (-6+6 i\right ) x \sqrt {3}+12 x^{2}+\left (6+6 i\right ) x +12 i\right )}\right )}{6}+\frac {\left (\left (\left (-1+i\right ) x^{2}-1-i\right ) \sqrt {3}+\left (-1-i\right ) x^{2}+4 i x +1-i\right ) \sqrt {2}\, \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (-2 i+\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \sqrt {9 i-3 \sqrt {3}}\, \arctan \left (\frac {3^{\frac {3}{4}} \left (\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (-1-i\right ) x +2 i\right )}{\sqrt {\frac {i \left (-x^{4}+1\right )}{\left (-2 i+\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \left (\left (-6+6 i\right ) x \sqrt {3}-12 x^{2}+\left (6+6 i\right ) x -12 i\right )}\right )}{6}-\frac {2 \left (\left (\left (1+i\right ) x^{2}-i x -1+i\right ) 3^{\frac {3}{4}}+3 \,3^{\frac {1}{4}} x \right ) \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (-2 i+\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \operatorname {arctanh}\left (2 \,3^{\frac {1}{4}} \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (-2 i+\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (1+i\right ) x \right )^{2}}}\right )}{3}+\frac {2 \left (\left (\left (1+i\right ) x^{2}+i x -1+i\right ) 3^{\frac {3}{4}}-3 \,3^{\frac {1}{4}} x \right ) \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (2 i+\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \operatorname {arctanh}\left (2 \,3^{\frac {1}{4}} \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (2 i+\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (1+i\right ) x \right )^{2}}}\right )}{3}+x \sqrt {2}\, \left (\left (1-i\right ) \arctan \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {x^{4}-1}\, \sqrt {2}}{x}\right )+\left (1+i\right ) \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {x^{4}-1}\, \sqrt {2}}{x}\right )\right ) \sqrt {\frac {x^{4}-1}{x^{2}}}}{8 \sqrt {\frac {x^{4}-1}{x^{2}}}\, x}\) | \(615\) |
| trager | \(\text {Expression too large to display}\) | \(850\) |
1/2*(-1/8*ln((x^4-1)/x^2+(x^4-1)^(1/2)/x*2^(1/2)+1)+1/4*arctan(1+(x^4-1)^( 1/2)/x*2^(1/2))+1/24*3^(3/4)*(ln((1/2*(x^4-1)/x^2-1/2*3^(1/4)*(x^4-1)^(1/2 )/x*2^(1/2)+1/2*3^(1/2))/(1/2*(x^4-1)/x^2+1/2*3^(1/4)*(x^4-1)^(1/2)/x*2^(1 /2)+1/2*3^(1/2)))+2*arctan(1/3*3^(3/4)*(x^4-1)^(1/2)/x*2^(1/2)+1)+2*arctan (1/3*3^(3/4)*(x^4-1)^(1/2)/x*2^(1/2)-1))+1/8*ln((x^4-1)/x^2-(x^4-1)^(1/2)/ x*2^(1/2)+1)+1/4*arctan(-1+(x^4-1)^(1/2)/x*2^(1/2)))*2^(1/2)
