Integrand size = 32, antiderivative size = 337 \[ \int \frac {1}{1-12 x-39 x^2-58 x^3-45 x^4-18 x^5-3 x^6} \, dx=-\frac {\sqrt [6]{3} \arctan \left (\frac {1-2 \sqrt [3]{\frac {3}{-1+\sqrt {7}}} (1+x)}{\sqrt {3}}\right )}{2 \sqrt {7} \left (-1+\sqrt {7}\right )^{2/3}}+\frac {\sqrt [6]{3} \arctan \left (\frac {1+2 \sqrt [3]{\frac {3}{1+\sqrt {7}}} (1+x)}{\sqrt {3}}\right )}{2 \sqrt {7} \left (1+\sqrt {7}\right )^{2/3}}-\frac {\log \left (\sqrt [3]{1+\sqrt {7}}-\sqrt [3]{3} (1+x)\right )}{2 \sqrt [3]{3} \sqrt {7} \left (1+\sqrt {7}\right )^{2/3}}+\frac {\log \left (\sqrt [3]{-1+\sqrt {7}}+\sqrt [3]{3} (1+x)\right )}{2 \sqrt [3]{3} \sqrt {7} \left (-1+\sqrt {7}\right )^{2/3}}-\frac {\log \left (\left (-1+\sqrt {7}\right )^{2/3}-\sqrt [3]{3 \left (-1+\sqrt {7}\right )} (1+x)+3^{2/3} (1+x)^2\right )}{4 \sqrt [3]{3} \sqrt {7} \left (-1+\sqrt {7}\right )^{2/3}}+\frac {\log \left (\left (1+\sqrt {7}\right )^{2/3}+\sqrt [3]{3 \left (1+\sqrt {7}\right )} (1+x)+3^{2/3} (1+x)^2\right )}{4 \sqrt [3]{3} \sqrt {7} \left (1+\sqrt {7}\right )^{2/3}} \] Output:
-1/14*3^(1/6)*arctan(1/3*(1-2*3^(1/3)*(1/(-1+7^(1/2)))^(1/3)*(1+x))*3^(1/2 ))*7^(1/2)/(-1+7^(1/2))^(2/3)+1/14*3^(1/6)*arctan(1/3*(1+2*3^(1/3)*(1/(1+7 ^(1/2)))^(1/3)*(1+x))*3^(1/2))*7^(1/2)/(1+7^(1/2))^(2/3)-1/42*ln((1+7^(1/2 ))^(1/3)-3^(1/3)*(1+x))*3^(2/3)*7^(1/2)/(1+7^(1/2))^(2/3)+1/42*ln((-1+7^(1 /2))^(1/3)+3^(1/3)*(1+x))*3^(2/3)*7^(1/2)/(-1+7^(1/2))^(2/3)-1/84*ln((-1+7 ^(1/2))^(2/3)-(-3+3*7^(1/2))^(1/3)*(1+x)+3^(2/3)*(1+x)^2)*3^(2/3)*7^(1/2)/ (-1+7^(1/2))^(2/3)+1/84*ln((1+7^(1/2))^(2/3)+(3+3*7^(1/2))^(1/3)*(1+x)+3^( 2/3)*(1+x)^2)*3^(2/3)*7^(1/2)/(1+7^(1/2))^(2/3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.25 \[ \int \frac {1}{1-12 x-39 x^2-58 x^3-45 x^4-18 x^5-3 x^6} \, dx=-\frac {1}{6} \text {RootSum}\left [-1+12 \text {$\#$1}+39 \text {$\#$1}^2+58 \text {$\#$1}^3+45 \text {$\#$1}^4+18 \text {$\#$1}^5+3 \text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1})}{2+13 \text {$\#$1}+29 \text {$\#$1}^2+30 \text {$\#$1}^3+15 \text {$\#$1}^4+3 \text {$\#$1}^5}\&\right ] \] Input:
Integrate[(1 - 12*x - 39*x^2 - 58*x^3 - 45*x^4 - 18*x^5 - 3*x^6)^(-1),x]
Output:
-1/6*RootSum[-1 + 12*#1 + 39*#1^2 + 58*#1^3 + 45*#1^4 + 18*#1^5 + 3*#1^6 & , Log[x - #1]/(2 + 13*#1 + 29*#1^2 + 30*#1^3 + 15*#1^4 + 3*#1^5) & ]
Time = 0.88 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.92, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2458, 1685, 750, 16, 25, 1142, 25, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{-3 x^6-18 x^5-45 x^4-58 x^3-39 x^2-12 x+1} \, dx\) |
