Integrand size = 33, antiderivative size = 173 \[ \int \frac {10-9 x-4 x^2}{-4-8 x+7 x^2+8 x^3-4 x^4} \, dx=\sqrt {\frac {21557}{5772}+\frac {1405}{481 \sqrt {3}}} \text {arctanh}\left (\sqrt {\frac {113}{481}+\frac {64 \sqrt {3}}{481}}-4 \sqrt {\frac {1}{481} \left (23-4 \sqrt {3}\right )} x\right )-\frac {1}{2} \sqrt {\frac {21557-5620 \sqrt {3}}{1443}} \text {arctanh}\left (\sqrt {\frac {1}{481} \left (113-64 \sqrt {3}\right )}-\sqrt {\frac {368}{481}+\frac {64 \sqrt {3}}{481}} x\right )-\frac {1}{4} \sqrt {3} \log \left (-2-2 x-\sqrt {3} x+2 x^2\right )+\frac {1}{4} \sqrt {3} \log \left (-2-2 x+\sqrt {3} x+2 x^2\right ) \] Output:
-1/2886*(31106751+8109660*3^(1/2))^(1/2)*arctanh(-1/481*(54353+30784*3^(1/ 2))^(1/2)+4/481*(11063-1924*3^(1/2))^(1/2)*x)+1/2886*(31106751-8109660*3^( 1/2))^(1/2)*arctanh(-1/481*(54353-30784*3^(1/2))^(1/2)+4/481*(11063+1924*3 ^(1/2))^(1/2)*x)-1/4*3^(1/2)*ln(-2-2*x-x*3^(1/2)+2*x^2)+1/4*3^(1/2)*ln(-2- 2*x+x*3^(1/2)+2*x^2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.50 \[ \int \frac {10-9 x-4 x^2}{-4-8 x+7 x^2+8 x^3-4 x^4} \, dx=\frac {1}{2} \text {RootSum}\left [4+8 \text {$\#$1}-7 \text {$\#$1}^2-8 \text {$\#$1}^3+4 \text {$\#$1}^4\&,\frac {-10 \log (x-\text {$\#$1})+9 \log (x-\text {$\#$1}) \text {$\#$1}+4 \log (x-\text {$\#$1}) \text {$\#$1}^2}{4-7 \text {$\#$1}-12 \text {$\#$1}^2+8 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[(10 - 9*x - 4*x^2)/(-4 - 8*x + 7*x^2 + 8*x^3 - 4*x^4),x]
Output:
RootSum[4 + 8*#1 - 7*#1^2 - 8*#1^3 + 4*#1^4 & , (-10*Log[x - #1] + 9*Log[x - #1]*#1 + 4*Log[x - #1]*#1^2)/(4 - 7*#1 - 12*#1^2 + 8*#1^3) & ]/2
Time = 0.49 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.83, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, 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.
\(\displaystyle \int \frac {-4 x^2-9 x+10}{-4 x^4+8 x^3+7 x^2-8 x-4} \, dx\) |
\(\Big \downarrow \) 2492 |
\(\displaystyle -\frac {1}{4} \int \left (\frac {2 \left (-6 x+5 \sqrt {3}+19\right )}{\sqrt {3} \left (-2 x^2+\left (2+\sqrt {3}\right ) x+2\right )}-\frac {2 \left (-6 x-5 \sqrt {3}+19\right )}{\sqrt {3} \left (-2 x^2+\left (2-\sqrt {3}\right ) x+2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (-2 \sqrt {\frac {21557-5620 \sqrt {3}}{1443}} \text {arctanh}\left (\frac {-4 x-\sqrt {3}+2}{\sqrt {23-4 \sqrt {3}}}\right )+2 \sqrt {\frac {21557+5620 \sqrt {3}}{1443}} \text {arctanh}\left (\frac {-4 x+\sqrt {3}+2}{\sqrt {23+4 \sqrt {3}}}\right )+\sqrt {3} \log \left (-2 x^2+\left (2-\sqrt {3}\right ) x+2\right )-\sqrt {3} \log \left (-2 x^2+\left (2+\sqrt {3}\right ) x+2\right )\right )\) |
Input:
Int[(10 - 9*x - 4*x^2)/(-4 - 8*x + 7*x^2 + 8*x^3 - 4*x^4),x]
Output:
(-2*Sqrt[(21557 - 5620*Sqrt[3])/1443]*ArcTanh[(2 - Sqrt[3] - 4*x)/Sqrt[23 - 4*Sqrt[3]]] + 2*Sqrt[(21557 + 5620*Sqrt[3])/1443]*ArcTanh[(2 + Sqrt[3] - 4*x)/Sqrt[23 + 4*Sqrt[3]]] + Sqrt[3]*Log[2 + (2 - Sqrt[3])*x - 2*x^2] - S qrt[3]*Log[2 + (2 + Sqrt[3])*x - 2*x^2])/4
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 = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.35
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{4}-8 \textit {\_Z}^{3}-7 \textit {\_Z}^{2}+8 \textit {\_Z} +4\right )}{\sum }\frac {\left (4 \textit {\_R}^{2}+9 \textit {\_R} -10\right ) \ln \left (x -\textit {\_R} \right )}{8 \textit {\_R}^{3}-12 \textit {\_R}^{2}-7 \textit {\_R} +4}\right )}{2}\) | \(61\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{4}-8 \textit {\_Z}^{3}-7 \textit {\_Z}^{2}+8 \textit {\_Z} +4\right )}{\sum }\frac {\left (4 \textit {\_R}^{2}+9 \textit {\_R} -10\right ) \ln \left (x -\textit {\_R} \right )}{8 \textit {\_R}^{3}-12 \textit {\_R}^{2}-7 \textit {\_R} +4}\right )}{2}\) | \(61\) |
Input:
int((-4*x^2-9*x+10)/(-4*x^4+8*x^3+7*x^2-8*x-4),x,method=_RETURNVERBOSE)
Output:
1/2*sum((4*_R^2+9*_R-10)/(8*_R^3-12*_R^2-7*_R+4)*ln(x-_R),_R=RootOf(4*_Z^4 -8*_Z^3-7*_Z^2+8*_Z+4))
Leaf count of result is larger than twice the leaf count of optimal. 471 vs. \(2 (116) = 232\).
