Integrand size = 22, antiderivative size = 45 \[ \int \frac {-1+x^{16}}{\sqrt {1+x^4} \left (1+x^{16}\right )} \, dx=\frac {1}{8} \text {RootSum}\left [2-4 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt {1+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
Time = 0.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^{16}}{\sqrt {1+x^4} \left (1+x^{16}\right )} \, dx=\frac {1}{8} \text {RootSum}\left [2-4 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt {1+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
Result contains complex when optimal does not.
Time = 5.76 (sec) , antiderivative size = 1790, normalized size of antiderivative = 39.78, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2019, 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.
| \(\displaystyle \int \frac {x^{16}-1}{\sqrt {x^4+1} \left (x^{16}+1\right )} \, dx\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle \int \frac {\sqrt {x^4+1} \left (x^{12}-x^8+x^4-1\right )}{x^{16}+1}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {x^4+1}}{16 \left (\sqrt [16]{-1}-x\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {x^4+1}}{16 \left (\sqrt [16]{-1}-i x\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {x^4+1}}{16 \left (i x+\sqrt [16]{-1}\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {x^4+1}}{16 \left (x+\sqrt [16]{-1}\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {x^4+1}}{16 \left (\sqrt [16]{-1}-\sqrt [8]{-1} x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {x^4+1}}{16 \left (\sqrt [8]{-1} x+\sqrt [16]{-1}\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {x^4+1}}{16 \left (\sqrt [16]{-1}-\sqrt [4]{-1} x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {x^4+1}}{16 \left (\sqrt [4]{-1} x+\sqrt [16]{-1}\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {x^4+1}}{16 \left (\sqrt [16]{-1}-(-1)^{3/8} x\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {x^4+1}}{16 \left ((-1)^{3/8} x+\sqrt [16]{-1}\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {x^4+1}}{16 \left (\sqrt [16]{-1}-(-1)^{5/8} x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {x^4+1}}{16 \left ((-1)^{5/8} x+\sqrt [16]{-1}\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {x^4+1}}{16 \left (\sqrt [16]{-1}-(-1)^{3/4} x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {x^4+1}}{16 \left ((-1)^{3/4} x+\sqrt [16]{-1}\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {x^4+1}}{16 \left (\sqrt [16]{-1}-(-1)^{7/8} x\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {x^4+1}}{16 \left ((-1)^{7/8} x+\sqrt [16]{-1}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{16} (-1)^{7/8} \sqrt {(-1)^{3/8}-(-1)^{5/8}} \left ((1+i)+\sqrt {2}\right ) \arctan \left (\frac {\sqrt {(-1)^{3/8}-(-1)^{5/8}} x}{\sqrt {x^4+1}}\right )-\frac {(-1)^{5/8} \sqrt {(-1)^{3/8}-(-1)^{5/8}} \left (1+(-1)^{5/8}\right ) \left ((1-i)+\sqrt {2}\right ) \arctan \left (\frac {\sqrt {(-1)^{3/8}-(-1)^{5/8}} x}{\sqrt {x^4+1}}\right )}{16 \left (i-\sqrt [8]{-1}\right )}+\frac {1}{16} (-1)^{5/8} \sqrt {\sqrt [8]{-1}-(-1)^{7/8}} \left ((-1+i)+\sqrt {2}\right ) \arctan \left (\frac {\sqrt {\sqrt [8]{-1}-(-1)^{7/8}} x}{\sqrt {x^4+1}}\right )-\frac {1}{16} (-1)^{3/8} \sqrt {\sqrt [8]{-1}-(-1)^{7/8}} \left ((-1-i)+\sqrt {2}\right ) \arctan \left (\frac {\sqrt {\sqrt [8]{-1}-(-1)^{7/8}} x}{\sqrt {x^4+1}}\right )+\frac {1}{16} (-1)^{7/8} \sqrt {(-1)^{3/8}-(-1)^{5/8}} \left ((1+i)+\sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {(-1)^{3/8}-(-1)^{5/8}} x}{\sqrt {x^4+1}}\right )+\frac {(-1)^{3/4} \left (1+(-1)^{3/8}\right ) \sqrt {(-1)^{3/8}-(-1)^{5/8}} \left ((1+i)+i \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {(-1)^{3/8}-(-1)^{5/8}} x}{\sqrt {x^4+1}}\right )}{16 \left (i+\sqrt [8]{-1}\right )}+\frac {1}{16} (-1)^{5/8} \sqrt {\sqrt [8]{-1}-(-1)^{7/8}} \left ((-1+i)+\sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt [8]{-1}-(-1)^{7/8}} x}{\sqrt {x^4+1}}\right )-\frac {1}{16} (-1)^{3/8} \sqrt {\sqrt [8]{-1}-(-1)^{7/8}} \left ((-1-i)+\sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt [8]{-1}-(-1)^{7/8}} x}{\sqrt {x^4+1}}\right )+\frac {(-1)^{7/8} \left ((1+i)+\sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{8 \left (1+(-1)^{5/8}\right ) \sqrt {x^4+1}}-\frac {(-1)^{7/8} \left ((1+i)+\sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{8 \left (1-(-1)^{5/8}\right ) \sqrt {x^4+1}}+\frac {\left ((1+i)+\sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{8 \left (i+\sqrt [8]{-1}\right ) \sqrt {x^4+1}}+\frac {\left ((1+i)+\sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{8 \left (i-\sqrt [8]{-1}\right ) \sqrt {x^4+1}}+\frac {(-1)^{3/8} \left ((-1-i)+\sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{8 \left (1+\sqrt [8]{-1}\right ) \sqrt {x^4+1}}+\frac {i \left ((-1-i)+\sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{8 \left (1+\sqrt [8]{-1}\right ) \sqrt {x^4+1}}-\frac {(-1)^{3/8} \left ((-1-i)+\sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{8 \left (1-\sqrt [8]{-1}\right ) \sqrt {x^4+1}}-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (i-\sqrt [4]{-1}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{\left (1-\sqrt [8]{-1}\right ) \sqrt {x^4+1}}-\frac {\left (1-(-1)^{5/8}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} (-1)^{3/8} \left (i-\sqrt [8]{-1}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{16 \left (1+(-1)^{5/8}\right ) \sqrt {x^4+1}}-\frac {\left (2+(2+i) \sqrt [8]{-1}-(-1)^{5/8}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} (-1)^{7/8} \left (1-\sqrt [8]{-1}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{32 \left (1-\sqrt [8]{-1}\right ) \sqrt {x^4+1}}-\frac {\left (2-(11-12 i) \sqrt [8]{-1}-(10+11 i) (-1)^{5/8}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {1}{4} (-1)^{3/8} \left (i+\sqrt [8]{-1}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{32 \left (1-(-1)^{5/8}\right ) \sqrt {x^4+1}}-\frac {\left (2-(2+i) \sqrt [8]{-1}+(-1)^{5/8}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {1}{4} (-1)^{7/8} \left (1+\sqrt [8]{-1}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{32 \left (1+\sqrt [8]{-1}\right ) \sqrt {x^4+1}}+\frac {\left (2-\sqrt [8]{-1}+(2-i) (-1)^{5/8}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (2+(-1)^{3/8}-(-1)^{5/8}\right ),2 \arctan (x),\frac {1}{2}\right )}{32 \left (1-(-1)^{5/8}\right ) \sqrt {x^4+1}}+\frac {\left (2+\sqrt [8]{-1}-(2-i) (-1)^{5/8}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (2-(-1)^{3/8}+(-1)^{5/8}\right ),2 \arctan (x),\frac {1}{2}\right )}{32 \left (1+(-1)^{5/8}\right ) \sqrt {x^4+1}}+\frac {\left (2-(2-i) \sqrt [8]{-1}-(-1)^{5/8}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (2+\sqrt [8]{-1}-(-1)^{7/8}\right ),2 \arctan (x),\frac {1}{2}\right )}{32 \left (1+\sqrt [8]{-1}\right ) \sqrt {x^4+1}}+\frac {\left (2+(2-i) \sqrt [8]{-1}+(-1)^{5/8}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (2-\sqrt [8]{-1}+(-1)^{7/8}\right ),2 \arctan (x),\frac {1}{2}\right )}{32 \left (1-\sqrt [8]{-1}\right ) \sqrt {x^4+1}}\) |
-1/16*((-1)^(5/8)*Sqrt[(-1)^(3/8) - (-1)^(5/8)]*(1 + (-1)^(5/8))*((1 - I) + Sqrt[2])*ArcTan[(Sqrt[(-1)^(3/8) - (-1)^(5/8)]*x)/Sqrt[1 + x^4]])/(I - ( -1)^(1/8)) + ((-1)^(7/8)*Sqrt[(-1)^(3/8) - (-1)^(5/8)]*((1 + I) + Sqrt[2]) *ArcTan[(Sqrt[(-1)^(3/8) - (-1)^(5/8)]*x)/Sqrt[1 + x^4]])/16 - ((-1)^(3/8) *Sqrt[(-1)^(1/8) - (-1)^(7/8)]*((-1 - I) + Sqrt[2])*ArcTan[(Sqrt[(-1)^(1/8 ) - (-1)^(7/8)]*x)/Sqrt[1 + x^4]])/16 + ((-1)^(5/8)*Sqrt[(-1)^(1/8) - (-1) ^(7/8)]*((-1 + I) + Sqrt[2])*ArcTan[(Sqrt[(-1)^(1/8) - (-1)^(7/8)]*x)/Sqrt [1 + x^4]])/16 + ((-1)^(3/4)*(1 + (-1)^(3/8))*Sqrt[(-1)^(3/8) - (-1)^(5/8) ]*((1 + I) + I*Sqrt[2])*ArcTanh[(Sqrt[(-1)^(3/8) - (-1)^(5/8)]*x)/Sqrt[1 + x^4]])/(16*(I + (-1)^(1/8))) + ((-1)^(7/8)*Sqrt[(-1)^(3/8) - (-1)^(5/8)]* ((1 + I) + Sqrt[2])*ArcTanh[(Sqrt[(-1)^(3/8) - (-1)^(5/8)]*x)/Sqrt[1 + x^4 ]])/16 - ((-1)^(3/8)*Sqrt[(-1)^(1/8) - (-1)^(7/8)]*((-1 - I) + Sqrt[2])*Ar cTanh[(Sqrt[(-1)^(1/8) - (-1)^(7/8)]*x)/Sqrt[1 + x^4]])/16 + ((-1)^(5/8)*S qrt[(-1)^(1/8) - (-1)^(7/8)]*((-1 + I) + Sqrt[2])*ArcTanh[(Sqrt[(-1)^(1/8) - (-1)^(7/8)]*x)/Sqrt[1 + x^4]])/16 - ((1/8 + I/8)*(I - (-1)^(1/4))*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/((1 - (-1)^( 1/8))*Sqrt[1 + x^4]) - ((-1)^(3/8)*((-1 - I) + Sqrt[2])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(8*(1 - (-1)^(1/8))*Sqrt[ 1 + x^4]) + ((I/8)*((-1 - I) + Sqrt[2])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2) ^2]*EllipticF[2*ArcTan[x], 1/2])/((1 + (-1)^(1/8))*Sqrt[1 + x^4]) + ((-...