| \(\Big \downarrow \) 2458 |
| \(\displaystyle \int \frac {1}{-3 (x+1)^6+2 (x+1)^3+2}d(x+1)\) |
| \(\Big \downarrow \) 1685 |
| \(\displaystyle \frac {3 \int \frac {1}{-3 (x+1)^3+\sqrt {7}+1}d(x+1)}{2 \sqrt {7}}-\frac {3 \int \frac {1}{-3 (x+1)^3-\sqrt {7}+1}d(x+1)}{2 \sqrt {7}}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {1}{\sqrt [3]{1+\sqrt {7}}-\sqrt [3]{3} (x+1)}d(x+1)}{3 \left (1+\sqrt {7}\right )^{2/3}}+\frac {\int \frac {\sqrt [3]{3} (x+1)+2 \sqrt [3]{1+\sqrt {7}}}{3^{2/3} (x+1)^2+\sqrt [3]{3 \left (1+\sqrt {7}\right )} (x+1)+\left (1+\sqrt {7}\right )^{2/3}}d(x+1)}{3 \left (1+\sqrt {7}\right )^{2/3}}\right )}{2 \sqrt {7}}-\frac {3 \left (\frac {\int \frac {1}{-\sqrt [3]{3} (x+1)-\sqrt [3]{-1+\sqrt {7}}}d(x+1)}{3 \left (\sqrt {7}-1\right )^{2/3}}+\frac {\int -\frac {2 \sqrt [3]{-1+\sqrt {7}}-\sqrt [3]{3} (x+1)}{3^{2/3} (x+1)^2-\sqrt [3]{3 \left (-1+\sqrt {7}\right )} (x+1)+\left (-1+\sqrt {7}\right )^{2/3}}d(x+1)}{3 \left (\sqrt {7}-1\right )^{2/3}}\right )}{2 \sqrt {7}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {\sqrt [3]{3} (x+1)+2 \sqrt [3]{1+\sqrt {7}}}{3^{2/3} (x+1)^2+\sqrt [3]{3 \left (1+\sqrt {7}\right )} (x+1)+\left (1+\sqrt {7}\right )^{2/3}}d(x+1)}{3 \left (1+\sqrt {7}\right )^{2/3}}-\frac {\log \left (\sqrt [3]{1+\sqrt {7}}-\sqrt [3]{3} (x+1)\right )}{3 \sqrt [3]{3} \left (1+\sqrt {7}\right )^{2/3}}\right )}{2 \sqrt {7}}-\frac {3 \left (\frac {\int -\frac {2 \sqrt [3]{-1+\sqrt {7}}-\sqrt [3]{3} (x+1)}{3^{2/3} (x+1)^2-\sqrt [3]{3 \left (-1+\sqrt {7}\right )} (x+1)+\left (-1+\sqrt {7}\right )^{2/3}}d(x+1)}{3 \left (\sqrt {7}-1\right )^{2/3}}-\frac {\log \left (\sqrt [3]{3} (x+1)+\sqrt [3]{\sqrt {7}-1}\right )}{3 \sqrt [3]{3} \left (\sqrt {7}-1\right )^{2/3}}\right )}{2 \sqrt {7}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {\sqrt [3]{3} (x+1)+2 \sqrt [3]{1+\sqrt {7}}}{3^{2/3} (x+1)^2+\sqrt [3]{3 \left (1+\sqrt {7}\right )} (x+1)+\left (1+\sqrt {7}\right )^{2/3}}d(x+1)}{3 \left (1+\sqrt {7}\right )^{2/3}}-\frac {\log \left (\sqrt [3]{1+\sqrt {7}}-\sqrt [3]{3} (x+1)\right )}{3 \sqrt [3]{3} \left (1+\sqrt {7}\right )^{2/3}}\right )}{2 \sqrt {7}}-\frac {3 \left (-\frac {\int \frac {2 \sqrt [3]{-1+\sqrt {7}}-\sqrt [3]{3} (x+1)}{3^{2/3} (x+1)^2-\sqrt [3]{3 \left (-1+\sqrt {7}\right )} (x+1)+\left (-1+\sqrt {7}\right )^{2/3}}d(x+1)}{3 \left (\sqrt {7}-1\right )^{2/3}}-\frac {\log \left (\sqrt [3]{3} (x+1)+\sqrt [3]{\sqrt {7}-1}\right )}{3 \sqrt [3]{3} \left (\sqrt {7}-1\right )^{2/3}}\right )}{2 \sqrt {7}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {3 \left (\frac {\frac {3}{2} \sqrt [3]{1+\sqrt {7}} \int \frac {1}{3^{2/3} (x+1)^2+\sqrt [3]{3 \left (1+\sqrt {7}\right )} (x+1)+\left (1+\sqrt {7}\right )^{2/3}}d(x+1)+\frac {\int \frac {2\ 3^{2/3} (x+1)+\sqrt [3]{3 \left (1+\sqrt {7}\right )}}{3^{2/3} (x+1)^2+\sqrt [3]{3 \left (1+\sqrt {7}\right )} (x+1)+\left (1+\sqrt {7}\right )^{2/3}}d(x+1)}{2 \sqrt [3]{3}}}{3 \left (1+\sqrt {7}\right )^{2/3}}-\frac {\log \left (\sqrt [3]{1+\sqrt {7}}-\sqrt [3]{3} (x+1)\right )}{3 \sqrt [3]{3} \left (1+\sqrt {7}\right )^{2/3}}\right )}{2 \sqrt {7}}-\frac {3 \left (-\frac {\frac {3}{2} \sqrt [3]{\sqrt {7}-1} \int \frac {1}{3^{2/3} (x+1)^2-\sqrt [3]{3 \left (-1+\sqrt {7}\right )} (x+1)+\left (-1+\sqrt {7}\right )^{2/3}}d(x+1)-\frac {\int -\frac {\sqrt [3]{3 \left (-1+\sqrt {7}\right )}-2\ 3^{2/3} (x+1)}{3^{2/3} (x+1)^2-\sqrt [3]{3 \left (-1+\sqrt {7}\right )} (x+1)+\left (-1+\sqrt {7}\right )^{2/3}}d(x+1)}{2 \sqrt [3]{3}}}{3 \left (\sqrt {7}-1\right )^{2/3}}-\frac {\log \left (\sqrt [3]{3} (x+1)+\sqrt [3]{\sqrt {7}-1}\right )}{3 \sqrt [3]{3} \left (\sqrt {7}-1\right )^{2/3}}\right )}{2 \sqrt {7}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \left (\frac {\frac {3}{2} \sqrt [3]{1+\sqrt {7}} \int \frac {1}{3^{2/3} (x+1)^2+\sqrt [3]{3 \left (1+\sqrt {7}\right )} (x+1)+\left (1+\sqrt {7}\right )^{2/3}}d(x+1)+\frac {\int \frac {2\ 3^{2/3} (x+1)+\sqrt [3]{3 \left (1+\sqrt {7}\right )}}{3^{2/3} (x+1)^2+\sqrt [3]{3 \left (1+\sqrt {7}\right )} (x+1)+\left (1+\sqrt {7}\right )^{2/3}}d(x+1)}{2 \sqrt [3]{3}}}{3 \left (1+\sqrt {7}\right )^{2/3}}-\frac {\log \left (\sqrt [3]{1+\sqrt {7}}-\sqrt [3]{3} (x+1)\right )}{3 \sqrt [3]{3} \left (1+\sqrt {7}\right )^{2/3}}\right )}{2 \sqrt {7}}-\frac {3 \left (-\frac {\frac {3}{2} \sqrt [3]{\sqrt {7}-1} \int \frac {1}{3^{2/3} (x+1)^2-\sqrt [3]{3 \left (-1+\sqrt {7}\right )} (x+1)+\left (-1+\sqrt {7}\right )^{2/3}}d(x+1)+\frac {\int \frac {\sqrt [3]{3 \left (-1+\sqrt {7}\right )}-2\ 3^{2/3} (x+1)}{3^{2/3} (x+1)^2-\sqrt [3]{3 \left (-1+\sqrt {7}\right )} (x+1)+\left (-1+\sqrt {7}\right )^{2/3}}d(x+1)}{2 \sqrt [3]{3}}}{3 \left (\sqrt {7}-1\right )^{2/3}}-\frac {\log \left (\sqrt [3]{3} (x+1)+\sqrt [3]{\sqrt {7}-1}\right )}{3 \sqrt [3]{3} \left (\sqrt {7}-1\right )^{2/3}}\right )}{2 \sqrt {7}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3 \left (\frac {\frac {\int \frac {2\ 3^{2/3} (x+1)+\sqrt [3]{3 \left (1+\sqrt {7}\right )}}{3^{2/3} (x+1)^2+\sqrt [3]{3 \left (1+\sqrt {7}\right )} (x+1)+\left (1+\sqrt {7}\right )^{2/3}}d(x+1)}{2 \sqrt [3]{3}}-3^{2/3} \int \frac {1}{-\left (2 \sqrt [3]{\frac {3}{1+\sqrt {7}}} (x+1)+1\right )^2-3}d\left (2 \sqrt [3]{\frac {3}{1+\sqrt {7}}} (x+1)+1\right )}{3 \left (1+\sqrt {7}\right )^{2/3}}-\frac {\log \left (\sqrt [3]{1+\sqrt {7}}-\sqrt [3]{3} (x+1)\right )}{3 \sqrt [3]{3} \left (1+\sqrt {7}\right )^{2/3}}\right )}{2 \sqrt {7}}-\frac {3 \left (-\frac {\frac {\int \frac {\sqrt [3]{3 \left (-1+\sqrt {7}\right )}-2\ 3^{2/3} (x+1)}{3^{2/3} (x+1)^2-\sqrt [3]{3 \left (-1+\sqrt {7}\right )} (x+1)+\left (-1+\sqrt {7}\right )^{2/3}}d(x+1)}{2 \sqrt [3]{3}}+3^{2/3} \int \frac {1}{-\left (1-2 \sqrt [3]{\frac {3}{-1+\sqrt {7}}} (x+1)\right )^2-3}d\left (1-2 \sqrt [3]{\frac {3}{-1+\sqrt {7}}} (x+1)\right )}{3 \left (\sqrt {7}-1\right )^{2/3}}-\frac {\log \left (\sqrt [3]{3} (x+1)+\sqrt [3]{\sqrt {7}-1}\right )}{3 \sqrt [3]{3} \left (\sqrt {7}-1\right )^{2/3}}\right )}{2 \sqrt {7}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 \left (\frac {\frac {\int \frac {2\ 3^{2/3} (x+1)+\sqrt [3]{3 \left (1+\sqrt {7}\right )}}{3^{2/3} (x+1)^2+\sqrt [3]{3 \left (1+\sqrt {7}\right )} (x+1)+\left (1+\sqrt {7}\right )^{2/3}}d(x+1)}{2 \sqrt [3]{3}}+\sqrt [6]{3} \arctan \left (\frac {2 \sqrt [3]{\frac {3}{1+\sqrt {7}}} (x+1)+1}{\sqrt {3}}\right )}{3 \left (1+\sqrt {7}\right )^{2/3}}-\frac {\log \left (\sqrt [3]{1+\sqrt {7}}-\sqrt [3]{3} (x+1)\right )}{3 \sqrt [3]{3} \left (1+\sqrt {7}\right )^{2/3}}\right )}{2 \sqrt {7}}-\frac {3 \left (-\frac {\frac {\int \frac {\sqrt [3]{3 \left (-1+\sqrt {7}\right )}-2\ 3^{2/3} (x+1)}{3^{2/3} (x+1)^2-\sqrt [3]{3 \left (-1+\sqrt {7}\right )} (x+1)+\left (-1+\sqrt {7}\right )^{2/3}}d(x+1)}{2 \sqrt [3]{3}}-\sqrt [6]{3} \arctan \left (\frac {1-2 \sqrt [3]{\frac {3}{\sqrt {7}-1}} (x+1)}{\sqrt {3}}\right )}{3 \left (\sqrt {7}-1\right )^{2/3}}-\frac {\log \left (\sqrt [3]{3} (x+1)+\sqrt [3]{\sqrt {7}-1}\right )}{3 \sqrt [3]{3} \left (\sqrt {7}-1\right )^{2/3}}\right )}{2 \sqrt {7}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt [6]{3} \arctan \left (\frac {2 \sqrt [3]{\frac {3}{1+\sqrt {7}}} (x+1)+1}{\sqrt {3}}\right )+\frac {\log \left (3^{2/3} (x+1)^2+\sqrt [3]{3 \left (1+\sqrt {7}\right )} (x+1)+\left (1+\sqrt {7}\right )^{2/3}\right )}{2 \sqrt [3]{3}}}{3 \left (1+\sqrt {7}\right )^{2/3}}-\frac {\log \left (\sqrt [3]{1+\sqrt {7}}-\sqrt [3]{3} (x+1)\right )}{3 \sqrt [3]{3} \left (1+\sqrt {7}\right )^{2/3}}\right )}{2 \sqrt {7}}-\frac {3 \left (-\frac {-\sqrt [6]{3} \arctan \left (\frac {1-2 \sqrt [3]{\frac {3}{\sqrt {7}-1}} (x+1)}{\sqrt {3}}\right )-\frac {\log \left (3^{2/3} (x+1)^2-\sqrt [3]{3 \left (\sqrt {7}-1\right )} (x+1)+\left (\sqrt {7}-1\right )^{2/3}\right )}{2 \sqrt [3]{3}}}{3 \left (\sqrt {7}-1\right )^{2/3}}-\frac {\log \left (\sqrt [3]{3} (x+1)+\sqrt [3]{\sqrt {7}-1}\right )}{3 \sqrt [3]{3} \left (\sqrt {7}-1\right )^{2/3}}\right )}{2 \sqrt {7}}\) |
Input:
Int[(1 - 12*x - 39*x^2 - 58*x^3 - 45*x^4 - 18*x^5 - 3*x^6)^(-1),x]
Output:
(-3*(-1/3*Log[(-1 + Sqrt[7])^(1/3) + 3^(1/3)*(1 + x)]/(3^(1/3)*(-1 + Sqrt[ 7])^(2/3)) - (-(3^(1/6)*ArcTan[(1 - 2*(3/(-1 + Sqrt[7]))^(1/3)*(1 + x))/Sq rt[3]]) - Log[(-1 + Sqrt[7])^(2/3) - (3*(-1 + Sqrt[7]))^(1/3)*(1 + x) + 3^ (2/3)*(1 + x)^2]/(2*3^(1/3)))/(3*(-1 + Sqrt[7])^(2/3))))/(2*Sqrt[7]) + (3* (-1/3*Log[(1 + Sqrt[7])^(1/3) - 3^(1/3)*(1 + x)]/(3^(1/3)*(1 + Sqrt[7])^(2 /3)) + (3^(1/6)*ArcTan[(1 + 2*(3/(1 + Sqrt[7]))^(1/3)*(1 + x))/Sqrt[3]] + Log[(1 + Sqrt[7])^(2/3) + (3*(1 + Sqrt[7]))^(1/3)*(1 + x) + 3^(2/3)*(1 + x )^2]/(2*3^(1/3)))/(3*(1 + Sqrt[7])^(2/3))))/(2*Sqrt[7])
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^n), x], x] - Simp[ c/q Int[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.21
| method | result | size |
| default | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{6}+18 \textit {\_Z}^{5}+45 \textit {\_Z}^{4}+58 \textit {\_Z}^{3}+39 \textit {\_Z}^{2}+12 \textit {\_Z} -1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{5}+15 \textit {\_R}^{4}+30 \textit {\_R}^{3}+29 \textit {\_R}^{2}+13 \textit {\_R} +2}\right )}{6}\) | \(71\) |
| risch | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{6}+18 \textit {\_Z}^{5}+45 \textit {\_Z}^{4}+58 \textit {\_Z}^{3}+39 \textit {\_Z}^{2}+12 \textit {\_Z} -1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{5}+15 \textit {\_R}^{4}+30 \textit {\_R}^{3}+29 \textit {\_R}^{2}+13 \textit {\_R} +2}\right )}{6}\) | \(71\) |
Input:
int(1/(-3*x^6-18*x^5-45*x^4-58*x^3-39*x^2-12*x+1),x,method=_RETURNVERBOSE)
Output:
-1/6*sum(1/(3*_R^5+15*_R^4+30*_R^3+29*_R^2+13*_R+2)*ln(x-_R),_R=RootOf(3*_ Z^6+18*_Z^5+45*_Z^4+58*_Z^3+39*_Z^2+12*_Z-1))