Time = 0.08 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.72 \[ \int \frac {10-9 x-4 x^2}{-4-8 x+7 x^2+8 x^3-4 x^4} \, dx=\text {Too large to display} \] Input:
integrate((-4*x^2-9*x+10)/(-4*x^4+8*x^3+7*x^2-8*x-4),x, algorithm="fricas" )
Output:
1/4*(sqrt(1/2810*(877*sqrt(481) - 21557)*sqrt(877/2886*sqrt(481) + 21557/2 886) - 877/2886*sqrt(481) + 30215/2886) + sqrt(877/2886*sqrt(481) + 21557/ 2886))*log((1443*(1477387*sqrt(481) + 6680613)*sqrt(877/2886*sqrt(481) + 2 1557/2886) + 4151457470*sqrt(481) + 66441092770)*sqrt(1/2810*(877*sqrt(481 ) - 21557)*sqrt(877/2886*sqrt(481) + 21557/2886) - 877/2886*sqrt(481) + 30 215/2886) + 33927808*(163*sqrt(481) - 3848)*sqrt(877/2886*sqrt(481) + 2155 7/2886) + 190674280960*x - 95337140480) - 1/4*(sqrt(1/2810*(877*sqrt(481) - 21557)*sqrt(877/2886*sqrt(481) + 21557/2886) - 877/2886*sqrt(481) + 3021 5/2886) - sqrt(877/2886*sqrt(481) + 21557/2886))*log(-(1443*(1477387*sqrt( 481) + 6680613)*sqrt(877/2886*sqrt(481) + 21557/2886) + 4151457470*sqrt(48 1) + 66441092770)*sqrt(1/2810*(877*sqrt(481) - 21557)*sqrt(877/2886*sqrt(4 81) + 21557/2886) - 877/2886*sqrt(481) + 30215/2886) + 33927808*(163*sqrt( 481) - 3848)*sqrt(877/2886*sqrt(481) + 21557/2886) + 190674280960*x - 9533 7140480) - 1/4*(sqrt(-1/2810*(877*sqrt(481) - 21557)*sqrt(877/2886*sqrt(48 1) + 21557/2886) - 877/2886*sqrt(481) + 30215/2886) + sqrt(877/2886*sqrt(4 81) + 21557/2886))*log((1443*(1477387*sqrt(481) + 6680613)*sqrt(877/2886*s qrt(481) + 21557/2886) - 4151457470*sqrt(481) - 66441092770)*sqrt(-1/2810* (877*sqrt(481) - 21557)*sqrt(877/2886*sqrt(481) + 21557/2886) - 877/2886*s qrt(481) + 30215/2886) - 33927808*(163*sqrt(481) - 3848)*sqrt(877/2886*sqr t(481) + 21557/2886) + 190674280960*x - 95337140480) + 1/4*(sqrt(-1/281...
Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.24 \[ \int \frac {10-9 x-4 x^2}{-4-8 x+7 x^2+8 x^3-4 x^4} \, dx=\operatorname {RootSum} {\left (34632 t^{4} - 77658 t^{2} + 25290 t + 13127, \left ( t \mapsto t \log {\left (\frac {2131869441 t^{3}}{929833988} + \frac {4840845087 t^{2}}{1859667976} - \frac {3084714377 t}{929833988} + x - \frac {8044277969}{3719335952} \right )} \right )\right )} \] Input:
integrate((-4*x**2-9*x+10)/(-4*x**4+8*x**3+7*x**2-8*x-4),x)
Output:
RootSum(34632*_t**4 - 77658*_t**2 + 25290*_t + 13127, Lambda(_t, _t*log(21 31869441*_t**3/929833988 + 4840845087*_t**2/1859667976 - 3084714377*_t/929 833988 + x - 8044277969/3719335952)))
\[ \int \frac {10-9 x-4 x^2}{-4-8 x+7 x^2+8 x^3-4 x^4} \, dx=\int { \frac {4 \, x^{2} + 9 \, x - 10}{4 \, x^{4} - 8 \, x^{3} - 7 \, x^{2} + 8 \, x + 4} \,d x } \] Input:
integrate((-4*x^2-9*x+10)/(-4*x^4+8*x^3+7*x^2-8*x-4),x, algorithm="maxima" )
Output:
integrate((4*x^2 + 9*x - 10)/(4*x^4 - 8*x^3 - 7*x^2 + 8*x + 4), x)
Time = 0.12 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.14 \[ \int \frac {10-9 x-4 x^2}{-4-8 x+7 x^2+8 x^3-4 x^4} \, dx=1.14860948933000 \, \log \left (x + 0.935253839597000\right ) - 1.59718546836000 \, \log \left (x + 0.434654180756000\right ) - 0.282584085545000 \, \log \left (x - 1.06922843581000\right ) + 0.731160064576000 \, \log \left (x - 2.30067958454000\right ) \] Input:
integrate((-4*x^2-9*x+10)/(-4*x^4+8*x^3+7*x^2-8*x-4),x, algorithm="giac")
Output:
1.14860948933000*log(x + 0.935253839597000) - 1.59718546836000*log(x + 0.4 34654180756000) - 0.282584085545000*log(x - 1.06922843581000) + 0.73116006 4576000*log(x - 2.30067958454000)
Time = 10.04 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.84 \[ \int \frac {10-9 x-4 x^2}{-4-8 x+7 x^2+8 x^3-4 x^4} \, dx=\sum _{k=1}^4\ln \left (\frac {163\,\mathrm {root}\left (z^4-\frac {12943\,z^2}{5772}+\frac {1405\,z}{1924}+\frac {13127}{34632},z,k\right )}{4}+\frac {2037\,x}{64}-\frac {\mathrm {root}\left (z^4-\frac {12943\,z^2}{5772}+\frac {1405\,z}{1924}+\frac {13127}{34632},z,k\right )\,x\,1611}{32}-\frac {{\mathrm {root}\left (z^4-\frac {12943\,z^2}{5772}+\frac {1405\,z}{1924}+\frac {13127}{34632},z,k\right )}^2\,x\,1221}{16}+\frac {{\mathrm {root}\left (z^4-\frac {12943\,z^2}{5772}+\frac {1405\,z}{1924}+\frac {13127}{34632},z,k\right )}^3\,x\,891}{8}-\frac {117\,{\mathrm {root}\left (z^4-\frac {12943\,z^2}{5772}+\frac {1405\,z}{1924}+\frac {13127}{34632},z,k\right )}^2}{8}+\frac {51\,{\mathrm {root}\left (z^4-\frac {12943\,z^2}{5772}+\frac {1405\,z}{1924}+\frac {13127}{34632},z,k\right )}^3}{2}-\frac {857}{32}\right )\,\mathrm {root}\left (z^4-\frac {12943\,z^2}{5772}+\frac {1405\,z}{1924}+\frac {13127}{34632},z,k\right ) \] Input:
int((9*x + 4*x^2 - 10)/(8*x - 7*x^2 - 8*x^3 + 4*x^4 + 4),x)
Output:
symsum(log((163*root(z^4 - (12943*z^2)/5772 + (1405*z)/1924 + 13127/34632, z, k))/4 + (2037*x)/64 - (1611*root(z^4 - (12943*z^2)/5772 + (1405*z)/192 4 + 13127/34632, z, k)*x)/32 - (1221*root(z^4 - (12943*z^2)/5772 + (1405*z )/1924 + 13127/34632, z, k)^2*x)/16 + (891*root(z^4 - (12943*z^2)/5772 + ( 1405*z)/1924 + 13127/34632, z, k)^3*x)/8 - (117*root(z^4 - (12943*z^2)/577 2 + (1405*z)/1924 + 13127/34632, z, k)^2)/8 + (51*root(z^4 - (12943*z^2)/5 772 + (1405*z)/1924 + 13127/34632, z, k)^3)/2 - 857/32)*root(z^4 - (12943* z^2)/5772 + (1405*z)/1924 + 13127/34632, z, k), k, 1, 4)
\[ \int \frac {10-9 x-4 x^2}{-4-8 x+7 x^2+8 x^3-4 x^4} \, dx=4 \left (\int \frac {x^{2}}{4 x^{4}-8 x^{3}-7 x^{2}+8 x +4}d x \right )+9 \left (\int \frac {x}{4 x^{4}-8 x^{3}-7 x^{2}+8 x +4}d x \right )-10 \left (\int \frac {1}{4 x^{4}-8 x^{3}-7 x^{2}+8 x +4}d x \right ) \] Input:
int((-4*x^2-9*x+10)/(-4*x^4+8*x^3+7*x^2-8*x-4),x)
Output:
4*int(x**2/(4*x**4 - 8*x**3 - 7*x**2 + 8*x + 4),x) + 9*int(x/(4*x**4 - 8*x **3 - 7*x**2 + 8*x + 4),x) - 10*int(1/(4*x**4 - 8*x**3 - 7*x**2 + 8*x + 4) ,x)