Integrand size = 46, antiderivative size = 196 \[ \int \frac {1}{x^2 \sqrt [3]{-4-8 x+11 x^2+17 x^3-20 x^4-7 x^5+16 x^6-7 x^7+x^8}} \, dx=\frac {(-2+x)^{2/3} \left (-1-x+x^2\right ) \left (-\frac {\sqrt [3]{-2+x}}{2 x}-\frac {\sqrt [3]{2} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} \sqrt [3]{-2+x}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt [3]{2} \log \left (2+2^{2/3} \sqrt [3]{-2+x}\right )-\frac {\log \left (-2+2^{2/3} \sqrt [3]{-2+x}-\sqrt [3]{2} (-2+x)^{2/3}\right )}{3\ 2^{2/3}}-\text {RootSum}\left [1+3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log \left (\sqrt [3]{-2+x}-\text {$\#$1}\right ) \text {$\#$1}}{3+2 \text {$\#$1}^3}\&\right ]\right )}{\sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}} \] Output:
Unintegrable
Time = 0.25 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^2 \sqrt [3]{-4-8 x+11 x^2+17 x^3-20 x^4-7 x^5+16 x^6-7 x^7+x^8}} \, dx=-\frac {\left (-1-x+x^2\right ) \left (-6+3 x+2 \sqrt [3]{2} \sqrt {3} (-2+x)^{2/3} x \arctan \left (\frac {1-2^{2/3} \sqrt [3]{-2+x}}{\sqrt {3}}\right )-2 \sqrt [3]{2} (-2+x)^{2/3} x \log \left (2+2^{2/3} \sqrt [3]{-2+x}\right )+\sqrt [3]{2} (-2+x)^{2/3} x \log \left (-2+2^{2/3} \sqrt [3]{-2+x}-\sqrt [3]{2} (-2+x)^{2/3}\right )+6 (-2+x)^{2/3} x \text {RootSum}\left [1+3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log \left (\sqrt [3]{-2+x}-\text {$\#$1}\right ) \text {$\#$1}}{3+2 \text {$\#$1}^3}\&\right ]\right )}{6 x \sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}} \] Input:
Integrate[1/(x^2*(-4 - 8*x + 11*x^2 + 17*x^3 - 20*x^4 - 7*x^5 + 16*x^6 - 7 *x^7 + x^8)^(1/3)),x]
Output:
-1/6*((-1 - x + x^2)*(-6 + 3*x + 2*2^(1/3)*Sqrt[3]*(-2 + x)^(2/3)*x*ArcTan [(1 - 2^(2/3)*(-2 + x)^(1/3))/Sqrt[3]] - 2*2^(1/3)*(-2 + x)^(2/3)*x*Log[2 + 2^(2/3)*(-2 + x)^(1/3)] + 2^(1/3)*(-2 + x)^(2/3)*x*Log[-2 + 2^(2/3)*(-2 + x)^(1/3) - 2^(1/3)*(-2 + x)^(2/3)] + 6*(-2 + x)^(2/3)*x*RootSum[1 + 3*#1 ^3 + #1^6 & , (Log[(-2 + x)^(1/3) - #1]*#1)/(3 + 2*#1^3) & ]))/(x*((-2 + x )^2*(-1 - x + x^2)^3)^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(518\) vs. \(2(196)=392\).
Time = 1.24 (sec) , antiderivative size = 518, normalized size of antiderivative = 2.64, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {7239, 7270, 25, 1199, 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.