3.6.86.3.1 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_)), 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 = 13.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.84
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\sqrt {x^{4}+1}}{x}\right )}{\textit {\_R}}\right )}{8}\) | \(38\) |
| pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\sqrt {x^{4}+1}}{x}\right )}{\textit {\_R}}\right )}{8}\) | \(38\) |
| elliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{8}-8 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (2 \textit {\_R}^{6}-\textit {\_R}^{2}\right ) \ln \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}-\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right ) \sqrt {2}}{16}\) | \(67\) |
| trager | \(\text {Expression too large to display}\) | \(1528\) |
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.45 (sec) , antiderivative size = 1957, normalized size of antiderivative = 43.49 \[ \int \frac {-1+x^{16}}{\sqrt {1+x^4} \left (1+x^{16}\right )} \, dx=\text {Too large to display} \]
1/32*sqrt(2)*sqrt(-sqrt(2)*sqrt(sqrt(2) + 2))*log(-(2*(4*x^11 + 6*x^7 + 4* x^3 - sqrt(2)*(3*x^11 + 4*x^7 + 3*x^3))*sqrt(x^4 + 1)*sqrt(sqrt(2) + 2) + 2*(2*x^13 + 4*x^9 + 4*x^5 - sqrt(2)*(x^13 + 3*x^9 + 3*x^5 + x) + 2*x)*sqrt (x^4 + 1) + (x^16 + 8*x^12 + 12*x^8 + 8*x^4 - sqrt(2)*(x^16 + 6*x^12 + 8*x ^8 + 6*x^4 + 1) + 2*(3*x^14 + 7*x^10 + 7*x^6 + 3*x^2 - sqrt(2)*(2*x^14 + 5 *x^10 + 5*x^6 + 2*x^2))*sqrt(sqrt(2) + 2) + 1)*sqrt(-sqrt(2)*sqrt(sqrt(2) + 2)))/(x^16 + 1)) - 1/32*sqrt(2)*sqrt(-sqrt(2)*sqrt(sqrt(2) + 2))*log(-(2 *(4*x^11 + 6*x^7 + 4*x^3 - sqrt(2)*(3*x^11 + 4*x^7 + 3*x^3))*sqrt(x^4 + 1) *sqrt(sqrt(2) + 2) + 2*(2*x^13 + 4*x^9 + 4*x^5 - sqrt(2)*(x^13 + 3*x^9 + 3 *x^5 + x) + 2*x)*sqrt(x^4 + 1) - (x^16 + 8*x^12 + 12*x^8 + 8*x^4 - sqrt(2) *(x^16 + 6*x^12 + 8*x^8 + 6*x^4 + 1) + 2*(3*x^14 + 7*x^10 + 7*x^6 + 3*x^2 - sqrt(2)*(2*x^14 + 5*x^10 + 5*x^6 + 2*x^2))*sqrt(sqrt(2) + 2) + 1)*sqrt(- sqrt(2)*sqrt(sqrt(2) + 2)))/(x^16 + 1)) - 1/32*sqrt(2)*sqrt(sqrt(2)*sqrt(s qrt(2) + 2))*log((2*(4*x^11 + 6*x^7 + 4*x^3 - sqrt(2)*(3*x^11 + 4*x^7 + 3* x^3))*sqrt(x^4 + 1)*sqrt(sqrt(2) + 2) - 2*(2*x^13 + 4*x^9 + 4*x^5 - sqrt(2 )*(x^13 + 3*x^9 + 3*x^5 + x) + 2*x)*sqrt(x^4 + 1) + (x^16 + 8*x^12 + 12*x^ 8 + 8*x^4 - sqrt(2)*(x^16 + 6*x^12 + 8*x^8 + 6*x^4 + 1) - 2*(3*x^14 + 7*x^ 10 + 7*x^6 + 3*x^2 - sqrt(2)*(2*x^14 + 5*x^10 + 5*x^6 + 2*x^2))*sqrt(sqrt( 2) + 2) + 1)*sqrt(sqrt(2)*sqrt(sqrt(2) + 2)))/(x^16 + 1)) + 1/32*sqrt(2)*s qrt(sqrt(2)*sqrt(sqrt(2) + 2))*log((2*(4*x^11 + 6*x^7 + 4*x^3 - sqrt(2)...
Timed out. \[ \int \frac {-1+x^{16}}{\sqrt {1+x^4} \left (1+x^{16}\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.49 \[ \int \frac {-1+x^{16}}{\sqrt {1+x^4} \left (1+x^{16}\right )} \, dx=\int { \frac {x^{16} - 1}{{\left (x^{16} + 1\right )} \sqrt {x^{4} + 1}} \,d x } \]
Not integrable
Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.49 \[ \int \frac {-1+x^{16}}{\sqrt {1+x^4} \left (1+x^{16}\right )} \, dx=\int { \frac {x^{16} - 1}{{\left (x^{16} + 1\right )} \sqrt {x^{4} + 1}} \,d x } \]
Not integrable
Time = 5.47 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.49 \[ \int \frac {-1+x^{16}}{\sqrt {1+x^4} \left (1+x^{16}\right )} \, dx=\int \frac {x^{16}-1}{\sqrt {x^4+1}\,\left (x^{16}+1\right )} \,d x \]