\(\int \frac {8-9 x-8 x^2}{-5-6 x^2+2 x^4} \, dx\) [87]

Optimal result

Integrand size = 25, antiderivative size = 106 \[ \int \frac {8-9 x-8 x^2}{-5-6 x^2+2 x^4} \, dx=-\sqrt {\frac {88}{95}+\frac {56}{5 \sqrt {19}}} \arctan \left (\sqrt {\frac {1}{5} \left (3+\sqrt {19}\right )} x\right )+\sqrt {-\frac {88}{95}+\frac {56}{5 \sqrt {19}}} \text {arctanh}\left (\sqrt {\frac {1}{5} \left (-3+\sqrt {19}\right )} x\right )+\frac {9 \text {arctanh}\left (\frac {1}{19} \left (-3 \sqrt {19}+2 \sqrt {19} x^2\right )\right )}{2 \sqrt {19}} \] Output:

-2/95*(2090+1330*19^(1/2))^(1/2)*arctan(1/5*(15+5*19^(1/2))^(1/2)*x)+2/95* 
(-2090+1330*19^(1/2))^(1/2)*arctanh(1/5*(-15+5*19^(1/2))^(1/2)*x)+9/38*arc 
tanh(-3/19*19^(1/2)+2/19*19^(1/2)*x^2)*19^(1/2)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.20 \[ \int \frac {8-9 x-8 x^2}{-5-6 x^2+2 x^4} \, dx=\frac {1}{76} \left (8 \left (-19+\sqrt {19}\right ) \sqrt {\frac {2}{-3+\sqrt {19}}} \arctan \left (\sqrt {\frac {2}{-3+\sqrt {19}}} x\right )+8 \sqrt {\frac {2}{3+\sqrt {19}}} \left (19+\sqrt {19}\right ) \text {arctanh}\left (\sqrt {\frac {2}{3+\sqrt {19}}} x\right )+9 \sqrt {19} \left (\log \left (3-\sqrt {19}-2 x^2\right )-\log \left (3+\sqrt {19}-2 x^2\right )\right )\right ) \] Input:

Integrate[(8 - 9*x - 8*x^2)/(-5 - 6*x^2 + 2*x^4),x]
 

Output:

(8*(-19 + Sqrt[19])*Sqrt[2/(-3 + Sqrt[19])]*ArcTan[Sqrt[2/(-3 + Sqrt[19])] 
*x] + 8*Sqrt[2/(3 + Sqrt[19])]*(19 + Sqrt[19])*ArcTanh[Sqrt[2/(3 + Sqrt[19 
])]*x] + 9*Sqrt[19]*(Log[3 - Sqrt[19] - 2*x^2] - Log[3 + Sqrt[19] - 2*x^2] 
))/76
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.32, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2202, 27, 25, 1432, 1081, 1480, 216, 220, 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.

\(\displaystyle \int \frac {-8 x^2-9 x+8}{2 x^4-6 x^2-5} \, dx\)

\(\Big \downarrow \) 2202

\(\displaystyle \int -\frac {9 x}{2 x^4-6 x^2-5}dx+\int \frac {8-8 x^2}{2 x^4-6 x^2-5}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {8-8 x^2}{2 x^4-6 x^2-5}dx-9 \int -\frac {x}{-2 x^4+6 x^2+5}dx\)

\(\Big \downarrow \) 25

\(\displaystyle 9 \int \frac {x}{-2 x^4+6 x^2+5}dx+\int \frac {8-8 x^2}{2 x^4-6 x^2-5}dx\)

\(\Big \downarrow \) 1432

\(\displaystyle \frac {9}{2} \int \frac {1}{-2 x^4+6 x^2+5}dx^2+\int \frac {8-8 x^2}{2 x^4-6 x^2-5}dx\)

\(\Big \downarrow \) 1081

\(\displaystyle \int \frac {8-8 x^2}{2 x^4-6 x^2-5}dx-9 \int \left (\frac {1}{2 \sqrt {19} \left (-2 x^2-\sqrt {19}+3\right )}-\frac {1}{2 \sqrt {19} \left (-2 x^2+\sqrt {19}+3\right )}\right )dx^2\)

