Integrand size = 27, antiderivative size = 45 \[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1-x^8+x^{16}\right )} \, dx=\frac {1}{8} \text {RootSum}\left [1+4 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt {-1+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
Time = 0.35 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1-x^8+x^{16}\right )} \, dx=\frac {1}{8} \text {RootSum}\left [1+4 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt {-1+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
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 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4-1} x^8}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^4}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1}}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^{12}}{x^{16}-x^8+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {x^4-1} \left (x^{12}+x^8+x^4+1\right )}{x^{16}-x^8+1}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4-1} x^8}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^4}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1}}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^{12}}{x^{16}-x^8+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {x^4-1} \left (x^{12}+x^8+x^4+1\right )}{x^{16}-x^8+1}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4-1} x^8}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^4}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1}}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^{12}}{x^{16}-x^8+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {x^4-1} \left (x^{12}+x^8+x^4+1\right )}{x^{16}-x^8+1}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4-1} x^8}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^4}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1}}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^{12}}{x^{16}-x^8+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {x^4-1} \left (x^{12}+x^8+x^4+1\right )}{x^{16}-x^8+1}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4-1} x^8}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^4}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1}}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^{12}}{x^{16}-x^8+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {x^4-1} \left (x^{12}+x^8+x^4+1\right )}{x^{16}-x^8+1}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4-1} x^8}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^4}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1}}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^{12}}{x^{16}-x^8+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {x^4-1} \left (x^{12}+x^8+x^4+1\right )}{x^{16}-x^8+1}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4-1} x^8}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^4}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1}}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^{12}}{x^{16}-x^8+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {x^4-1} \left (x^{12}+x^8+x^4+1\right )}{x^{16}-x^8+1}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4-1} x^8}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^4}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1}}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^{12}}{x^{16}-x^8+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {x^4-1} \left (x^{12}+x^8+x^4+1\right )}{x^{16}-x^8+1}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4-1} x^8}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^4}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1}}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^{12}}{x^{16}-x^8+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {x^4-1} \left (x^{12}+x^8+x^4+1\right )}{x^{16}-x^8+1}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4-1} x^8}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^4}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1}}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^{12}}{x^{16}-x^8+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {x^4-1} \left (x^{12}+x^8+x^4+1\right )}{x^{16}-x^8+1}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4-1} x^8}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^4}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1}}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^{12}}{x^{16}-x^8+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {x^4-1} \left (x^{12}+x^8+x^4+1\right )}{x^{16}-x^8+1}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4-1} x^8}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^4}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1}}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^{12}}{x^{16}-x^8+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {x^4-1} \left (x^{12}+x^8+x^4+1\right )}{x^{16}-x^8+1}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4-1} x^8}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^4}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1}}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^{12}}{x^{16}-x^8+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {x^4-1} \left (x^{12}+x^8+x^4+1\right )}{x^{16}-x^8+1}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4-1} x^8}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^4}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1}}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^{12}}{x^{16}-x^8+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {x^4-1} \left (x^{12}+x^8+x^4+1\right )}{x^{16}-x^8+1}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4-1} x^8}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^4}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1}}{x^{16}-x^8+1}+\frac {\sqrt {x^4-1} x^{12}}{x^{16}-x^8+1}\right )dx\) |
