Integrand size = 44, antiderivative size = 175 \[ \int \frac {1}{x^2 \sqrt [3]{-1+5 x-7 x^2-2 x^3+10 x^4-2 x^5-5 x^6+x^7+x^8}} \, dx=\frac {(-1+x)^{2/3} \left (-1+x+x^2\right ) \left (-\frac {\sqrt [3]{-1+x}}{x}+\frac {5 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {5}{3} \log \left (1+\sqrt [3]{-1+x}\right )+\frac {5}{6} \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )+\text {RootSum}\left [1+3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {3 \log \left (\sqrt [3]{-1+x}-\text {$\#$1}\right )+\log \left (\sqrt [3]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}^3}{3 \text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]\right )}{\sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}} \] Output:
Unintegrable
Time = 0.16 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x^2 \sqrt [3]{-1+5 x-7 x^2-2 x^3+10 x^4-2 x^5-5 x^6+x^7+x^8}} \, dx=\frac {\left (-1+x+x^2\right ) \left (6-6 x+10 \sqrt {3} (-1+x)^{2/3} x \arctan \left (\frac {1-2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )-10 (-1+x)^{2/3} x \log \left (1+\sqrt [3]{-1+x}\right )+5 (-1+x)^{2/3} x \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )+6 (-1+x)^{2/3} x \text {RootSum}\left [1+3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {3 \log \left (\sqrt [3]{-1+x}-\text {$\#$1}\right )+\log \left (\sqrt [3]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}^3}{3 \text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]\right )}{6 x \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}} \] Input:
Integrate[1/(x^2*(-1 + 5*x - 7*x^2 - 2*x^3 + 10*x^4 - 2*x^5 - 5*x^6 + x^7 + x^8)^(1/3)),x]
Output:
((-1 + x + x^2)*(6 - 6*x + 10*Sqrt[3]*(-1 + x)^(2/3)*x*ArcTan[(1 - 2*(-1 + x)^(1/3))/Sqrt[3]] - 10*(-1 + x)^(2/3)*x*Log[1 + (-1 + x)^(1/3)] + 5*(-1 + x)^(2/3)*x*Log[1 - (-1 + x)^(1/3) + (-1 + x)^(2/3)] + 6*(-1 + x)^(2/3)*x *RootSum[1 + 3*#1^3 + #1^6 & , (3*Log[(-1 + x)^(1/3) - #1] + Log[(-1 + x)^ (1/3) - #1]*#1^3)/(3*#1^2 + 2*#1^5) & ]))/(6*x*((-1 + x)^2*(-1 + x + x^2)^ 3)^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(529\) vs. \(2(175)=350\).
Time = 1.15 (sec) , antiderivative size = 529, normalized size of antiderivative = 3.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, 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.
\(\displaystyle \int \frac {1}{x^2 \sqrt [3]{x^8+x^7-5 x^6-2 x^5+10 x^4-2 x^3-7 x^2+5 x-1}} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {1}{x^2 \sqrt [3]{(x-1)^2 \left (x^2+x-1\right )^3}}dx\) |
\(\Big \downarrow \) 7270 |
\(\displaystyle -\frac {(x-1)^{2/3} \left (-x^2-x+1\right ) \int -\frac {1}{(x-1)^{2/3} x^2 \left (-x^2-x+1\right )}dx}{\sqrt [3]{-(1-x)^2 \left (-x^2-x+1\right )^3}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {(x-1)^{2/3} \left (-x^2-x+1\right ) \int \frac {1}{(x-1)^{2/3} x^2 \left (-x^2-x+1\right )}dx}{\sqrt [3]{-(1-x)^2 \left (-x^2-x+1\right )^3}}\) |