\(\Big \downarrow \) 1480

\(\displaystyle -\frac {4}{19} \left (19+\sqrt {19}\right ) \int \frac {1}{2 x^2-\sqrt {19}-3}dx-\frac {4}{19} \left (19-\sqrt {19}\right ) \int \frac {1}{2 x^2+\sqrt {19}-3}dx-9 \int \left (\frac {1}{2 \sqrt {19} \left (-2 x^2-\sqrt {19}+3\right )}-\frac {1}{2 \sqrt {19} \left (-2 x^2+\sqrt {19}+3\right )}\right )dx^2\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {4}{19} \left (19+\sqrt {19}\right ) \int \frac {1}{2 x^2-\sqrt {19}-3}dx-9 \int \left (\frac {1}{2 \sqrt {19} \left (-2 x^2-\sqrt {19}+3\right )}-\frac {1}{2 \sqrt {19} \left (-2 x^2+\sqrt {19}+3\right )}\right )dx^2-\frac {2}{19} \left (19-\sqrt {19}\right ) \sqrt {\frac {2}{\sqrt {19}-3}} \arctan \left (\sqrt {\frac {2}{\sqrt {19}-3}} x\right )\)

\(\Big \downarrow \) 220

\(\displaystyle -9 \int \left (\frac {1}{2 \sqrt {19} \left (-2 x^2-\sqrt {19}+3\right )}-\frac {1}{2 \sqrt {19} \left (-2 x^2+\sqrt {19}+3\right )}\right )dx^2-\frac {2}{19} \left (19-\sqrt {19}\right ) \sqrt {\frac {2}{\sqrt {19}-3}} \arctan \left (\sqrt {\frac {2}{\sqrt {19}-3}} x\right )+\frac {2}{19} \sqrt {\frac {2}{3+\sqrt {19}}} \left (19+\sqrt {19}\right ) \text {arctanh}\left (\sqrt {\frac {2}{3+\sqrt {19}}} x\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2}{19} \left (19-\sqrt {19}\right ) \sqrt {\frac {2}{\sqrt {19}-3}} \arctan \left (\sqrt {\frac {2}{\sqrt {19}-3}} x\right )+\frac {2}{19} \sqrt {\frac {2}{3+\sqrt {19}}} \left (19+\sqrt {19}\right ) \text {arctanh}\left (\sqrt {\frac {2}{3+\sqrt {19}}} x\right )-9 \left (\frac {\log \left (-2 x^2+\sqrt {19}+3\right )}{4 \sqrt {19}}-\frac {\log \left (-2 x^2-\sqrt {19}+3\right )}{4 \sqrt {19}}\right )\)

Input:

Int[(8 - 9*x - 8*x^2)/(-5 - 6*x^2 + 2*x^4),x]
 

Output:

(-2*(19 - Sqrt[19])*Sqrt[2/(-3 + Sqrt[19])]*ArcTan[Sqrt[2/(-3 + Sqrt[19])] 
*x])/19 + (2*Sqrt[2/(3 + Sqrt[19])]*(19 + Sqrt[19])*ArcTanh[Sqrt[2/(3 + Sq 
rt[19])]*x])/19 - 9*(-1/4*Log[3 - Sqrt[19] - 2*x^2]/Sqrt[19] + Log[3 + Sqr 
t[19] - 2*x^2]/(4*Sqrt[19]))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 220

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 
1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && 
 (LtQ[a, 0] || GtQ[b, 0])
 

rule 1081

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 
- 4*a*c, 2]}, Simp[c   Int[ExpandIntegrand[1/((b/2 - q/2 + c*x)*(b/2 + q/2 
+ c*x)), x], x], x]] /; FreeQ[{a, b, c}, x] && NiceSqrtQ[b^2 - 4*a*c]
 

rule 1432

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 
 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

rule 2009

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

rule 2202

Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n 
 = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b 
*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 
1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] 
 &&  !PolyQ[Pn, x^2]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.44