3.6.87.3.1 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_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, 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 = 12.41 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.40
| method | result | size |
| elliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (16 \textit {\_Z}^{8}+16 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (2 \textit {\_R}^{6}+\textit {\_R}^{2}\right ) \ln \left (\frac {\sqrt {x^{4}-1}\, \sqrt {2}}{2 x}-\textit {\_R} \right )}{2 \textit {\_R}^{7}+\textit {\_R}^{3}}\right ) \sqrt {2}}{16}\) | \(63\) |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-16 i \textit {\_Z}^{14}-48 \textit {\_Z}^{12}-64 i \textit {\_Z}^{10}-160 \textit {\_Z}^{8}+256 i \textit {\_Z}^{6}-768 \textit {\_Z}^{4}+1024 i \textit {\_Z}^{2}+256\right )}{\sum }\frac {\left (-\textit {\_R}^{14}+10 i \textit {\_R}^{12}+4 \textit {\_R}^{10}+24 i \textit {\_R}^{8}-48 \textit {\_R}^{6}-32 i \textit {\_R}^{4}-320 \textit {\_R}^{2}+128 i\right ) \ln \left (\frac {-\textit {\_R} x -x^{2}-i+\sqrt {x^{4}-1}}{x}\right )}{\textit {\_R} \left (-\textit {\_R}^{14}+14 i \textit {\_R}^{12}+36 \textit {\_R}^{10}+40 i \textit {\_R}^{8}+80 \textit {\_R}^{6}-96 i \textit {\_R}^{4}+192 \textit {\_R}^{2}-128 i\right )}\right )}{8}\) | \(163\) |
| pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-16 i \textit {\_Z}^{14}-48 \textit {\_Z}^{12}-64 i \textit {\_Z}^{10}-160 \textit {\_Z}^{8}+256 i \textit {\_Z}^{6}-768 \textit {\_Z}^{4}+1024 i \textit {\_Z}^{2}+256\right )}{\sum }\frac {\left (-\textit {\_R}^{14}+10 i \textit {\_R}^{12}+4 \textit {\_R}^{10}+24 i \textit {\_R}^{8}-48 \textit {\_R}^{6}-32 i \textit {\_R}^{4}-320 \textit {\_R}^{2}+128 i\right ) \ln \left (\frac {-\textit {\_R} x -x^{2}-i+\sqrt {x^{4}-1}}{x}\right )}{\textit {\_R} \left (-\textit {\_R}^{14}+14 i \textit {\_R}^{12}+36 \textit {\_R}^{10}+40 i \textit {\_R}^{8}+80 \textit {\_R}^{6}-96 i \textit {\_R}^{4}+192 \textit {\_R}^{2}-128 i\right )}\right )}{8}\) | \(163\) |
| trager | \(\text {Expression too large to display}\) | \(920\) |
1/16*sum((2*_R^6+_R^2)/(2*_R^7+_R^3)*ln(1/2*(x^4-1)^(1/2)*2^(1/2)/x-_R),_R =RootOf(16*_Z^8+16*_Z^4+1))*2^(1/2)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.46 (sec) , antiderivative size = 1933, normalized size of antiderivative = 42.96 \[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1-x^8+x^{16}\right )} \, dx=\text {Too large to display} \]
-1/16*sqrt(-sqrt(sqrt(3) - 2))*log(-(2*(7*x^11 - 12*x^7 + 7*x^3 + sqrt(3)* (4*x^11 - 7*x^7 + 4*x^3))*sqrt(x^4 - 1)*sqrt(sqrt(3) - 2) + 2*(2*x^13 - 5* x^9 + 5*x^5 + sqrt(3)*(x^13 - 3*x^9 + 3*x^5 - x) - 2*x)*sqrt(x^4 - 1) + (2 *x^16 - 14*x^12 + 22*x^8 - 14*x^4 + sqrt(3)*(x^16 - 8*x^12 + 13*x^8 - 8*x^ 4 + 1) + 2*(7*x^14 - 19*x^10 + 19*x^6 - 7*x^2 + sqrt(3)*(4*x^14 - 11*x^10 + 11*x^6 - 4*x^2))*sqrt(sqrt(3) - 2) + 2)*sqrt(-sqrt(sqrt(3) - 2)))/(x^16 - x^8 + 1)) + 1/16*sqrt(-sqrt(sqrt(3) - 2))*log(-(2*(7*x^11 - 12*x^7 + 7*x ^3 + sqrt(3)*(4*x^11 - 7*x^7 + 4*x^3))*sqrt(x^4 - 1)*sqrt(sqrt(3) - 2) + 2 *(2*x^13 - 5*x^9 + 5*x^5 + sqrt(3)*(x^13 - 3*x^9 + 3*x^5 - x) - 2*x)*sqrt( x^4 - 1) - (2*x^16 - 14*x^12 + 22*x^8 - 14*x^4 + sqrt(3)*(x^16 - 8*x^12 + 13*x^8 - 8*x^4 + 1) + 2*(7*x^14 - 19*x^10 + 19*x^6 - 7*x^2 + sqrt(3)*(4*x^ 14 - 11*x^10 + 11*x^6 - 4*x^2))*sqrt(sqrt(3) - 2) + 2)*sqrt(-sqrt(sqrt(3) - 2)))/(x^16 - x^8 + 1)) - 1/16*sqrt(-sqrt(-sqrt(3) - 2))*log(-(2*(2*x^13 - 5*x^9 + 5*x^5 - sqrt(3)*(x^13 - 3*x^9 + 3*x^5 - x) + (7*x^11 - 12*x^7 + 7*x^3 - sqrt(3)*(4*x^11 - 7*x^7 + 4*x^3))*sqrt(-sqrt(3) - 2) - 2*x)*sqrt(x ^4 - 1) + (2*x^16 - 14*x^12 + 22*x^8 - 14*x^4 - sqrt(3)*(x^16 - 8*x^12 + 1 3*x^8 - 8*x^4 + 1) + 2*(7*x^14 - 19*x^10 + 19*x^6 - 7*x^2 - sqrt(3)*(4*x^1 4 - 11*x^10 + 11*x^6 - 4*x^2))*sqrt(-sqrt(3) - 2) + 2)*sqrt(-sqrt(-sqrt(3) - 2)))/(x^16 - x^8 + 1)) + 1/16*sqrt(-sqrt(-sqrt(3) - 2))*log(-(2*(2*x^13 - 5*x^9 + 5*x^5 - sqrt(3)*(x^13 - 3*x^9 + 3*x^5 - x) + (7*x^11 - 12*x^...
Timed out. \[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1-x^8+x^{16}\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 0.33 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.60 \[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1-x^8+x^{16}\right )} \, dx=\int { \frac {x^{16} - 1}{{\left (x^{16} - x^{8} + 1\right )} \sqrt {x^{4} - 1}} \,d x } \]
Not integrable
Time = 0.59 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.60 \[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1-x^8+x^{16}\right )} \, dx=\int { \frac {x^{16} - 1}{{\left (x^{16} - x^{8} + 1\right )} \sqrt {x^{4} - 1}} \,d x } \]
Not integrable
Time = 5.79 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.60 \[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1-x^8+x^{16}\right )} \, dx=\int \frac {x^{16}-1}{\sqrt {x^4-1}\,\left (x^{16}-x^8+1\right )} \,d x \]