Integrand size = 31, antiderivative size = 101 \[ \int \frac {6-14 x+5 x^2}{9-42 x+43 x^2-14 x^3+x^4} \, dx=\frac {1}{2} \sqrt {\frac {1}{2} \left (5+\sqrt {21}\right )} \text {arctanh}\left (\frac {1}{2} \left (\sqrt {7}-\sqrt {\frac {1}{2} \left (5-\sqrt {21}\right )} x\right )\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (5-\sqrt {21}\right )} \text {arctanh}\left (\frac {1}{2} \left (\sqrt {7}-\sqrt {\frac {1}{2} \left (5+\sqrt {21}\right )} x\right )\right ) \] Output:
1/2*(1/2*7^(1/2)+1/2*3^(1/2))*arctanh(1/2*7^(1/2)-1/2*(1/2*7^(1/2)-1/2*3^( 1/2))*x)-1/2*(1/2*7^(1/2)-1/2*3^(1/2))*arctanh(-1/2*7^(1/2)+1/2*(1/2*7^(1/ 2)+1/2*3^(1/2))*x)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.83 \[ \int \frac {6-14 x+5 x^2}{9-42 x+43 x^2-14 x^3+x^4} \, dx=\frac {1}{2} \text {RootSum}\left [9-42 \text {$\#$1}+43 \text {$\#$1}^2-14 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {6 \log (x-\text {$\#$1})-14 \log (x-\text {$\#$1}) \text {$\#$1}+5 \log (x-\text {$\#$1}) \text {$\#$1}^2}{-21+43 \text {$\#$1}-21 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[(6 - 14*x + 5*x^2)/(9 - 42*x + 43*x^2 - 14*x^3 + x^4),x]
Output:
RootSum[9 - 42*#1 + 43*#1^2 - 14*#1^3 + #1^4 & , (6*Log[x - #1] - 14*Log[x - #1]*#1 + 5*Log[x - #1]*#1^2)/(-21 + 43*#1 - 21*#1^2 + 2*#1^3) & ]/2
Time = 0.85 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.30, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2492, 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.
\(\displaystyle \int \frac {5 x^2-14 x+6}{x^4-14 x^3+43 x^2-42 x+9} \, dx\) |
\(\Big \downarrow \) 2492 |
\(\displaystyle \int \left (\frac {4-\sqrt {3} x}{4 \left (x^2-\left (7-2 \sqrt {3}\right ) x+3\right )}+\frac {\sqrt {3} x+4}{4 \left (x^2-\left (7+2 \sqrt {3}\right ) x+3\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \sqrt {7} \text {arctanh}\left (\frac {-2 x-2 \sqrt {3}+7}{\sqrt {7 \left (7-4 \sqrt {3}\right )}}\right )+\frac {1}{4} \sqrt {7} \text {arctanh}\left (\frac {-2 x+2 \sqrt {3}+7}{\sqrt {7 \left (7+4 \sqrt {3}\right )}}\right )-\frac {1}{8} \sqrt {3} \log \left (x^2-\left (7-2 \sqrt {3}\right ) x+3\right )+\frac {1}{8} \sqrt {3} \log \left (x^2-\left (7+2 \sqrt {3}\right ) x+3\right )\) |
Input:
Int[(6 - 14*x + 5*x^2)/(9 - 42*x + 43*x^2 - 14*x^3 + x^4),x]
Output:
(Sqrt[7]*ArcTanh[(7 - 2*Sqrt[3] - 2*x)/Sqrt[7*(7 - 4*Sqrt[3])]])/4 + (Sqrt [7]*ArcTanh[(7 + 2*Sqrt[3] - 2*x)/Sqrt[7*(7 + 4*Sqrt[3])]])/4 - (Sqrt[3]*L og[3 - (7 - 2*Sqrt[3])*x + x^2])/8 + (Sqrt[3]*Log[3 - (7 + 2*Sqrt[3])*x + x^2])/8
Int[(Px_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*(x_)^3 + (e_.)*(x_)^4) ^(p_), x_Symbol] :> Simp[e^p Int[ExpandIntegrand[Px*(b/d + ((d + Sqrt[e*( (b^2 - 4*a*c)/a) + 8*a*d*(e/b)])/(2*e))*x + x^2)^p*(b/d + ((d - Sqrt[e*((b^ 2 - 4*a*c)/a) + 8*a*d*(e/b)])/(2*e))*x + x^2)^p, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && ILtQ[p, 0] && EqQ[a*d^2 - b^2*e, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.58
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-14 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-42 \textit {\_Z} +9\right )}{\sum }\frac {\left (5 \textit {\_R}^{2}-14 \textit {\_R} +6\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}-21 \textit {\_R}^{2}+43 \textit {\_R} -21}\right )}{2}\) | \(59\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-14 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-42 \textit {\_Z} +9\right )}{\sum }\frac {\left (5 \textit {\_R}^{2}-14 \textit {\_R} +6\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}-21 \textit {\_R}^{2}+43 \textit {\_R} -21}\right )}{2}\) | \(59\) |
Input:
int((5*x^2-14*x+6)/(x^4-14*x^3+43*x^2-42*x+9),x,method=_RETURNVERBOSE)
Output:
1/2*sum((5*_R^2-14*_R+6)/(2*_R^3-21*_R^2+43*_R-21)*ln(x-_R),_R=RootOf(_Z^4 -14*_Z^3+43*_Z^2-42*_Z+9))
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (57) = 114\).