\(\displaystyle \int \frac {1}{x^2 \sqrt [3]{x^8-7 x^7+16 x^6-7 x^5-20 x^4+17 x^3+11 x^2-8 x-4}} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {1}{x^2 \sqrt [3]{(x-2)^2 \left (x^2-x-1\right )^3}}dx\) |
\(\Big \downarrow \) 7270 |
\(\displaystyle -\frac {(x-2)^{2/3} \left (-x^2+x+1\right ) \int -\frac {1}{(x-2)^{2/3} x^2 \left (-x^2+x+1\right )}dx}{\sqrt [3]{-(2-x)^2 \left (-x^2+x+1\right )^3}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {(x-2)^{2/3} \left (-x^2+x+1\right ) \int \frac {1}{(x-2)^{2/3} x^2 \left (-x^2+x+1\right )}dx}{\sqrt [3]{-(2-x)^2 \left (-x^2+x+1\right )^3}}\) |
\(\Big \downarrow \) 1199 |
\(\displaystyle \frac {3 (x-2)^{2/3} \left (-x^2+x+1\right ) \int \left (\frac {x-2}{(x-2)^2+3 (x-2)+1}-\frac {1}{x}+\frac {1}{x^2}\right )d\sqrt [3]{x-2}}{\sqrt [3]{-(2-x)^2 \left (-x^2+x+1\right )^3}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 (x-2)^{2/3} \left (-x^2+x+1\right ) \left (\frac {\sqrt [3]{2} \arctan \left (\frac {1-2^{2/3} \sqrt [3]{x-2}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\sqrt [3]{\frac {1}{2} \left (3+\sqrt {5}\right )} \arctan \left (\frac {1-2 \sqrt [3]{\frac {2}{3+\sqrt {5}}} \sqrt [3]{x-2}}{\sqrt {3}}\right )}{\sqrt {15}}+\frac {\sqrt [3]{\frac {1}{2} \left (3-\sqrt {5}\right )} \arctan \left (\frac {1-2^{2/3} \sqrt [3]{3+\sqrt {5}} \sqrt [3]{x-2}}{\sqrt {3}}\right )}{\sqrt {15}}+\frac {\sqrt [3]{x-2}}{6 x}-\frac {1}{9} \sqrt [3]{2} \log \left (\sqrt [3]{x-2}+\sqrt [3]{2}\right )-\frac {\sqrt [3]{\frac {1}{2} \left (3-\sqrt {5}\right )} \log \left (\sqrt [3]{2} \sqrt [3]{x-2}+\sqrt [3]{3-\sqrt {5}}\right )}{3 \sqrt {5}}+\frac {\sqrt [3]{\frac {1}{2} \left (3+\sqrt {5}\right )} \log \left (\sqrt [3]{2} \sqrt [3]{x-2}+\sqrt [3]{3+\sqrt {5}}\right )}{3 \sqrt {5}}+\frac {\log \left ((x-2)^{2/3}-\sqrt [3]{2} \sqrt [3]{x-2}+2^{2/3}\right )}{9\ 2^{2/3}}+\frac {\sqrt [3]{\frac {1}{2} \left (3-\sqrt {5}\right )} \log \left (2^{2/3} (x-2)^{2/3}-\sqrt [3]{2 \left (3-\sqrt {5}\right )} \sqrt [3]{x-2}+\left (3-\sqrt {5}\right )^{2/3}\right )}{6 \sqrt {5}}-\frac {\sqrt [3]{\frac {1}{2} \left (3+\sqrt {5}\right )} \log \left (2^{2/3} (x-2)^{2/3}-\sqrt [3]{2 \left (3+\sqrt {5}\right )} \sqrt [3]{x-2}+\left (3+\sqrt {5}\right )^{2/3}\right )}{6 \sqrt {5}}\right )}{\sqrt [3]{-(2-x)^2 \left (-x^2+x+1\right )^3}}\) |
Input:
Int[1/(x^2*(-4 - 8*x + 11*x^2 + 17*x^3 - 20*x^4 - 7*x^5 + 16*x^6 - 7*x^7 + x^8)^(1/3)),x]
Output:
(3*(-2 + x)^(2/3)*(1 + x - x^2)*((-2 + x)^(1/3)/(6*x) + (2^(1/3)*ArcTan[(1 - 2^(2/3)*(-2 + x)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) - (((3 + Sqrt[5])/2)^(1/3 )*ArcTan[(1 - 2*(2/(3 + Sqrt[5]))^(1/3)*(-2 + x)^(1/3))/Sqrt[3]])/Sqrt[15] + (((3 - Sqrt[5])/2)^(1/3)*ArcTan[(1 - 2^(2/3)*(3 + Sqrt[5])^(1/3)*(-2 + x)^(1/3))/Sqrt[3]])/Sqrt[15] - (2^(1/3)*Log[2^(1/3) + (-2 + x)^(1/3)])/9 - (((3 - Sqrt[5])/2)^(1/3)*Log[(3 - Sqrt[5])^(1/3) + 2^(1/3)*(-2 + x)^(1/3) ])/(3*Sqrt[5]) + (((3 + Sqrt[5])/2)^(1/3)*Log[(3 + Sqrt[5])^(1/3) + 2^(1/3 )*(-2 + x)^(1/3)])/(3*Sqrt[5]) + Log[2^(2/3) - 2^(1/3)*(-2 + x)^(1/3) + (- 2 + x)^(2/3)]/(9*2^(2/3)) + (((3 - Sqrt[5])/2)^(1/3)*Log[(3 - Sqrt[5])^(2/ 3) - (2*(3 - Sqrt[5]))^(1/3)*(-2 + x)^(1/3) + 2^(2/3)*(-2 + x)^(2/3)])/(6* Sqrt[5]) - (((3 + Sqrt[5])/2)^(1/3)*Log[(3 + Sqrt[5])^(2/3) - (2*(3 + Sqrt [5]))^(1/3)*(-2 + x)^(1/3) + 2^(2/3)*(-2 + x)^(2/3)])/(6*Sqrt[5])))/(-((2 - x)^2*(1 + x - x^2)^3))^(1/3)