\(\Big \downarrow \) 1199 |
\(\displaystyle \frac {3 (x-1)^{2/3} \left (-x^2-x+1\right ) \int \left (\frac {9-5 \sqrt [3]{x-1}}{9 \left ((x-1)^{2/3}-\sqrt [3]{x-1}+1\right )}-\frac {x+2}{(x-1)^2+3 (x-1)+1}+\frac {5}{9 \left (\sqrt [3]{x-1}+1\right )}+\frac {1}{9 \left (\sqrt [3]{x-1}+1\right )^2}+\frac {1-\sqrt [3]{x-1}}{3 \left ((x-1)^{2/3}-\sqrt [3]{x-1}+1\right )^2}\right )d\sqrt [3]{x-1}}{\sqrt [3]{-(1-x)^2 \left (-x^2-x+1\right )^3}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 (x-1)^{2/3} \left (-x^2-x+1\right ) \left (-\frac {5 \arctan \left (\frac {1-2 \sqrt [3]{x-1}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\sqrt [3]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} \arctan \left (\frac {1-2 \sqrt [3]{\frac {2}{3+\sqrt {5}}} \sqrt [3]{x-1}}{\sqrt {3}}\right )}{\sqrt {15}}+\frac {\left (3+\sqrt {5}\right )^{5/3} \arctan \left (\frac {1-2^{2/3} \sqrt [3]{3+\sqrt {5}} \sqrt [3]{x-1}}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {15}}+\frac {\sqrt [3]{x-1}+1}{9 \left ((x-1)^{2/3}-\sqrt [3]{x-1}+1\right )}-\frac {1}{9 \left (\sqrt [3]{x-1}+1\right )}+\frac {5}{9} \log \left (\sqrt [3]{x-1}+1\right )-\frac {\left (3+\sqrt {5}\right )^{5/3} \log \left (\sqrt [3]{2} \sqrt [3]{x-1}+\sqrt [3]{3-\sqrt {5}}\right )}{6\ 2^{2/3} \sqrt {5}}+\frac {\sqrt [3]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} \log \left (\sqrt [3]{2} \sqrt [3]{x-1}+\sqrt [3]{3+\sqrt {5}}\right )}{3 \sqrt {5}}-\frac {5}{18} \log \left ((x-1)^{2/3}-\sqrt [3]{x-1}+1\right )+\frac {\left (3+\sqrt {5}\right )^{5/3} \log \left (2^{2/3} (x-1)^{2/3}-\sqrt [3]{2 \left (3-\sqrt {5}\right )} \sqrt [3]{x-1}+\left (3-\sqrt {5}\right )^{2/3}\right )}{12\ 2^{2/3} \sqrt {5}}-\frac {\sqrt [3]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} \log \left (2^{2/3} (x-1)^{2/3}-\sqrt [3]{2 \left (3+\sqrt {5}\right )} \sqrt [3]{x-1}+\left (3+\sqrt {5}\right )^{2/3}\right )}{6 \sqrt {5}}\right )}{\sqrt [3]{-(1-x)^2 \left (-x^2-x+1\right )^3}}\) |
Input:
Int[1/(x^2*(-1 + 5*x - 7*x^2 - 2*x^3 + 10*x^4 - 2*x^5 - 5*x^6 + x^7 + x^8) ^(1/3)),x]
Output:
(3*(-1 + x)^(2/3)*(1 - x - x^2)*(-1/9*1/(1 + (-1 + x)^(1/3)) + (1 + (-1 + x)^(1/3))/(9*(1 - (-1 + x)^(1/3) + (-1 + x)^(2/3))) - (5*ArcTan[(1 - 2*(-1 + x)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) - (((123 - 55*Sqrt[5])/2)^(1/3)*ArcTan[ (1 - 2*(2/(3 + Sqrt[5]))^(1/3)*(-1 + x)^(1/3))/Sqrt[3]])/Sqrt[15] + ((3 + Sqrt[5])^(5/3)*ArcTan[(1 - 2^(2/3)*(3 + Sqrt[5])^(1/3)*(-1 + x)^(1/3))/Sqr t[3]])/(2*2^(2/3)*Sqrt[15]) + (5*Log[1 + (-1 + x)^(1/3)])/9 - ((3 + Sqrt[5 ])^(5/3)*Log[(3 - Sqrt[5])^(1/3) + 2^(1/3)*(-1 + x)^(1/3)])/(6*2^(2/3)*Sqr t[5]) + (((123 - 55*Sqrt[5])/2)^(1/3)*Log[(3 + Sqrt[5])^(1/3) + 2^(1/3)*(- 1 + x)^(1/3)])/(3*Sqrt[5]) - (5*Log[1 - (-1 + x)^(1/3) + (-1 + x)^(2/3)])/ 18 + ((3 + Sqrt[5])^(5/3)*Log[(3 - Sqrt[5])^(2/3) - (2*(3 - Sqrt[5]))^(1/3 )*(-1 + x)^(1/3) + 2^(2/3)*(-1 + x)^(2/3)])/(12*2^(2/3)*Sqrt[5]) - (((123 - 55*Sqrt[5])/2)^(1/3)*Log[(3 + Sqrt[5])^(2/3) - (2*(3 + Sqrt[5]))^(1/3)*( -1 + x)^(1/3) + 2^(2/3)*(-1 + x)^(2/3)])/(6*Sqrt[5])))/(-((1 - 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 = 35.40 (sec) , antiderivative size = 11593, normalized size of antiderivative = 66.25
method | result | size |
risch | \(\text {Expression too large to display}\) | \(11593\) |
trager | \(\text {Expression too large to display}\) | \(98808\) |
Input:
int(1/x^2/(x^8+x^7-5*x^6-2*x^5+10*x^4-2*x^3-7*x^2+5*x-1)^(1/3),x,method=_R ETURNVERBOSE)
Output:
result too large to display
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.10 (sec) , antiderivative size = 1375, normalized size of antiderivative = 7.86 \[ \int \frac {1}{x^2 \sqrt [3]{-1+5 x-7 x^2-2 x^3+10 x^4-2 x^5-5 x^6+x^7+x^8}} \, dx=\text {Too large to display} \] Input:
integrate(1/x^2/(x^8+x^7-5*x^6-2*x^5+10*x^4-2*x^3-7*x^2+5*x-1)^(1/3),x, al gorithm="fricas")
Output:
1/6*(10*sqrt(3)*(x^6 + x^5 - 3*x^4 - x^3 + 3*x^2 - x)*arctan(-1/3*(sqrt(3) *(x^3 - 2*x + 1) - 2*sqrt(3)*(x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1)^(1/3))/(x^3 - 2*x + 1)) - 3*(x^6 + x^5 - 3*x^4 - x^3 + 3 *x^2 - sqrt(-3)*(x^6 + x^5 - 3*x^4 - x^3 + 3*x^2 - x) - x)*(123/50*sqrt(5) + 11/2)^(1/3)*log(-(5*(9*x^3 - 4*sqrt(5)*(x^3 + sqrt(-3)*(x^3 - 2*x + 1) - 2*x + 1) + 9*sqrt(-3)*(x^3 - 2*x + 1) - 18*x + 9)*(123/50*sqrt(5) + 11/2 )^(2/3) - 2*(x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1) ^(1/3))/(x^3 - 2*x + 1)) - 3*(x^6 + x^5 - 3*x^4 - x^3 + 3*x^2 + sqrt(-3)*( x^6 + x^5 - 3*x^4 - x^3 + 3*x^2 - x) - x)*(123/50*sqrt(5) + 11/2)^(1/3)*lo g(-(5*(9*x^3 - 4*sqrt(5)*(x^3 - sqrt(-3)*(x^3 - 2*x + 1) - 2*x + 1) - 9*sq rt(-3)*(x^3 - 2*x + 1) - 18*x + 9)*(123/50*sqrt(5) + 11/2)^(2/3) - 2*(x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1)^(1/3))/(x^3 - 2* x + 1)) + 6*(x^6 + x^5 - 3*x^4 - x^3 + 3*x^2 - x)*(123/50*sqrt(5) + 11/2)^ (1/3)*log((5*(9*x^3 - 4*sqrt(5)*(x^3 - 2*x + 1) - 18*x + 9)*(123/50*sqrt(5 ) + 11/2)^(2/3) + (x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^2 + 5* x - 1)^(1/3))/(x^3 - 2*x + 1)) - 3*(x^6 + x^5 - 3*x^4 - x^3 + 3*x^2 - sqrt (-3)*(x^6 + x^5 - 3*x^4 - x^3 + 3*x^2 - x) - x)*(-123/50*sqrt(5) + 11/2)^( 1/3)*log(-(5*(9*x^3 + 4*sqrt(5)*(x^3 + sqrt(-3)*(x^3 - 2*x + 1) - 2*x + 1) + 9*sqrt(-3)*(x^3 - 2*x + 1) - 18*x + 9)*(-123/50*sqrt(5) + 11/2)^(2/3) - 2*(x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1)^(1/3)...
