\(\int \frac {-9+4 x}{-160+100 x+528 x^2+562 x^3+360 x^4+132 x^5+18 x^6} \, dx\) [12]

Optimal result

Integrand size = 38, antiderivative size = 136 \[ \int \frac {-9+4 x}{-160+100 x+528 x^2+562 x^3+360 x^4+132 x^5+18 x^6} \, dx=-\frac {4643 \sqrt {\frac {3}{29}} \arctan \left (\frac {1}{29} \left (\sqrt {87}+2 \sqrt {87} x\right )\right )}{235924}-\frac {11657 \text {arctanh}\left (\frac {1}{5} \left (5 \sqrt {5}+2 \sqrt {5} x\right )\right )}{28618 \sqrt {5}}-\frac {1385 \sqrt {\frac {5}{2}} \text {arctanh}\left (\frac {1}{10} \left (2 \sqrt {10}+3 \sqrt {10} x\right )\right )}{13858}+\frac {5575 \log \left (5+5 x+x^2\right )}{57236}-\frac {4599 \log \left (8+3 x+3 x^2\right )}{471848}-\frac {4859 \log \left (-2+4 x+3 x^2\right )}{55432} \] Output:

-4643/6841796*87^(1/2)*arctan(1/29*87^(1/2)+2/29*87^(1/2)*x)-11657/143090* 
arctanh(5^(1/2)+2/5*x*5^(1/2))*5^(1/2)-1385/27716*10^(1/2)*arctanh(1/5*10^ 
(1/2)+3/10*10^(1/2)*x)+5575/57236*ln(x^2+5*x+5)-4599/471848*ln(3*x^2+3*x+8 
)-4859/55432*ln(3*x^2+4*x-2)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.93 \[ \int \frac {-9+4 x}{-160+100 x+528 x^2+562 x^3+360 x^4+132 x^5+18 x^6} \, dx=\frac {-1903630 \sqrt {87} \arctan \left (\sqrt {\frac {3}{29}} (1+2 x)\right )+50605 \left (-4859+1385 \sqrt {10}\right ) \log \left (-2+\sqrt {10}-3 x\right )+9802 \left (27875+11657 \sqrt {5}\right ) \log \left (-5+\sqrt {5}-2 x\right )+9802 \left (27875-11657 \sqrt {5}\right ) \log \left (5+\sqrt {5}+2 x\right )-50605 \left (4859+1385 \sqrt {10}\right ) \log \left (2+\sqrt {10}+3 x\right )-27341055 \log \left (8+3 x+3 x^2\right )}{2805136360} \] Input:

Integrate[(-9 + 4*x)/(-160 + 100*x + 528*x^2 + 562*x^3 + 360*x^4 + 132*x^5 
 + 18*x^6),x]
 

Output:

(-1903630*Sqrt[87]*ArcTan[Sqrt[3/29]*(1 + 2*x)] + 50605*(-4859 + 1385*Sqrt 
[10])*Log[-2 + Sqrt[10] - 3*x] + 9802*(27875 + 11657*Sqrt[5])*Log[-5 + Sqr 
t[5] - 2*x] + 9802*(27875 - 11657*Sqrt[5])*Log[5 + Sqrt[5] + 2*x] - 50605* 
(4859 + 1385*Sqrt[10])*Log[2 + Sqrt[10] + 3*x] - 27341055*Log[8 + 3*x + 3* 
x^2])/2805136360
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2462, 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.

\(\displaystyle \int \frac {4 x-9}{18 x^6+132 x^5+360 x^4+562 x^3+528 x^2+100 x-160} \, dx\)

\(\Big \downarrow \) 2462

\(\displaystyle \int \left (\frac {4132-14577 x}{27716 \left (3 x^2+4 x-2\right )}+\frac {5575 x+19766}{28618 \left (x^2+5 x+5\right )}-\frac {3 (4599 x+4621)}{235924 \left (3 x^2+3 x+8\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4643 \sqrt {\frac {3}{29}} \arctan \left (\sqrt {\frac {3}{29}} (2 x+1)\right )}{235924}-\frac {4599 \log \left (3 x^2+3 x+8\right )}{471848}+\frac {\left (27875+11657 \sqrt {5}\right ) \log \left (2 x-\sqrt {5}+5\right )}{286180}+\frac {\left (27875-11657 \sqrt {5}\right ) \log \left (2 x+\sqrt {5}+5\right )}{286180}-\frac {\left (4859-1385 \sqrt {10}\right ) \log \left (3 x-\sqrt {10}+2\right )}{55432}-\frac {\left (4859+1385 \sqrt {10}\right ) \log \left (3 x+\sqrt {10}+2\right )}{55432}\)