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{4}-6 \textit {\_Z}^{2}-5\right )}{\sum }\frac {\left (-8 \textit {\_R}^{2}-9 \textit {\_R} +8\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}-3 \textit {\_R}}\right )}{4}\)	\(47\)
default	\(-\frac {9 \sqrt {19}\, \ln \left (2 x^{2}-\sqrt {19}-3\right )}{76}+\frac {\left (4 \sqrt {19}+76\right ) \operatorname {arctanh}\left (\frac {2 x}{\sqrt {6+2 \sqrt {19}}}\right )}{19 \sqrt {6+2 \sqrt {19}}}+\frac {9 \sqrt {19}\, \ln \left (2 x^{2}-3+\sqrt {19}\right )}{76}+\frac {\left (4 \sqrt {19}-76\right ) \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {19}-6}}\right )}{19 \sqrt {2 \sqrt {19}-6}}\)	\(98\)

Input:

int((-8*x^2-9*x+8)/(2*x^4-6*x^2-5),x,method=_RETURNVERBOSE)
 

Output:

1/4*sum((-8*_R^2-9*_R+8)/(2*_R^3-3*_R)*ln(x-_R),_R=RootOf(2*_Z^4-6*_Z^2-5) 
)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.25 \[ \int \frac {8-9 x-8 x^2}{-5-6 x^2+2 x^4} \, dx=-\frac {1}{76} \, {\left (9 \, \sqrt {19} - 76 \, \sqrt {\frac {14}{95} \, \sqrt {19} - \frac {22}{95}}\right )} \log \left ({\left (8 \, \sqrt {19} + 19\right )} \sqrt {\frac {14}{95} \, \sqrt {19} - \frac {22}{95}} + 18 \, x\right ) - \frac {1}{76} \, {\left (9 \, \sqrt {19} + 76 \, \sqrt {\frac {14}{95} \, \sqrt {19} - \frac {22}{95}}\right )} \log \left (-{\left (8 \, \sqrt {19} + 19\right )} \sqrt {\frac {14}{95} \, \sqrt {19} - \frac {22}{95}} + 18 \, x\right ) - 2 \, \sqrt {\frac {14}{95} \, \sqrt {19} + \frac {22}{95}} \arctan \left (\frac {1}{18} \, {\left (\sqrt {19} x + 19 \, x\right )} \sqrt {\frac {14}{95} \, \sqrt {19} + \frac {22}{95}}\right ) + \frac {9}{76} \, \sqrt {19} \log \left (2 \, x^{2} + \sqrt {19} - 3\right ) \] Input:

integrate((-8*x^2-9*x+8)/(2*x^4-6*x^2-5),x, algorithm="fricas")
 

Output:

-1/76*(9*sqrt(19) - 76*sqrt(14/95*sqrt(19) - 22/95))*log((8*sqrt(19) + 19) 
*sqrt(14/95*sqrt(19) - 22/95) + 18*x) - 1/76*(9*sqrt(19) + 76*sqrt(14/95*s 
qrt(19) - 22/95))*log(-(8*sqrt(19) + 19)*sqrt(14/95*sqrt(19) - 22/95) + 18 
*x) - 2*sqrt(14/95*sqrt(19) + 22/95)*arctan(1/18*(sqrt(19)*x + 19*x)*sqrt( 
14/95*sqrt(19) + 22/95)) + 9/76*sqrt(19)*log(2*x^2 + sqrt(19) - 3)
 

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.40 \[ \int \frac {8-9 x-8 x^2}{-5-6 x^2+2 x^4} \, dx=- \operatorname {RootSum} {\left (462080 t^{4} - 32224 t^{2} - 612864 t - 76059, \left ( t \mapsto t \log {\left (- \frac {9393220 t^{3}}{4904289} - \frac {373160 t^{2}}{544921} + \frac {625499 t}{19617156} + x + \frac {2102419}{1089842} \right )} \right )\right )} \] Input:

integrate((-8*x**2-9*x+8)/(2*x**4-6*x**2-5),x)
 