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x _) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Denominator[m]}, Simp[q/e Subs t[Int[ExpandIntegrand[x^(q*(m + 1) - 1)*(((e*f - d*g)/e + g*(x^q/e))^n/((c* d^2 - b*d*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))), x], x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Integer Q[n] && FractionQ[m]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
Time = 46.78 (sec) , antiderivative size = 19088, normalized size of antiderivative = 97.39
method | result | size |
risch | \(\text {Expression too large to display}\) | \(19088\) |
trager | \(\text {Expression too large to display}\) | \(70041\) |
Input:
int(1/x^2/(x^8-7*x^7+16*x^6-7*x^5-20*x^4+17*x^3+11*x^2-8*x-4)^(1/3),x,meth od=_RETURNVERBOSE)
Output:
result too large to display
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.14 (sec) , antiderivative size = 1570, normalized size of antiderivative = 8.01 \[ \int \frac {1}{x^2 \sqrt [3]{-4-8 x+11 x^2+17 x^3-20 x^4-7 x^5+16 x^6-7 x^7+x^8}} \, dx=\text {Too large to display} \] Input:
integrate(1/x^2/(x^8-7*x^7+16*x^6-7*x^5-20*x^4+17*x^3+11*x^2-8*x-4)^(1/3), x, algorithm="fricas")
Output:
-1/6*(2*sqrt(3)*2^(1/3)*(x^6 - 4*x^5 + 3*x^4 + 4*x^3 - 3*x^2 - 2*x)*arctan (-1/3*(sqrt(3)*(x^3 - 3*x^2 + x + 2) - 2*sqrt(3)*2^(1/3)*(x^8 - 7*x^7 + 16 *x^6 - 7*x^5 - 20*x^4 + 17*x^3 + 11*x^2 - 8*x - 4)^(1/3))/(x^3 - 3*x^2 + x + 2)) + 2^(1/3)*(x^6 - 4*x^5 + 3*x^4 + 4*x^3 - 3*x^2 - 2*x)*log(-(2^(2/3) *(x^8 - 7*x^7 + 16*x^6 - 7*x^5 - 20*x^4 + 17*x^3 + 11*x^2 - 8*x - 4)^(1/3) *(x^3 - 3*x^2 + x + 2) - 2^(1/3)*(x^6 - 6*x^5 + 11*x^4 - 2*x^3 - 11*x^2 + 4*x + 4) - 2*(x^8 - 7*x^7 + 16*x^6 - 7*x^5 - 20*x^4 + 17*x^3 + 11*x^2 - 8* x - 4)^(2/3))/(x^6 - 6*x^5 + 11*x^4 - 2*x^3 - 11*x^2 + 4*x + 4)) - 2*2^(1/ 3)*(x^6 - 4*x^5 + 3*x^4 + 4*x^3 - 3*x^2 - 2*x)*log((2^(2/3)*(x^3 - 3*x^2 + x + 2) + 2*(x^8 - 7*x^7 + 16*x^6 - 7*x^5 - 20*x^4 + 17*x^3 + 11*x^2 - 8*x - 4)^(1/3))/(x^3 - 3*x^2 + x + 2)) + 3*(x^6 - 4*x^5 + 3*x^4 + 4*x^3 - 3*x ^2 - sqrt(-3)*(x^6 - 4*x^5 + 3*x^4 + 4*x^3 - 3*x^2 - 2*x) - 2*x)*(3/50*sqr t(5) - 1/10)^(1/3)*log(-(5*(3*x^3 - 9*x^2 + sqrt(5)*(x^3 - 3*x^2 + sqrt(-3 )*(x^3 - 3*x^2 + x + 2) + x + 2) + 3*sqrt(-3)*(x^3 - 3*x^2 + x + 2) + 3*x + 6)*(3/50*sqrt(5) - 1/10)^(2/3) - 4*(x^8 - 7*x^7 + 16*x^6 - 7*x^5 - 20*x^ 4 + 17*x^3 + 11*x^2 - 8*x - 4)^(1/3))/(x^3 - 3*x^2 + x + 2)) + 3*(x^6 - 4* x^5 + 3*x^4 + 4*x^3 - 3*x^2 + sqrt(-3)*(x^6 - 4*x^5 + 3*x^4 + 4*x^3 - 3*x^ 2 - 2*x) - 2*x)*(3/50*sqrt(5) - 1/10)^(1/3)*log(-(5*(3*x^3 - 9*x^2 + sqrt( 5)*(x^3 - 3*x^2 - sqrt(-3)*(x^3 - 3*x^2 + x + 2) + x + 2) - 3*sqrt(-3)*(x^ 3 - 3*x^2 + x + 2) + 3*x + 6)*(3/50*sqrt(5) - 1/10)^(2/3) - 4*(x^8 - 7*...