Input:

Int[(-9 + 4*x)/(-160 + 100*x + 528*x^2 + 562*x^3 + 360*x^4 + 132*x^5 + 18* 
x^6),x]
 

Output:

(-4643*Sqrt[3/29]*ArcTan[Sqrt[3/29]*(1 + 2*x)])/235924 + ((27875 + 11657*S 
qrt[5])*Log[5 - Sqrt[5] + 2*x])/286180 + ((27875 - 11657*Sqrt[5])*Log[5 + 
Sqrt[5] + 2*x])/286180 - ((4859 - 1385*Sqrt[10])*Log[2 - Sqrt[10] + 3*x])/ 
55432 - ((4859 + 1385*Sqrt[10])*Log[2 + Sqrt[10] + 3*x])/55432 - (4599*Log 
[8 + 3*x + 3*x^2])/471848
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2462

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr 
and[u*Qx^p, x], x] /;  !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ 
[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x] && ILtQ[p, 0 
] && RationalFunctionQ[u, x]
 

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.64

method	result	size
default	\(-\frac {4599 \ln \left (3 x^{2}+3 x +8\right )}{471848}-\frac {4643 \sqrt {87}\, \arctan \left (\frac {\left (6 x +3\right ) \sqrt {87}}{87}\right )}{6841796}-\frac {4859 \ln \left (3 x^{2}+4 x -2\right )}{55432}-\frac {1385 \sqrt {10}\, \operatorname {arctanh}\left (\frac {\left (6 x +4\right ) \sqrt {10}}{20}\right )}{27716}+\frac {5575 \ln \left (x^{2}+5 x +5\right )}{57236}-\frac {11657 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 x +5\right ) \sqrt {5}}{5}\right )}{143090}\)	\(87\)
risch	\(-\frac {4859 \ln \left (3 x +2-\sqrt {10}\right )}{55432}+\frac {1385 \ln \left (3 x +2-\sqrt {10}\right ) \sqrt {10}}{55432}-\frac {4859 \ln \left (3 x +2+\sqrt {10}\right )}{55432}-\frac {1385 \ln \left (3 x +2+\sqrt {10}\right ) \sqrt {10}}{55432}-\frac {4599 \ln \left (36 x^{2}+36 x +96\right )}{471848}-\frac {4643 \sqrt {87}\, \arctan \left (\frac {\left (6 x +3\right ) \sqrt {87}}{87}\right )}{6841796}+\frac {5575 \ln \left (2 x +5-\sqrt {5}\right )}{57236}+\frac {11657 \ln \left (2 x +5-\sqrt {5}\right ) \sqrt {5}}{286180}+\frac {5575 \ln \left (2 x +5+\sqrt {5}\right )}{57236}-\frac {11657 \ln \left (2 x +5+\sqrt {5}\right ) \sqrt {5}}{286180}\)	\(139\)

Input:

int((-9+4*x)/(18*x^6+132*x^5+360*x^4+562*x^3+528*x^2+100*x-160),x,method=_ 
RETURNVERBOSE)
 

Output:

-4599/471848*ln(3*x^2+3*x+8)-4643/6841796*87^(1/2)*arctan(1/87*(6*x+3)*87^ 
(1/2))-4859/55432*ln(3*x^2+4*x-2)-1385/27716*10^(1/2)*arctanh(1/20*(6*x+4) 
*10^(1/2))+5575/57236*ln(x^2+5*x+5)-11657/143090*5^(1/2)*arctanh(1/5*(2*x+ 
5)*5^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.95 \[ \int \frac {-9+4 x}{-160+100 x+528 x^2+562 x^3+360 x^4+132 x^5+18 x^6} \, dx=-\frac {4643}{235924} \, \sqrt {\frac {3}{29}} \arctan \left (\sqrt {\frac {3}{29}} {\left (2 \, x + 1\right )}\right ) + \frac {1385}{27716} \, \sqrt {\frac {5}{2}} \log \left (\frac {9 \, x^{2} - 4 \, \sqrt {\frac {5}{2}} {\left (3 \, x + 2\right )} + 12 \, x + 14}{3 \, x^{2} + 4 \, x - 2}\right ) + \frac {11657}{286180} \, \sqrt {5} \log \left (\frac {2 \, x^{2} - \sqrt {5} {\left (2 \, x + 5\right )} + 10 \, x + 15}{x^{2} + 5 \, x + 5}\right ) - \frac {4859}{55432} \, \log \left (3 \, x^{2} + 4 \, x - 2\right ) - \frac {4599}{471848} \, \log \left (3 \, x^{2} + 3 \, x + 8\right ) + \frac {5575}{57236} \, \log \left (x^{2} + 5 \, x + 5\right ) \] Input:

integrate((-9+4*x)/(18*x^6+132*x^5+360*x^4+562*x^3+528*x^2+100*x-160),x, a 
lgorithm="fricas")
 

Output:

-4643/235924*sqrt(3/29)*arctan(sqrt(3/29)*(2*x + 1)) + 1385/27716*sqrt(5/2 
)*log((9*x^2 - 4*sqrt(5/2)*(3*x + 2) + 12*x + 14)/(3*x^2 + 4*x - 2)) + 116 
57/286180*sqrt(5)*log((2*x^2 - sqrt(5)*(2*x + 5) + 10*x + 15)/(x^2 + 5*x + 
 5)) - 4859/55432*log(3*x^2 + 4*x - 2) - 4599/471848*log(3*x^2 + 3*x + 8) 
+ 5575/57236*log(x^2 + 5*x + 5)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.46 (sec) , antiderivative size = 607, normalized size of antiderivative = 4.46 \[ \int \frac {-9+4 x}{-160+100 x+528 x^2+562 x^3+360 x^4+132 x^5+18 x^6} \, dx=\text {Too large to display} \] Input:

integrate((-9+4*x)/(18*x**6+132*x**5+360*x**4+562*x**3+528*x**2+100*x-160) 
,x)
                                                                                    
                                                                                    
 

Output:

(-4859/55432 + 1385*sqrt(10)/55432)*log(x - 565732077701203774025061970019 
4805156569203471427645811546509113/252100455284058577303741432538493999692 
939149495175852498358660 + 10129684555407887655447955898819029261969643296 
41065155913001056*(-4859/55432 + 1385*sqrt(10)/55432)**3/20830180650528857 
3610641935712026233112192876847260919505 + 1115822485043302606994913180995 
6848865243153859426216115022336*(-4859/55432 + 1385*sqrt(10)/55432)**4/181 
9169110146186876199606238551695769179817791132745363677 - 8334560637989254 
78043887426932196724464384639955568721749230454784*(-4859/55432 + 1385*sqr 
t(10)/55432)**5/5457507330438560628598818715655087307539453373398236091031 
 + 2859308122032441774983768929838611722250974892588615417986265808*(-4859 
/55432 + 1385*sqrt(10)/55432)**2/27287536652192803142994093578275436537697 
266866991180455155 + 99903036978949839125398451490478271133212666483986468 
6241723*sqrt(10)/182021989374771535959380095695663537684432598913484370034 
916) + (-4599/471848 - 4643*sqrt(87)*I/13683592)*log(x - 11430107533137461 
758559528951195647416490440834837303840396269261/2145928265710644962902579 
511120351363239896662776009085900662740 + 10129684555407887655447955898819 
02926196964329641065155913001056*(-4599/471848 - 4643*sqrt(87)*I/13683592) 
**3/208301806505288573610641935712026233112192876847260919505 - 9990303697 
89498391253984514904782711332126664839864686241723*sqrt(87)*I/134033856785 
71764790905622619532670586680379758885260282386220 + 111582248504330260...
 