Output:

-RootSum(462080*_t**4 - 32224*_t**2 - 612864*_t - 76059, Lambda(_t, _t*log 
(-9393220*_t**3/4904289 - 373160*_t**2/544921 + 625499*_t/19617156 + x + 2 
102419/1089842)))
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {8-9 x-8 x^2}{-5-6 x^2+2 x^4} \, dx=\int { -\frac {8 \, x^{2} + 9 \, x - 8}{2 \, x^{4} - 6 \, x^{2} - 5} \,d x } \] Input:

integrate((-8*x^2-9*x+8)/(2*x^4-6*x^2-5),x, algorithm="maxima")
 

Output:

-integrate((8*x^2 + 9*x - 8)/(2*x^4 - 6*x^2 - 5), x)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.26 \[ \int \frac {8-9 x-8 x^2}{-5-6 x^2+2 x^4} \, dx=\frac {1}{380} \, {\left (32 \, \sqrt {19} \sqrt {2 \, \sqrt {19} + 6} - 45 \, \sqrt {19} - 76 \, \sqrt {2 \, \sqrt {19} + 6}\right )} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {19} + \frac {3}{2}} \right |}\right ) - \frac {1}{380} \, {\left (32 \, \sqrt {19} \sqrt {2 \, \sqrt {19} + 6} + 45 \, \sqrt {19} - 76 \, \sqrt {2 \, \sqrt {19} + 6}\right )} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {19} + \frac {3}{2}} \right |}\right ) - \frac {2}{95} \, \sqrt {1330 \, \sqrt {19} + 2090} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {19} - \frac {3}{2}}}\right ) + \frac {9}{76} \, \sqrt {19} \log \left (x^{2} + \frac {1}{2} \, \sqrt {19} - \frac {3}{2}\right ) \] Input:

integrate((-8*x^2-9*x+8)/(2*x^4-6*x^2-5),x, algorithm="giac")
 

Output:

1/380*(32*sqrt(19)*sqrt(2*sqrt(19) + 6) - 45*sqrt(19) - 76*sqrt(2*sqrt(19) 
 + 6))*log(abs(x + sqrt(1/2*sqrt(19) + 3/2))) - 1/380*(32*sqrt(19)*sqrt(2* 
sqrt(19) + 6) + 45*sqrt(19) - 76*sqrt(2*sqrt(19) + 6))*log(abs(x - sqrt(1/ 
2*sqrt(19) + 3/2))) - 2/95*sqrt(1330*sqrt(19) + 2090)*arctan(x/sqrt(1/2*sq 
rt(19) - 3/2)) + 9/76*sqrt(19)*log(x^2 + 1/2*sqrt(19) - 3/2)
 

Mupad [B] (verification not implemented)

Time = 10.20 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.19 \[ \int \frac {8-9 x-8 x^2}{-5-6 x^2+2 x^4} \, dx=\sum _{k=1}^4\ln \left (576\,\mathrm {root}\left (z^4-\frac {53\,z^2}{760}+\frac {126\,z}{95}-\frac {76059}{462080},z,k\right )-\frac {153\,x}{8}-\frac {\mathrm {root}\left (z^4-\frac {53\,z^2}{760}+\frac {126\,z}{95}-\frac {76059}{462080},z,k\right )\,x\,883}{2}+{\mathrm {root}\left (z^4-\frac {53\,z^2}{760}+\frac {126\,z}{95}-\frac {76059}{462080},z,k\right )}^2\,x\,342+{\mathrm {root}\left (z^4-\frac {53\,z^2}{760}+\frac {126\,z}{95}-\frac {76059}{462080},z,k\right )}^3\,x\,456+304\,{\mathrm {root}\left (z^4-\frac {53\,z^2}{760}+\frac {126\,z}{95}-\frac {76059}{462080},z,k\right )}^2-207\right )\,\mathrm {root}\left (z^4-\frac {53\,z^2}{760}+\frac {126\,z}{95}-\frac {76059}{462080},z,k\right ) \] Input:

int((9*x + 8*x^2 - 8)/(6*x^2 - 2*x^4 + 5),x)
 