Not integrable
Time = 0.91 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.11 \[ \int \frac {1}{x^2 \sqrt [3]{-4-8 x+11 x^2+17 x^3-20 x^4-7 x^5+16 x^6-7 x^7+x^8}} \, dx=\int \frac {1}{x^{2} \sqrt [3]{\left (x - 2\right )^{2} \left (x^{2} - x - 1\right )^{3}}}\, dx \] Input:
integrate(1/x**2/(x**8-7*x**7+16*x**6-7*x**5-20*x**4+17*x**3+11*x**2-8*x-4 )**(1/3),x)
Output:
Integral(1/(x**2*((x - 2)**2*(x**2 - x - 1)**3)**(1/3)), x)
Not integrable
Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.23 \[ \int \frac {1}{x^2 \sqrt [3]{-4-8 x+11 x^2+17 x^3-20 x^4-7 x^5+16 x^6-7 x^7+x^8}} \, dx=\int { \frac {1}{{\left (x^{8} - 7 \, x^{7} + 16 \, x^{6} - 7 \, x^{5} - 20 \, x^{4} + 17 \, x^{3} + 11 \, x^{2} - 8 \, x - 4\right )}^{\frac {1}{3}} x^{2}} \,d x } \] Input:
integrate(1/x^2/(x^8-7*x^7+16*x^6-7*x^5-20*x^4+17*x^3+11*x^2-8*x-4)^(1/3), x, algorithm="maxima")
Output:
integrate(1/((x^8 - 7*x^7 + 16*x^6 - 7*x^5 - 20*x^4 + 17*x^3 + 11*x^2 - 8* x - 4)^(1/3)*x^2), x)
Not integrable
Time = 0.71 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.23 \[ \int \frac {1}{x^2 \sqrt [3]{-4-8 x+11 x^2+17 x^3-20 x^4-7 x^5+16 x^6-7 x^7+x^8}} \, dx=\int { \frac {1}{{\left (x^{8} - 7 \, x^{7} + 16 \, x^{6} - 7 \, x^{5} - 20 \, x^{4} + 17 \, x^{3} + 11 \, x^{2} - 8 \, x - 4\right )}^{\frac {1}{3}} x^{2}} \,d x } \] Input:
integrate(1/x^2/(x^8-7*x^7+16*x^6-7*x^5-20*x^4+17*x^3+11*x^2-8*x-4)^(1/3), x, algorithm="giac")
Output:
integrate(1/((x^8 - 7*x^7 + 16*x^6 - 7*x^5 - 20*x^4 + 17*x^3 + 11*x^2 - 8* x - 4)^(1/3)*x^2), x)
Not integrable
Time = 8.54 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.23 \[ \int \frac {1}{x^2 \sqrt [3]{-4-8 x+11 x^2+17 x^3-20 x^4-7 x^5+16 x^6-7 x^7+x^8}} \, dx=\int \frac {1}{x^2\,{\left (x^8-7\,x^7+16\,x^6-7\,x^5-20\,x^4+17\,x^3+11\,x^2-8\,x-4\right )}^{1/3}} \,d x \] Input:
int(1/(x^2*(11*x^2 - 8*x + 17*x^3 - 20*x^4 - 7*x^5 + 16*x^6 - 7*x^7 + x^8 - 4)^(1/3)),x)
Output:
int(1/(x^2*(11*x^2 - 8*x + 17*x^3 - 20*x^4 - 7*x^5 + 16*x^6 - 7*x^7 + x^8 - 4)^(1/3)), x)
Not integrable
Time = 0.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.25 \[ \int \frac {1}{x^2 \sqrt [3]{-4-8 x+11 x^2+17 x^3-20 x^4-7 x^5+16 x^6-7 x^7+x^8}} \, dx=\int \frac {1}{\left (x^{2}-4 x +4\right )^{\frac {1}{3}} x^{4}-\left (x^{2}-4 x +4\right )^{\frac {1}{3}} x^{3}-\left (x^{2}-4 x +4\right )^{\frac {1}{3}} x^{2}}d x \] Input:
int(1/x^2/(x^8-7*x^7+16*x^6-7*x^5-20*x^4+17*x^3+11*x^2-8*x-4)^(1/3),x)
Output:
int(1/((x**2 - 4*x + 4)**(1/3)*x**4 - (x**2 - 4*x + 4)**(1/3)*x**3 - (x**2 - 4*x + 4)**(1/3)*x**2),x)