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.79 \[ \int \frac {-9+4 x}{-160+100 x+528 x^2+562 x^3+360 x^4+132 x^5+18 x^6} \, dx=-\frac {4643}{6841796} \, \sqrt {87} \arctan \left (\frac {1}{29} \, \sqrt {87} {\left (2 \, x + 1\right )}\right ) + \frac {1385}{55432} \, \sqrt {10} \log \left (\frac {3 \, x - \sqrt {10} + 2}{3 \, x + \sqrt {10} + 2}\right ) + \frac {11657}{286180} \, \sqrt {5} \log \left (\frac {2 \, x - \sqrt {5} + 5}{2 \, x + \sqrt {5} + 5}\right ) - \frac {4859}{55432} \, \log \left (3 \, x^{2} + 4 \, x - 2\right ) - \frac {4599}{471848} \, \log \left (3 \, x^{2} + 3 \, x + 8\right ) + \frac {5575}{57236} \, \log \left (x^{2} + 5 \, x + 5\right ) \] Input:

integrate((-9+4*x)/(18*x^6+132*x^5+360*x^4+562*x^3+528*x^2+100*x-160),x, a 
lgorithm="maxima")
 

Output:

-4643/6841796*sqrt(87)*arctan(1/29*sqrt(87)*(2*x + 1)) + 1385/55432*sqrt(1 
0)*log((3*x - sqrt(10) + 2)/(3*x + sqrt(10) + 2)) + 11657/286180*sqrt(5)*l 
og((2*x - sqrt(5) + 5)/(2*x + sqrt(5) + 5)) - 4859/55432*log(3*x^2 + 4*x - 
 2) - 4599/471848*log(3*x^2 + 3*x + 8) + 5575/57236*log(x^2 + 5*x + 5)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.85 \[ \int \frac {-9+4 x}{-160+100 x+528 x^2+562 x^3+360 x^4+132 x^5+18 x^6} \, dx=-\frac {4643}{6841796} \, \sqrt {87} \arctan \left (\frac {1}{29} \, \sqrt {87} {\left (2 \, x + 1\right )}\right ) + \frac {1385}{55432} \, \sqrt {10} \log \left (\frac {{\left | 6 \, x - 2 \, \sqrt {10} + 4 \right |}}{{\left | 6 \, x + 2 \, \sqrt {10} + 4 \right |}}\right ) + \frac {11657}{286180} \, \sqrt {5} \log \left (\frac {{\left | 2 \, x - \sqrt {5} + 5 \right |}}{{\left | 2 \, x + \sqrt {5} + 5 \right |}}\right ) - \frac {4599}{471848} \, \log \left (3 \, x^{2} + 3 \, x + 8\right ) - \frac {4859}{55432} \, \log \left ({\left | 3 \, x^{2} + 4 \, x - 2 \right |}\right ) + \frac {5575}{57236} \, \log \left ({\left | x^{2} + 5 \, x + 5 \right |}\right ) \] Input:

integrate((-9+4*x)/(18*x^6+132*x^5+360*x^4+562*x^3+528*x^2+100*x-160),x, a 
lgorithm="giac")
 

Output:

-4643/6841796*sqrt(87)*arctan(1/29*sqrt(87)*(2*x + 1)) + 1385/55432*sqrt(1 
0)*log(abs(6*x - 2*sqrt(10) + 4)/abs(6*x + 2*sqrt(10) + 4)) + 11657/286180 
*sqrt(5)*log(abs(2*x - sqrt(5) + 5)/abs(2*x + sqrt(5) + 5)) - 4599/471848* 
log(3*x^2 + 3*x + 8) - 4859/55432*log(abs(3*x^2 + 4*x - 2)) + 5575/57236*l 
og(abs(x^2 + 5*x + 5))
 