Output:

symsum(log(576*root(z^4 - (53*z^2)/760 + (126*z)/95 - 76059/462080, z, k) 
- (153*x)/8 - (883*root(z^4 - (53*z^2)/760 + (126*z)/95 - 76059/462080, z, 
 k)*x)/2 + 342*root(z^4 - (53*z^2)/760 + (126*z)/95 - 76059/462080, z, k)^ 
2*x + 456*root(z^4 - (53*z^2)/760 + (126*z)/95 - 76059/462080, z, k)^3*x + 
 304*root(z^4 - (53*z^2)/760 + (126*z)/95 - 76059/462080, z, k)^2 - 207)*r 
oot(z^4 - (53*z^2)/760 + (126*z)/95 - 76059/462080, z, k), k, 1, 4)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.69 \[ \int \frac {8-9 x-8 x^2}{-5-6 x^2+2 x^4} \, dx=-\frac {16 \sqrt {\sqrt {19}-3}\, \sqrt {38}\, \mathit {atan} \left (\frac {2 x}{\sqrt {\sqrt {19}-3}\, \sqrt {2}}\right )}{95}-\frac {2 \sqrt {\sqrt {19}-3}\, \sqrt {2}\, \mathit {atan} \left (\frac {2 x}{\sqrt {\sqrt {19}-3}\, \sqrt {2}}\right )}{5}-\frac {8 \sqrt {\sqrt {19}+3}\, \sqrt {38}\, \mathrm {log}\left (-\sqrt {\sqrt {19}+3}+\sqrt {2}\, x \right )}{95}+\frac {8 \sqrt {\sqrt {19}+3}\, \sqrt {38}\, \mathrm {log}\left (\sqrt {\sqrt {19}+3}+\sqrt {2}\, x \right )}{95}+\frac {\sqrt {\sqrt {19}+3}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {\sqrt {19}+3}+\sqrt {2}\, x \right )}{5}-\frac {\sqrt {\sqrt {19}+3}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {19}+3}+\sqrt {2}\, x \right )}{5}-\frac {9 \sqrt {19}\, \mathrm {log}\left (-\sqrt {\sqrt {19}+3}+\sqrt {2}\, x \right )}{76}-\frac {9 \sqrt {19}\, \mathrm {log}\left (\sqrt {\sqrt {19}+3}+\sqrt {2}\, x \right )}{76}+\frac {9 \sqrt {19}\, \mathrm {log}\left (\sqrt {19}+2 x^{2}-3\right )}{76} \] Input:

int((-8*x^2-9*x+8)/(2*x^4-6*x^2-5),x)
 

Output:

( - 64*sqrt(sqrt(19) - 3)*sqrt(38)*atan((2*x)/(sqrt(sqrt(19) - 3)*sqrt(2)) 
) - 152*sqrt(sqrt(19) - 3)*sqrt(2)*atan((2*x)/(sqrt(sqrt(19) - 3)*sqrt(2)) 
) - 32*sqrt(sqrt(19) + 3)*sqrt(38)*log( - sqrt(sqrt(19) + 3) + sqrt(2)*x) 
+ 32*sqrt(sqrt(19) + 3)*sqrt(38)*log(sqrt(sqrt(19) + 3) + sqrt(2)*x) + 76* 
sqrt(sqrt(19) + 3)*sqrt(2)*log( - sqrt(sqrt(19) + 3) + sqrt(2)*x) - 76*sqr 
t(sqrt(19) + 3)*sqrt(2)*log(sqrt(sqrt(19) + 3) + sqrt(2)*x) - 45*sqrt(19)* 
log( - sqrt(sqrt(19) + 3) + sqrt(2)*x) - 45*sqrt(19)*log(sqrt(sqrt(19) + 3 
) + sqrt(2)*x) + 45*sqrt(19)*log(sqrt(19) + 2*x**2 - 3))/380