Time = 0.10 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.77 \[ \int \frac {1}{1-12 x-39 x^2-58 x^3-45 x^4-18 x^5-3 x^6} \, dx=-\frac {1}{12} \, {\left (\frac {2}{49} \, \sqrt {7} + \frac {1}{14}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left ({\left (\sqrt {7} {\left (\sqrt {-3} + 1\right )} - 7 \, \sqrt {-3} - 7\right )} {\left (\frac {2}{49} \, \sqrt {7} + \frac {1}{14}\right )}^{\frac {1}{3}} + 6 \, x + 6\right ) + \frac {1}{12} \, {\left (\frac {2}{49} \, \sqrt {7} + \frac {1}{14}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (-{\left (\sqrt {7} {\left (\sqrt {-3} - 1\right )} - 7 \, \sqrt {-3} + 7\right )} {\left (\frac {2}{49} \, \sqrt {7} + \frac {1}{14}\right )}^{\frac {1}{3}} + 6 \, x + 6\right ) - \frac {1}{12} \, {\left (-\frac {2}{49} \, \sqrt {7} + \frac {1}{14}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (-{\left (\sqrt {7} {\left (\sqrt {-3} + 1\right )} + 7 \, \sqrt {-3} + 7\right )} {\left (-\frac {2}{49} \, \sqrt {7} + \frac {1}{14}\right )}^{\frac {1}{3}} + 6 \, x + 6\right ) + \frac {1}{12} \, {\left (-\frac {2}{49} \, \sqrt {7} + \frac {1}{14}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left ({\left (\sqrt {7} {\left (\sqrt {-3} - 1\right )} + 7 \, \sqrt {-3} - 7\right )} {\left (-\frac {2}{49} \, \sqrt {7} + \frac {1}{14}\right )}^{\frac {1}{3}} + 6 \, x + 6\right ) + \frac {1}{6} \, {\left (\frac {2}{49} \, \sqrt {7} + \frac {1}{14}\right )}^{\frac {1}{3}} \log \left (-{\left (\sqrt {7} - 7\right )} {\left (\frac {2}{49} \, \sqrt {7} + \frac {1}{14}\right )}^{\frac {1}{3}} + 3 \, x + 3\right ) + \frac {1}{6} \, {\left (-\frac {2}{49} \, \sqrt {7} + \frac {1}{14}\right )}^{\frac {1}{3}} \log \left ({\left (\sqrt {7} + 7\right )} {\left (-\frac {2}{49} \, \sqrt {7} + \frac {1}{14}\right )}^{\frac {1}{3}} + 3 \, x + 3\right ) \] Input:
integrate(1/(-3*x^6-18*x^5-45*x^4-58*x^3-39*x^2-12*x+1),x, algorithm="fric as")
Output:
-1/12*(2/49*sqrt(7) + 1/14)^(1/3)*(sqrt(-3) + 1)*log((sqrt(7)*(sqrt(-3) + 1) - 7*sqrt(-3) - 7)*(2/49*sqrt(7) + 1/14)^(1/3) + 6*x + 6) + 1/12*(2/49*s qrt(7) + 1/14)^(1/3)*(sqrt(-3) - 1)*log(-(sqrt(7)*(sqrt(-3) - 1) - 7*sqrt( -3) + 7)*(2/49*sqrt(7) + 1/14)^(1/3) + 6*x + 6) - 1/12*(-2/49*sqrt(7) + 1/ 14)^(1/3)*(sqrt(-3) + 1)*log(-(sqrt(7)*(sqrt(-3) + 1) + 7*sqrt(-3) + 7)*(- 2/49*sqrt(7) + 1/14)^(1/3) + 6*x + 6) + 1/12*(-2/49*sqrt(7) + 1/14)^(1/3)* (sqrt(-3) - 1)*log((sqrt(7)*(sqrt(-3) - 1) + 7*sqrt(-3) - 7)*(-2/49*sqrt(7 ) + 1/14)^(1/3) + 6*x + 6) + 1/6*(2/49*sqrt(7) + 1/14)^(1/3)*log(-(sqrt(7) - 7)*(2/49*sqrt(7) + 1/14)^(1/3) + 3*x + 3) + 1/6*(-2/49*sqrt(7) + 1/14)^ (1/3)*log((sqrt(7) + 7)*(-2/49*sqrt(7) + 1/14)^(1/3) + 3*x + 3)
Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.09 \[ \int \frac {1}{1-12 x-39 x^2-58 x^3-45 x^4-18 x^5-3 x^6} \, dx=- \operatorname {RootSum} {\left (7112448 t^{6} + 4704 t^{3} - 1, \left ( t \mapsto t \log {\left (- 10584 t^{4} - \frac {35 t}{2} + x + 1 \right )} \right )\right )} \] Input:
integrate(1/(-3*x**6-18*x**5-45*x**4-58*x**3-39*x**2-12*x+1),x)
Output:
-RootSum(7112448*_t**6 + 4704*_t**3 - 1, Lambda(_t, _t*log(-10584*_t**4 - 35*_t/2 + x + 1)))
\[ \int \frac {1}{1-12 x-39 x^2-58 x^3-45 x^4-18 x^5-3 x^6} \, dx=\int { -\frac {1}{3 \, x^{6} + 18 \, x^{5} + 45 \, x^{4} + 58 \, x^{3} + 39 \, x^{2} + 12 \, x - 1} \,d x } \] Input:
integrate(1/(-3*x^6-18*x^5-45*x^4-58*x^3-39*x^2-12*x+1),x, algorithm="maxi ma")
Output:
-integrate(1/(3*x^6 + 18*x^5 + 45*x^4 + 58*x^3 + 39*x^2 + 12*x - 1), x)
\[ \int \frac {1}{1-12 x-39 x^2-58 x^3-45 x^4-18 x^5-3 x^6} \, dx=\int { -\frac {1}{3 \, x^{6} + 18 \, x^{5} + 45 \, x^{4} + 58 \, x^{3} + 39 \, x^{2} + 12 \, x - 1} \,d x } \] Input:
integrate(1/(-3*x^6-18*x^5-45*x^4-58*x^3-39*x^2-12*x+1),x, algorithm="giac ")
Output:
integrate(-1/(3*x^6 + 18*x^5 + 45*x^4 + 58*x^3 + 39*x^2 + 12*x - 1), x)
Time = 23.07 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.37 \[ \int \frac {1}{1-12 x-39 x^2-58 x^3-45 x^4-18 x^5-3 x^6} \, dx=\frac {\ln \left (x+\frac {2^{2/3}\,7^{1/3}\,{\left (7-4\,\sqrt {7}\right )}^{1/3}}{6}+\frac {2^{2/3}\,7^{5/6}\,{\left (7-4\,\sqrt {7}\right )}^{1/3}}{42}+1\right )\,{\left (196-112\,\sqrt {7}\right )}^{1/3}}{84}-\frac {\ln \left (x-\frac {2^{2/3}\,7^{1/3}\,{\left (4\,\sqrt {7}-7\right )}^{1/3}}{6}-\frac {2^{2/3}\,7^{5/6}\,{\left (4\,\sqrt {7}-7\right )}^{1/3}}{42}+1\right )\,{\left (112\,\sqrt {7}-196\right )}^{1/3}}{84}+\frac {\ln \left (x+\frac {2^{2/3}\,7^{1/3}\,{\left (4\,\sqrt {7}+7\right )}^{1/3}}{6}-\frac {2^{2/3}\,7^{5/6}\,{\left (4\,\sqrt {7}+7\right )}^{1/3}}{42}+1\right )\,{\left (112\,\sqrt {7}+196\right )}^{1/3}}{84}-\frac {2^{2/3}\,7^{1/3}\,\ln \left (x-\frac {2^{2/3}\,7^{1/3}\,{\left (7-4\,\sqrt {7}\right )}^{1/3}}{12}-\frac {2^{2/3}\,7^{5/6}\,{\left (7-4\,\sqrt {7}\right )}^{1/3}}{84}+1-\frac {2^{2/3}\,\sqrt {3}\,7^{1/3}\,{\left (7-4\,\sqrt {7}\right )}^{1/3}\,1{}\mathrm {i}}{12}-\frac {2^{2/3}\,\sqrt {3}\,7^{5/6}\,{\left (7-4\,\sqrt {7}\right )}^{1/3}\,1{}\mathrm {i}}{84}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (7-4\,\sqrt {7}\right )}^{1/3}}{168}-\frac {2^{2/3}\,7^{1/3}\,\ln \left (x-\frac {2^{2/3}\,7^{1/3}\,{\left (4\,\sqrt {7}+7\right )}^{1/3}}{12}+\frac {2^{2/3}\,7^{5/6}\,{\left (4\,\sqrt {7}+7\right )}^{1/3}}{84}+1-\frac {2^{2/3}\,\sqrt {3}\,7^{1/3}\,{\left (4\,\sqrt {7}+7\right )}^{1/3}\,1{}\mathrm {i}}{12}+\frac {2^{2/3}\,\sqrt {3}\,7^{5/6}\,{\left (4\,\sqrt {7}+7\right )}^{1/3}\,1{}\mathrm {i}}{84}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (4\,\sqrt {7}+7\right )}^{1/3}}{168}+\frac {2^{2/3}\,7^{1/3}\,\ln \left (x-\frac {2^{2/3}\,7^{1/3}\,{\left (4\,\sqrt {7}+7\right )}^{1/3}}{12}+\frac {2^{2/3}\,7^{5/6}\,{\left (4\,\sqrt {7}+7\right )}^{1/3}}{84}+1+\frac {2^{2/3}\,\sqrt {3}\,7^{1/3}\,{\left (4\,\sqrt {7}+7\right )}^{1/3}\,1{}\mathrm {i}}{12}-\frac {2^{2/3}\,\sqrt {3}\,7^{5/6}\,{\left (4\,\sqrt {7}+7\right )}^{1/3}\,1{}\mathrm {i}}{84}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (4\,\sqrt {7}+7\right )}^{1/3}}{168} \] Input:
int(-1/(12*x + 39*x^2 + 58*x^3 + 45*x^4 + 18*x^5 + 3*x^6 - 1),x)
Output:
(log(x + (2^(2/3)*7^(1/3)*(7 - 4*7^(1/2))^(1/3))/6 + (2^(2/3)*7^(5/6)*(7 - 4*7^(1/2))^(1/3))/42 + 1)*(196 - 112*7^(1/2))^(1/3))/84 - (log(x - (2^(2/ 3)*7^(1/3)*(4*7^(1/2) - 7)^(1/3))/6 - (2^(2/3)*7^(5/6)*(4*7^(1/2) - 7)^(1/ 3))/42 + 1)*(112*7^(1/2) - 196)^(1/3))/84 + (log(x + (2^(2/3)*7^(1/3)*(4*7 ^(1/2) + 7)^(1/3))/6 - (2^(2/3)*7^(5/6)*(4*7^(1/2) + 7)^(1/3))/42 + 1)*(11 2*7^(1/2) + 196)^(1/3))/84 - (2^(2/3)*7^(1/3)*log(x - (2^(2/3)*7^(1/3)*(7 - 4*7^(1/2))^(1/3))/12 - (2^(2/3)*7^(5/6)*(7 - 4*7^(1/2))^(1/3))/84 - (2^( 2/3)*3^(1/2)*7^(1/3)*(7 - 4*7^(1/2))^(1/3)*1i)/12 - (2^(2/3)*3^(1/2)*7^(5/ 6)*(7 - 4*7^(1/2))^(1/3)*1i)/84 + 1)*(3^(1/2)*1i + 1)*(7 - 4*7^(1/2))^(1/3 ))/168 - (2^(2/3)*7^(1/3)*log(x - (2^(2/3)*7^(1/3)*(4*7^(1/2) + 7)^(1/3))/ 12 + (2^(2/3)*7^(5/6)*(4*7^(1/2) + 7)^(1/3))/84 - (2^(2/3)*3^(1/2)*7^(1/3) *(4*7^(1/2) + 7)^(1/3)*1i)/12 + (2^(2/3)*3^(1/2)*7^(5/6)*(4*7^(1/2) + 7)^( 1/3)*1i)/84 + 1)*(3^(1/2)*1i + 1)*(4*7^(1/2) + 7)^(1/3))/168 + (2^(2/3)*7^ (1/3)*log(x - (2^(2/3)*7^(1/3)*(4*7^(1/2) + 7)^(1/3))/12 + (2^(2/3)*7^(5/6 )*(4*7^(1/2) + 7)^(1/3))/84 + (2^(2/3)*3^(1/2)*7^(1/3)*(4*7^(1/2) + 7)^(1/ 3)*1i)/12 - (2^(2/3)*3^(1/2)*7^(5/6)*(4*7^(1/2) + 7)^(1/3)*1i)/84 + 1)*(3^ (1/2)*1i - 1)*(4*7^(1/2) + 7)^(1/3))/168
\[ \int \frac {1}{1-12 x-39 x^2-58 x^3-45 x^4-18 x^5-3 x^6} \, dx=-\left (\int \frac {1}{3 x^{6}+18 x^{5}+45 x^{4}+58 x^{3}+39 x^{2}+12 x -1}d x \right ) \] Input:
int(1/(-3*x^6-18*x^5-45*x^4-58*x^3-39*x^2-12*x+1),x)
Output:
- int(1/(3*x**6 + 18*x**5 + 45*x**4 + 58*x**3 + 39*x**2 + 12*x - 1),x)