Mupad [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.81 \[ \int \frac {-9+4 x}{-160+100 x+528 x^2+562 x^3+360 x^4+132 x^5+18 x^6} \, dx=\ln \left (x-\frac {\sqrt {10}}{3}+\frac {2}{3}\right )\,\left (\frac {1385\,\sqrt {10}}{55432}-\frac {4859}{55432}\right )-\ln \left (x+\frac {\sqrt {10}}{3}+\frac {2}{3}\right )\,\left (\frac {1385\,\sqrt {10}}{55432}+\frac {4859}{55432}\right )+\ln \left (x-\frac {\sqrt {5}}{2}+\frac {5}{2}\right )\,\left (\frac {11657\,\sqrt {5}}{286180}+\frac {5575}{57236}\right )-\ln \left (x+\frac {\sqrt {5}}{2}+\frac {5}{2}\right )\,\left (\frac {11657\,\sqrt {5}}{286180}-\frac {5575}{57236}\right )+\ln \left (x+\frac {1}{2}-\frac {\sqrt {87}\,1{}\mathrm {i}}{6}\right )\,\left (-\frac {4599}{471848}+\frac {\sqrt {87}\,4643{}\mathrm {i}}{13683592}\right )-\ln \left (x+\frac {1}{2}+\frac {\sqrt {87}\,1{}\mathrm {i}}{6}\right )\,\left (\frac {4599}{471848}+\frac {\sqrt {87}\,4643{}\mathrm {i}}{13683592}\right ) \] Input:

int((4*x - 9)/(100*x + 528*x^2 + 562*x^3 + 360*x^4 + 132*x^5 + 18*x^6 - 16 
0),x)
 

Output:

log(x - 10^(1/2)/3 + 2/3)*((1385*10^(1/2))/55432 - 4859/55432) - log(x + 1 
0^(1/2)/3 + 2/3)*((1385*10^(1/2))/55432 + 4859/55432) + log(x - 5^(1/2)/2 
+ 5/2)*((11657*5^(1/2))/286180 + 5575/57236) - log(x + 5^(1/2)/2 + 5/2)*(( 
11657*5^(1/2))/286180 - 5575/57236) + log(x - (87^(1/2)*1i)/6 + 1/2)*((87^ 
(1/2)*4643i)/13683592 - 4599/471848) - log(x + (87^(1/2)*1i)/6 + 1/2)*((87 
^(1/2)*4643i)/13683592 + 4599/471848)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.92 \[ \int \frac {-9+4 x}{-160+100 x+528 x^2+562 x^3+360 x^4+132 x^5+18 x^6} \, dx=-\frac {4643 \sqrt {87}\, \mathit {atan} \left (\frac {6 x +3}{\sqrt {87}}\right )}{6841796}+\frac {1385 \sqrt {10}\, \mathrm {log}\left (-\sqrt {10}+3 x +2\right )}{55432}-\frac {1385 \sqrt {10}\, \mathrm {log}\left (\sqrt {10}+3 x +2\right )}{55432}+\frac {11657 \sqrt {5}\, \mathrm {log}\left (-\sqrt {5}+2 x +5\right )}{286180}-\frac {11657 \sqrt {5}\, \mathrm {log}\left (\sqrt {5}+2 x +5\right )}{286180}-\frac {4859 \,\mathrm {log}\left (-\sqrt {10}+3 x +2\right )}{55432}+\frac {5575 \,\mathrm {log}\left (-\sqrt {5}+2 x +5\right )}{57236}-\frac {4859 \,\mathrm {log}\left (\sqrt {10}+3 x +2\right )}{55432}+\frac {5575 \,\mathrm {log}\left (\sqrt {5}+2 x +5\right )}{57236}-\frac {4599 \,\mathrm {log}\left (3 x^{2}+3 x +8\right )}{471848} \] Input:

int((-9+4*x)/(18*x^6+132*x^5+360*x^4+562*x^3+528*x^2+100*x-160),x)
 

Output:

( - 1903630*sqrt(87)*atan((6*x + 3)/sqrt(87)) + 70087925*sqrt(10)*log( - s 
qrt(10) + 3*x + 2) - 70087925*sqrt(10)*log(sqrt(10) + 3*x + 2) + 114261914 
*sqrt(5)*log( - sqrt(5) + 2*x + 5) - 114261914*sqrt(5)*log(sqrt(5) + 2*x + 
 5) - 245889695*log( - sqrt(10) + 3*x + 2) + 273230750*log( - sqrt(5) + 2* 
x + 5) - 245889695*log(sqrt(10) + 3*x + 2) + 273230750*log(sqrt(5) + 2*x + 
 5) - 27341055*log(3*x**2 + 3*x + 8))/2805136360