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 601, normalized size of antiderivative = 2.51 \[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1+x^8+x^{16}\right )} \, dx=\left (\frac {1}{96} i + \frac {1}{96}\right ) \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (\frac {3^{\frac {3}{4}} \sqrt {2} {\left (\left (i + 1\right ) \, x^{8} - \left (5 i + 5\right ) \, x^{4} + i + 1\right )} - 6 \cdot 3^{\frac {1}{4}} \sqrt {2} {\left (\left (i - 1\right ) \, x^{6} - \left (i - 1\right ) \, x^{2}\right )} - 12 \, {\left (x^{5} - i \, \sqrt {3} x^{3} - x\right )} \sqrt {x^{4} - 1}}{x^{8} + x^{4} + 1}\right ) - \left (\frac {1}{96} i - \frac {1}{96}\right ) \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (\frac {3^{\frac {3}{4}} \sqrt {2} {\left (-\left (i - 1\right ) \, x^{8} + \left (5 i - 5\right ) \, x^{4} - i + 1\right )} - 6 \cdot 3^{\frac {1}{4}} \sqrt {2} {\left (-\left (i + 1\right ) \, x^{6} + \left (i + 1\right ) \, x^{2}\right )} - 12 \, {\left (x^{5} + i \, \sqrt {3} x^{3} - x\right )} \sqrt {x^{4} - 1}}{x^{8} + x^{4} + 1}\right ) + \left (\frac {1}{96} i - \frac {1}{96}\right ) \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (\frac {3^{\frac {3}{4}} \sqrt {2} {\left (\left (i - 1\right ) \, x^{8} - \left (5 i - 5\right ) \, x^{4} + i - 1\right )} - 6 \cdot 3^{\frac {1}{4}} \sqrt {2} {\left (\left (i + 1\right ) \, x^{6} - \left (i + 1\right ) \, x^{2}\right )} - 12 \, {\left (x^{5} + i \, \sqrt {3} x^{3} - x\right )} \sqrt {x^{4} - 1}}{x^{8} + x^{4} + 1}\right ) - \left (\frac {1}{96} i + \frac {1}{96}\right ) \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (\frac {3^{\frac {3}{4}} \sqrt {2} {\left (-\left (i + 1\right ) \, x^{8} + \left (5 i + 5\right ) \, x^{4} - i - 1\right )} - 6 \cdot 3^{\frac {1}{4}} \sqrt {2} {\left (-\left (i - 1\right ) \, x^{6} + \left (i - 1\right ) \, x^{2}\right )} - 12 \, {\left (x^{5} - i \, \sqrt {3} x^{3} - x\right )} \sqrt {x^{4} - 1}}{x^{8} + x^{4} + 1}\right ) + \left (\frac {1}{32} i + \frac {1}{32}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i + 1\right ) \, x^{8} - \left (2 i - 2\right ) \, x^{6} - \left (3 i + 3\right ) \, x^{4} + \left (2 i - 2\right ) \, x^{2} + i + 1\right )} - 4 \, {\left (x^{5} - i \, x^{3} - x\right )} \sqrt {x^{4} - 1}}{x^{8} - x^{4} + 1}\right ) - \left (\frac {1}{32} i - \frac {1}{32}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i - 1\right ) \, x^{8} + \left (2 i + 2\right ) \, x^{6} + \left (3 i - 3\right ) \, x^{4} - \left (2 i + 2\right ) \, x^{2} - i + 1\right )} - 4 \, {\left (x^{5} + i \, x^{3} - x\right )} \sqrt {x^{4} - 1}}{x^{8} - x^{4} + 1}\right ) + \left (\frac {1}{32} i - \frac {1}{32}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i - 1\right ) \, x^{8} - \left (2 i + 2\right ) \, x^{6} - \left (3 i - 3\right ) \, x^{4} + \left (2 i + 2\right ) \, x^{2} + i - 1\right )} - 4 \, {\left (x^{5} + i \, x^{3} - x\right )} \sqrt {x^{4} - 1}}{x^{8} - x^{4} + 1}\right ) - \left (\frac {1}{32} i + \frac {1}{32}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i + 1\right ) \, x^{8} + \left (2 i - 2\right ) \, x^{6} + \left (3 i + 3\right ) \, x^{4} - \left (2 i - 2\right ) \, x^{2} - i - 1\right )} - 4 \, {\left (x^{5} - i \, x^{3} - x\right )} \sqrt {x^{4} - 1}}{x^{8} - x^{4} + 1}\right ) \]
