Integrand size = 37, antiderivative size = 50 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )}{\sqrt [4]{x-x^4+x^7} \left (1+3 x^6+x^{12}\right )} \, dx=\frac {1}{6} \text {RootSum}\left [2+2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x-x^4+x^7}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
Time = 5.26 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.82 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )}{\sqrt [4]{x-x^4+x^7} \left (1+3 x^6+x^{12}\right )} \, dx=\frac {\sqrt [4]{x} \sqrt [4]{1-x^3+x^6} \text {RootSum}\left [2+2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-3 \log \left (\sqrt [4]{x}\right )+\log \left (\sqrt [4]{1-x^3+x^6}-x^{3/4} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 \sqrt [4]{x-x^4+x^7}} \]
(x^(1/4)*(1 - x^3 + x^6)^(1/4)*RootSum[2 + 2*#1^4 + #1^8 & , (-3*Log[x^(1/ 4)] + Log[(1 - x^3 + x^6)^(1/4) - x^(3/4)*#1])/#1 & ])/(6*(x - x^4 + x^7)^ (1/4))
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.
| \(\displaystyle \int \frac {\left (x^6-1\right ) \left (x^6+1\right )}{\sqrt [4]{x^7-x^4+x} \left (x^{12}+3 x^6+1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{x^6-x^3+1} \int -\frac {\left (1-x^6\right ) \left (x^6+1\right )}{\sqrt [4]{x} \sqrt [4]{x^6-x^3+1} \left (x^{12}+3 x^6+1\right )}dx}{\sqrt [4]{x^7-x^4+x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x} \sqrt [4]{x^6-x^3+1} \int \frac {\left (1-x^6\right ) \left (x^6+1\right )}{\sqrt [4]{x} \sqrt [4]{x^6-x^3+1} \left (x^{12}+3 x^6+1\right )}dx}{\sqrt [4]{x^7-x^4+x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{x^6-x^3+1} \int \frac {\sqrt {x} \left (1-x^6\right ) \left (x^6+1\right )}{\sqrt [4]{x^6-x^3+1} \left (x^{12}+3 x^6+1\right )}d\sqrt [4]{x}}{\sqrt [4]{x^7-x^4+x}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{x^6-x^3+1} \int \frac {\sqrt {x} \left (1-x^{12}\right )}{\sqrt [4]{x^6-x^3+1} \left (x^{12}+3 x^6+1\right )}d\sqrt [4]{x}}{\sqrt [4]{x^7-x^4+x}}\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{x^6-x^3+1} \int \frac {1-x^4}{\sqrt [4]{x^2-x+1} \left (x^4+3 x^2+1\right )}dx^{3/4}}{3 \sqrt [4]{x^7-x^4+x}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{x^6-x^3+1} \int \left (\frac {3 x^2+2}{\sqrt [4]{x^2-x+1} \left (x^4+3 x^2+1\right )}-\frac {1}{\sqrt [4]{x^2-x+1}}\right )dx^{3/4}}{3 \sqrt [4]{x^7-x^4+x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{x^6-x^3+1} \left (\frac {1}{2} i \sqrt {3-\sqrt {5}} \int \frac {1}{\left (i \sqrt {3-\sqrt {5}}-\sqrt {2} x\right ) \sqrt [4]{x^2-x+1}}dx^{3/4}+\frac {1}{2} i \sqrt {3+\sqrt {5}} \int \frac {1}{\left (i \sqrt {3+\sqrt {5}}-\sqrt {2} x\right ) \sqrt [4]{x^2-x+1}}dx^{3/4}+\frac {1}{2} i \sqrt {3-\sqrt {5}} \int \frac {1}{\left (\sqrt {2} x+i \sqrt {3-\sqrt {5}}\right ) \sqrt [4]{x^2-x+1}}dx^{3/4}+\frac {1}{2} i \sqrt {3+\sqrt {5}} \int \frac {1}{\left (\sqrt {2} x+i \sqrt {3+\sqrt {5}}\right ) \sqrt [4]{x^2-x+1}}dx^{3/4}-\frac {x^{3/4} \sqrt [4]{1-\frac {2 x}{1-i \sqrt {3}}} \sqrt [4]{1-\frac {2 x}{1+i \sqrt {3}}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{4},\frac {1}{4},\frac {5}{4},\frac {2 x}{1+i \sqrt {3}},\frac {2 x}{1-i \sqrt {3}}\right )}{\sqrt [4]{x^2-x+1}}\right )}{3 \sqrt [4]{x^7-x^4+x}}\) |
3.7.41.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
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]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1) Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function OfQ[x^(m + 1), 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]
Timed out.