Not integrable
Time = 0.86 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.13 \[ \int \frac {1}{x^2 \sqrt [3]{-1+5 x-7 x^2-2 x^3+10 x^4-2 x^5-5 x^6+x^7+x^8}} \, dx=\int \frac {1}{x^{2} \sqrt [3]{\left (x - 1\right )^{2} \left (x^{2} + x - 1\right )^{3}}}\, dx \] Input:
integrate(1/x**2/(x**8+x**7-5*x**6-2*x**5+10*x**4-2*x**3-7*x**2+5*x-1)**(1 /3),x)
Output:
Integral(1/(x**2*((x - 1)**2*(x**2 + x - 1)**3)**(1/3)), x)
Not integrable
Time = 0.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.25 \[ \int \frac {1}{x^2 \sqrt [3]{-1+5 x-7 x^2-2 x^3+10 x^4-2 x^5-5 x^6+x^7+x^8}} \, dx=\int { \frac {1}{{\left (x^{8} + x^{7} - 5 \, x^{6} - 2 \, x^{5} + 10 \, x^{4} - 2 \, x^{3} - 7 \, x^{2} + 5 \, x - 1\right )}^{\frac {1}{3}} x^{2}} \,d x } \] Input:
integrate(1/x^2/(x^8+x^7-5*x^6-2*x^5+10*x^4-2*x^3-7*x^2+5*x-1)^(1/3),x, al gorithm="maxima")
Output:
integrate(1/((x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1 )^(1/3)*x^2), x)
Not integrable
Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.25 \[ \int \frac {1}{x^2 \sqrt [3]{-1+5 x-7 x^2-2 x^3+10 x^4-2 x^5-5 x^6+x^7+x^8}} \, dx=\int { \frac {1}{{\left (x^{8} + x^{7} - 5 \, x^{6} - 2 \, x^{5} + 10 \, x^{4} - 2 \, x^{3} - 7 \, x^{2} + 5 \, x - 1\right )}^{\frac {1}{3}} x^{2}} \,d x } \] Input:
integrate(1/x^2/(x^8+x^7-5*x^6-2*x^5+10*x^4-2*x^3-7*x^2+5*x-1)^(1/3),x, al gorithm="giac")
Output:
integrate(1/((x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1 )^(1/3)*x^2), x)
Not integrable
Time = 9.31 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.25 \[ \int \frac {1}{x^2 \sqrt [3]{-1+5 x-7 x^2-2 x^3+10 x^4-2 x^5-5 x^6+x^7+x^8}} \, dx=\int \frac {1}{x^2\,{\left (x^8+x^7-5\,x^6-2\,x^5+10\,x^4-2\,x^3-7\,x^2+5\,x-1\right )}^{1/3}} \,d x \] Input:
int(1/(x^2*(5*x - 7*x^2 - 2*x^3 + 10*x^4 - 2*x^5 - 5*x^6 + x^7 + x^8 - 1)^ (1/3)),x)
Output:
int(1/(x^2*(5*x - 7*x^2 - 2*x^3 + 10*x^4 - 2*x^5 - 5*x^6 + x^7 + x^8 - 1)^ (1/3)), x)
Not integrable
Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.27 \[ \int \frac {1}{x^2 \sqrt [3]{-1+5 x-7 x^2-2 x^3+10 x^4-2 x^5-5 x^6+x^7+x^8}} \, dx=\int \frac {1}{\left (x^{2}-2 x +1\right )^{\frac {1}{3}} x^{4}+\left (x^{2}-2 x +1\right )^{\frac {1}{3}} x^{3}-\left (x^{2}-2 x +1\right )^{\frac {1}{3}} x^{2}}d x \] Input:
int(1/x^2/(x^8+x^7-5*x^6-2*x^5+10*x^4-2*x^3-7*x^2+5*x-1)^(1/3),x)
Output:
int(1/((x**2 - 2*x + 1)**(1/3)*x**4 + (x**2 - 2*x + 1)**(1/3)*x**3 - (x**2 - 2*x + 1)**(1/3)*x**2),x)