Time = 0.07 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.28 \[ \int \frac {6-14 x+5 x^2}{9-42 x+43 x^2-14 x^3+x^4} \, dx=-\frac {1}{4} \, \sqrt {-\frac {1}{2} \, \sqrt {21} + \frac {5}{2}} \log \left (2 \, x + \sqrt {21} + 4 \, \sqrt {-\frac {1}{2} \, \sqrt {21} + \frac {5}{2}} - 7\right ) + \frac {1}{4} \, \sqrt {-\frac {1}{2} \, \sqrt {21} + \frac {5}{2}} \log \left (2 \, x + \sqrt {21} - 4 \, \sqrt {-\frac {1}{2} \, \sqrt {21} + \frac {5}{2}} - 7\right ) - \frac {1}{4} \, \sqrt {\frac {1}{2} \, \sqrt {21} + \frac {5}{2}} \log \left (2 \, x - \sqrt {21} + 4 \, \sqrt {\frac {1}{2} \, \sqrt {21} + \frac {5}{2}} - 7\right ) + \frac {1}{4} \, \sqrt {\frac {1}{2} \, \sqrt {21} + \frac {5}{2}} \log \left (2 \, x - \sqrt {21} - 4 \, \sqrt {\frac {1}{2} \, \sqrt {21} + \frac {5}{2}} - 7\right ) \] Input:
integrate((5*x^2-14*x+6)/(x^4-14*x^3+43*x^2-42*x+9),x, algorithm="fricas")
Output:
-1/4*sqrt(-1/2*sqrt(21) + 5/2)*log(2*x + sqrt(21) + 4*sqrt(-1/2*sqrt(21) + 5/2) - 7) + 1/4*sqrt(-1/2*sqrt(21) + 5/2)*log(2*x + sqrt(21) - 4*sqrt(-1/ 2*sqrt(21) + 5/2) - 7) - 1/4*sqrt(1/2*sqrt(21) + 5/2)*log(2*x - sqrt(21) + 4*sqrt(1/2*sqrt(21) + 5/2) - 7) + 1/4*sqrt(1/2*sqrt(21) + 5/2)*log(2*x - sqrt(21) - 4*sqrt(1/2*sqrt(21) + 5/2) - 7)
Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.26 \[ \int \frac {6-14 x+5 x^2}{9-42 x+43 x^2-14 x^3+x^4} \, dx=\operatorname {RootSum} {\left (256 t^{4} - 80 t^{2} + 1, \left ( t \mapsto t \log {\left (- 16 t^{2} - 8 t + x - 1 \right )} \right )\right )} \] Input:
integrate((5*x**2-14*x+6)/(x**4-14*x**3+43*x**2-42*x+9),x)
Output:
RootSum(256*_t**4 - 80*_t**2 + 1, Lambda(_t, _t*log(-16*_t**2 - 8*_t + x - 1)))
\[ \int \frac {6-14 x+5 x^2}{9-42 x+43 x^2-14 x^3+x^4} \, dx=\int { \frac {5 \, x^{2} - 14 \, x + 6}{x^{4} - 14 \, x^{3} + 43 \, x^{2} - 42 \, x + 9} \,d x } \] Input:
integrate((5*x^2-14*x+6)/(x^4-14*x^3+43*x^2-42*x+9),x, algorithm="maxima")
Output:
integrate((5*x^2 - 14*x + 6)/(x^4 - 14*x^3 + 43*x^2 - 42*x + 9), x)
Time = 0.15 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.25 \[ \int \frac {6-14 x+5 x^2}{9-42 x+43 x^2-14 x^3+x^4} \, dx=-0.114212562937000 \, \log \left (x - 0.295011649026000\right ) - 0.547225264829000 \, \log \left (x - 1.41348572884000\right ) + 0.114212562937000 \, \log \left (x - 2.12241265602000\right ) + 0.547225264829000 \, \log \left (x - 10.1690899661000\right ) \] Input:
integrate((5*x^2-14*x+6)/(x^4-14*x^3+43*x^2-42*x+9),x, algorithm="giac")
Output:
-0.114212562937000*log(x - 0.295011649026000) - 0.547225264829000*log(x - 1.41348572884000) + 0.114212562937000*log(x - 2.12241265602000) + 0.547225 264829000*log(x - 10.1690899661000)
Time = 0.20 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.62 \[ \int \frac {6-14 x+5 x^2}{9-42 x+43 x^2-14 x^3+x^4} \, dx=\mathrm {atanh}\left (\frac {3087\,\sqrt {3}}{3528\,x-882\,\sqrt {3}\,\sqrt {7}+672\,\sqrt {3}\,\sqrt {7}\,x-5292}+\frac {1512\,\sqrt {7}}{3528\,x-882\,\sqrt {3}\,\sqrt {7}+672\,\sqrt {3}\,\sqrt {7}\,x-5292}-\frac {4263\,\sqrt {3}\,x}{2\,\left (3528\,x-882\,\sqrt {3}\,\sqrt {7}+672\,\sqrt {3}\,\sqrt {7}\,x-5292\right )}-\frac {2205\,\sqrt {7}\,x}{2\,\left (3528\,x-882\,\sqrt {3}\,\sqrt {7}+672\,\sqrt {3}\,\sqrt {7}\,x-5292\right )}\right )\,\left (\frac {\sqrt {3}}{4}+\frac {\sqrt {7}}{4}\right )+\mathrm {atanh}\left (\frac {3087\,\sqrt {3}}{3528\,x+882\,\sqrt {3}\,\sqrt {7}-672\,\sqrt {3}\,\sqrt {7}\,x-5292}-\frac {1512\,\sqrt {7}}{3528\,x+882\,\sqrt {3}\,\sqrt {7}-672\,\sqrt {3}\,\sqrt {7}\,x-5292}-\frac {4263\,\sqrt {3}\,x}{2\,\left (3528\,x+882\,\sqrt {3}\,\sqrt {7}-672\,\sqrt {3}\,\sqrt {7}\,x-5292\right )}+\frac {2205\,\sqrt {7}\,x}{2\,\left (3528\,x+882\,\sqrt {3}\,\sqrt {7}-672\,\sqrt {3}\,\sqrt {7}\,x-5292\right )}\right )\,\left (\frac {\sqrt {3}}{4}-\frac {\sqrt {7}}{4}\right ) \] Input:
int((5*x^2 - 14*x + 6)/(43*x^2 - 42*x - 14*x^3 + x^4 + 9),x)
Output:
atanh((3087*3^(1/2))/(3528*x - 882*3^(1/2)*7^(1/2) + 672*3^(1/2)*7^(1/2)*x - 5292) + (1512*7^(1/2))/(3528*x - 882*3^(1/2)*7^(1/2) + 672*3^(1/2)*7^(1 /2)*x - 5292) - (4263*3^(1/2)*x)/(2*(3528*x - 882*3^(1/2)*7^(1/2) + 672*3^ (1/2)*7^(1/2)*x - 5292)) - (2205*7^(1/2)*x)/(2*(3528*x - 882*3^(1/2)*7^(1/ 2) + 672*3^(1/2)*7^(1/2)*x - 5292)))*(3^(1/2)/4 + 7^(1/2)/4) + atanh((3087 *3^(1/2))/(3528*x + 882*3^(1/2)*7^(1/2) - 672*3^(1/2)*7^(1/2)*x - 5292) - (1512*7^(1/2))/(3528*x + 882*3^(1/2)*7^(1/2) - 672*3^(1/2)*7^(1/2)*x - 529 2) - (4263*3^(1/2)*x)/(2*(3528*x + 882*3^(1/2)*7^(1/2) - 672*3^(1/2)*7^(1/ 2)*x - 5292)) + (2205*7^(1/2)*x)/(2*(3528*x + 882*3^(1/2)*7^(1/2) - 672*3^ (1/2)*7^(1/2)*x - 5292)))*(3^(1/2)/4 - 7^(1/2)/4)
\[ \int \frac {6-14 x+5 x^2}{9-42 x+43 x^2-14 x^3+x^4} \, dx=5 \left (\int \frac {x^{2}}{x^{4}-14 x^{3}+43 x^{2}-42 x +9}d x \right )-14 \left (\int \frac {x}{x^{4}-14 x^{3}+43 x^{2}-42 x +9}d x \right )+6 \left (\int \frac {1}{x^{4}-14 x^{3}+43 x^{2}-42 x +9}d x \right ) \] Input:
int((5*x^2-14*x+6)/(x^4-14*x^3+43*x^2-42*x+9),x)
Output:
5*int(x**2/(x**4 - 14*x**3 + 43*x**2 - 42*x + 9),x) - 14*int(x/(x**4 - 14* x**3 + 43*x**2 - 42*x + 9),x) + 6*int(1/(x**4 - 14*x**3 + 43*x**2 - 42*x + 9),x)