\[\int \frac {\left (x^{6}-1\right ) \left (x^{6}+1\right )}{\left (x^{7}-x^{4}+x \right )^{\frac {1}{4}} \left (x^{12}+3 x^{6}+1\right )}d x\]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 25.44 (sec) , antiderivative size = 1281, normalized size of antiderivative = 25.62 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )}{\sqrt [4]{x-x^4+x^7} \left (1+3 x^6+x^{12}\right )} \, dx=\text {Too large to display} \]
-1/24*sqrt(2)*sqrt(-sqrt(2)*sqrt(-I - 1))*log((sqrt(2)*sqrt(-I - 1)*((3*I + 1)*x^12 - (4*I + 8)*x^9 + (I + 7)*x^6 - (4*I + 8)*x^3 + 3*I + 1) - 4*((3 *I + 1)*x^7 + (I - 3)*x^4 + (3*I + 1)*x)*sqrt(x^7 - x^4 + x) - 2*(sqrt(2)* (x^7 - x^4 + x)^(3/4)*((3*I + 1)*x^6 + (I - 3)*x^3 + 3*I + 1) + 2*sqrt(-I - 1)*((I + 2)*x^8 + (2*I - 1)*x^5 + (I + 2)*x^2)*(x^7 - x^4 + x)^(1/4))*sq rt(-sqrt(2)*sqrt(-I - 1)))/(x^12 + 3*x^6 + 1)) + 1/24*sqrt(2)*sqrt(-sqrt(2 )*sqrt(-I - 1))*log((sqrt(2)*sqrt(-I - 1)*((3*I + 1)*x^12 - (4*I + 8)*x^9 + (I + 7)*x^6 - (4*I + 8)*x^3 + 3*I + 1) - 4*((3*I + 1)*x^7 + (I - 3)*x^4 + (3*I + 1)*x)*sqrt(x^7 - x^4 + x) - 2*(sqrt(2)*(x^7 - x^4 + x)^(3/4)*(-(3 *I + 1)*x^6 - (I - 3)*x^3 - 3*I - 1) + 2*sqrt(-I - 1)*(-(I + 2)*x^8 - (2*I - 1)*x^5 - (I + 2)*x^2)*(x^7 - x^4 + x)^(1/4))*sqrt(-sqrt(2)*sqrt(-I - 1) ))/(x^12 + 3*x^6 + 1)) - 1/24*sqrt(2)*sqrt(-sqrt(2)*sqrt(I - 1))*log((sqrt (2)*sqrt(I - 1)*(-(3*I - 1)*x^12 + (4*I - 8)*x^9 - (I - 7)*x^6 + (4*I - 8) *x^3 - 3*I + 1) - 4*sqrt(x^7 - x^4 + x)*(-(3*I - 1)*x^7 - (I + 3)*x^4 - (3 *I - 1)*x) - 2*(sqrt(2)*(x^7 - x^4 + x)^(3/4)*(-(3*I - 1)*x^6 - (I + 3)*x^ 3 - 3*I + 1) + 2*sqrt(I - 1)*(-(I - 2)*x^8 - (2*I + 1)*x^5 - (I - 2)*x^2)* (x^7 - x^4 + x)^(1/4))*sqrt(-sqrt(2)*sqrt(I - 1)))/(x^12 + 3*x^6 + 1)) + 1 /24*sqrt(2)*sqrt(-sqrt(2)*sqrt(I - 1))*log((sqrt(2)*sqrt(I - 1)*(-(3*I - 1 )*x^12 + (4*I - 8)*x^9 - (I - 7)*x^6 + (4*I - 8)*x^3 - 3*I + 1) - 4*sqrt(x ^7 - x^4 + x)*(-(3*I - 1)*x^7 - (I + 3)*x^4 - (3*I - 1)*x) - 2*(sqrt(2)...
Timed out. \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )}{\sqrt [4]{x-x^4+x^7} \left (1+3 x^6+x^{12}\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.74 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )}{\sqrt [4]{x-x^4+x^7} \left (1+3 x^6+x^{12}\right )} \, dx=\int { \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}}{{\left (x^{12} + 3 \, x^{6} + 1\right )} {\left (x^{7} - x^{4} + x\right )}^{\frac {1}{4}}} \,d x } \]
Exception generated. \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )}{\sqrt [4]{x-x^4+x^7} \left (1+3 x^6+x^{12}\right )} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Invalid _EXT in replace_ext Error: Bad Argument Valuein tegrate((sageVARx^12+3*sageVARx^6+1)^-1*((sageVARx^7-sageVARx^4+sageVARx)^ (1/4))^-1*(s
Not integrable
Time = 6.37 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.74 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )}{\sqrt [4]{x-x^4+x^7} \left (1+3 x^6+x^{12}\right )} \, dx=\int \frac {\left (x^6-1\right )\,\left (x^6+1\right )}{\left (x^{12}+3\,x^6+1\right )\,{\left (x^7-x^4+x\right )}^{1/4}} \,d x \]