(1/96*I + 1/96)*3^(3/4)*sqrt(2)*log((3^(3/4)*sqrt(2)*((I + 1)*x^8 - (5*I + 5)*x^4 + I + 1) - 6*3^(1/4)*sqrt(2)*((I - 1)*x^6 - (I - 1)*x^2) - 12*(x^5 - I*sqrt(3)*x^3 - x)*sqrt(x^4 - 1))/(x^8 + x^4 + 1)) - (1/96*I - 1/96)*3^ (3/4)*sqrt(2)*log((3^(3/4)*sqrt(2)*(-(I - 1)*x^8 + (5*I - 5)*x^4 - I + 1) - 6*3^(1/4)*sqrt(2)*(-(I + 1)*x^6 + (I + 1)*x^2) - 12*(x^5 + I*sqrt(3)*x^3 - x)*sqrt(x^4 - 1))/(x^8 + x^4 + 1)) + (1/96*I - 1/96)*3^(3/4)*sqrt(2)*lo g((3^(3/4)*sqrt(2)*((I - 1)*x^8 - (5*I - 5)*x^4 + I - 1) - 6*3^(1/4)*sqrt( 2)*((I + 1)*x^6 - (I + 1)*x^2) - 12*(x^5 + I*sqrt(3)*x^3 - x)*sqrt(x^4 - 1 ))/(x^8 + x^4 + 1)) - (1/96*I + 1/96)*3^(3/4)*sqrt(2)*log((3^(3/4)*sqrt(2) *(-(I + 1)*x^8 + (5*I + 5)*x^4 - I - 1) - 6*3^(1/4)*sqrt(2)*(-(I - 1)*x^6 + (I - 1)*x^2) - 12*(x^5 - I*sqrt(3)*x^3 - x)*sqrt(x^4 - 1))/(x^8 + x^4 + 1)) + (1/32*I + 1/32)*sqrt(2)*log((sqrt(2)*((I + 1)*x^8 - (2*I - 2)*x^6 - (3*I + 3)*x^4 + (2*I - 2)*x^2 + I + 1) - 4*(x^5 - I*x^3 - x)*sqrt(x^4 - 1) )/(x^8 - x^4 + 1)) - (1/32*I - 1/32)*sqrt(2)*log((sqrt(2)*(-(I - 1)*x^8 + (2*I + 2)*x^6 + (3*I - 3)*x^4 - (2*I + 2)*x^2 - I + 1) - 4*(x^5 + I*x^3 - x)*sqrt(x^4 - 1))/(x^8 - x^4 + 1)) + (1/32*I - 1/32)*sqrt(2)*log((sqrt(2)* ((I - 1)*x^8 - (2*I + 2)*x^6 - (3*I - 3)*x^4 + (2*I + 2)*x^2 + I - 1) - 4* (x^5 + I*x^3 - x)*sqrt(x^4 - 1))/(x^8 - x^4 + 1)) - (1/32*I + 1/32)*sqrt(2 )*log((sqrt(2)*(-(I + 1)*x^8 + (2*I - 2)*x^6 + (3*I + 3)*x^4 - (2*I - 2)*x ^2 - I - 1) - 4*(x^5 - I*x^3 - x)*sqrt(x^4 - 1))/(x^8 - x^4 + 1))
\[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1+x^8+x^{16}\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right ) \left (x^{8} + 1\right )}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right ) \left (x^{8} - x^{4} + 1\right )}\, dx \]
Integral((x - 1)*(x + 1)*(x**2 + 1)*(x**4 + 1)*(x**8 + 1)/(sqrt((x - 1)*(x + 1)*(x**2 + 1))*(x**2 - x + 1)*(x**2 + x + 1)*(x**4 - x**2 + 1)*(x**8 - x**4 + 1)), x)
\[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1+x^8+x^{16}\right )} \, dx=\int { \frac {x^{16} - 1}{{\left (x^{16} + x^{8} + 1\right )} \sqrt {x^{4} - 1}} \,d x } \]
\[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1+x^8+x^{16}\right )} \, dx=\int { \frac {x^{16} - 1}{{\left (x^{16} + x^{8} + 1\right )} \sqrt {x^{4} - 1}} \,d x } \]
Timed out. \[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1+x^8+x^{16}\right )} \, dx=\int \frac {x^{16}-1}{\sqrt {x^4-1}\,\left (x^{16}+x^8+1\right )} \,d x \]