3.114 \(\int \frac {3+2 x^3}{-9 x+x^5} \, dx\)

Optimal. Leaf size=48 \[ \frac {1}{12} \log \left (9-x^4\right )-\frac {\log (x)}{3}+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {1593, 1831, 266, 36, 31, 29, 298, 203, 206} \[ \frac {1}{12} \log \left (9-x^4\right )-\frac {\log (x)}{3}+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 2*x^3)/(-9*x + x^5),x]

[Out]

ArcTan[x/Sqrt[3]]/Sqrt[3] - ArcTanh[x/Sqrt[3]]/Sqrt[3] - Log[x]/3 + Log[9 - x^4]/12

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 203

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

Rule 206

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1831

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[((c*x)^(m + ii)*(Coeff[Pq,
x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2)))/(c^ii*(a + b*x^n)), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; Fr
eeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n

Rubi steps

\begin {align*} \int \frac {3+2 x^3}{-9 x+x^5} \, dx &=\int \frac {3+2 x^3}{x \left (-9+x^4\right )} \, dx\\ &=\int \left (\frac {3}{x \left (-9+x^4\right )}+\frac {2 x^2}{-9+x^4}\right ) \, dx\\ &=2 \int \frac {x^2}{-9+x^4} \, dx+3 \int \frac {1}{x \left (-9+x^4\right )} \, dx\\ &=\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{(-9+x) x} \, dx,x,x^4\right )-\int \frac {1}{3-x^2} \, dx+\int \frac {1}{3+x^2} \, dx\\ &=\frac {\tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{-9+x} \, dx,x,x^4\right )-\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^4\right )\\ &=\frac {\tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{3}+\frac {1}{12} \log \left (9-x^4\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 67, normalized size = 1.40 \[ \frac {1}{12} \left (\log \left (9-x^4\right )-4 \log (x)+2 \sqrt {3} \log \left (3-\sqrt {3} x\right )-2 \sqrt {3} \log \left (\sqrt {3} x+3\right )+4 \sqrt {3} \tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 2*x^3)/(-9*x + x^5),x]

[Out]

(4*Sqrt[3]*ArcTan[x/Sqrt[3]] - 4*Log[x] + 2*Sqrt[3]*Log[3 - Sqrt[3]*x] - 2*Sqrt[3]*Log[3 + Sqrt[3]*x] + Log[9
- x^4])/12

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.04, size = 46, normalized size = 0.96 \[ \frac {1}{12} \log \left (x^4-9\right )-\frac {\log (x)}{3}+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(3 + 2*x^3)/(-9*x + x^5),x]

[Out]

ArcTan[x/Sqrt[3]]/Sqrt[3] - ArcTanh[x/Sqrt[3]]/Sqrt[3] - Log[x]/3 + Log[-9 + x^4]/12

________________________________________________________________________________________

fricas [A]  time = 0.89, size = 58, normalized size = 1.21 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) + \frac {1}{6} \, \sqrt {3} \log \left (\frac {x^{2} - 2 \, \sqrt {3} x + 3}{x^{2} - 3}\right ) + \frac {1}{12} \, \log \left (x^{2} + 3\right ) + \frac {1}{12} \, \log \left (x^{2} - 3\right ) - \frac {1}{3} \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+3)/(x^5-9*x),x, algorithm="fricas")

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*x) + 1/6*sqrt(3)*log((x^2 - 2*sqrt(3)*x + 3)/(x^2 - 3)) + 1/12*log(x^2 + 3) + 1
/12*log(x^2 - 3) - 1/3*log(x)

________________________________________________________________________________________

giac [A]  time = 1.12, size = 64, normalized size = 1.33 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) + \frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {3} \right |}}{{\left | 2 \, x + 2 \, \sqrt {3} \right |}}\right ) + \frac {1}{12} \, \log \left (x^{2} + 3\right ) + \frac {1}{12} \, \log \left ({\left | x^{2} - 3 \right |}\right ) - \frac {1}{3} \, \log \left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+3)/(x^5-9*x),x, algorithm="giac")

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*x) + 1/6*sqrt(3)*log(abs(2*x - 2*sqrt(3))/abs(2*x + 2*sqrt(3))) + 1/12*log(x^2
+ 3) + 1/12*log(abs(x^2 - 3)) - 1/3*log(abs(x))

________________________________________________________________________________________

maple [A]  time = 0.34, size = 46, normalized size = 0.96

method	result	size
default	\(\frac {\ln \left (x^{2}+3\right )}{12}+\frac {\arctan \left (\frac {x \sqrt {3}}{3}\right ) \sqrt {3}}{3}-\frac {\ln \relax (x )}{3}+\frac {\ln \left (x^{2}-3\right )}{12}-\frac {\arctanh \left (\frac {x \sqrt {3}}{3}\right ) \sqrt {3}}{3}\)	\(46\)
meijerg	\(\frac {\ln \left (1-\frac {x^{4}}{9}\right )}{12}-\frac {\ln \relax (x )}{3}+\frac {\ln \relax (3)}{6}-\frac {i \pi }{12}+\frac {x^{3} \sqrt {3}\, \left (\ln \left (1-\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{3}\right )-\ln \left (1+\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{3}\right )+2 \arctan \left (\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{3}\right )\right )}{6 \left (x^{4}\right )^{\frac {3}{4}}}\)	\(79\)
risch	\(-\frac {\ln \relax (x )}{3}+\frac {\ln \left (4 x^{2}+12\right )}{12}+\frac {\arctan \left (\frac {x \sqrt {3}}{3}\right ) \sqrt {3}}{3}+\frac {\ln \left (2 x -2 \sqrt {3}\right )}{12}+\frac {\ln \left (2 x -2 \sqrt {3}\right ) \sqrt {3}}{6}+\frac {\ln \left (2 x +2 \sqrt {3}\right )}{12}-\frac {\ln \left (2 x +2 \sqrt {3}\right ) \sqrt {3}}{6}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^3+3)/(x^5-9*x),x,method=_RETURNVERBOSE)

[Out]

1/12*ln(x^2+3)+1/3*arctan(1/3*x*3^(1/2))*3^(1/2)-1/3*ln(x)+1/12*ln(x^2-3)-1/3*arctanh(1/3*x*3^(1/2))*3^(1/2)

________________________________________________________________________________________

maxima [A]  time = 0.98, size = 54, normalized size = 1.12 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) + \frac {1}{6} \, \sqrt {3} \log \left (\frac {x - \sqrt {3}}{x + \sqrt {3}}\right ) + \frac {1}{12} \, \log \left (x^{2} + 3\right ) + \frac {1}{12} \, \log \left (x^{2} - 3\right ) - \frac {1}{3} \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+3)/(x^5-9*x),x, algorithm="maxima")

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*x) + 1/6*sqrt(3)*log((x - sqrt(3))/(x + sqrt(3))) + 1/12*log(x^2 + 3) + 1/12*lo
g(x^2 - 3) - 1/3*log(x)

________________________________________________________________________________________

mupad [B]  time = 0.17, size = 73, normalized size = 1.52 \[ \ln \left (x-\sqrt {3}\right )\,\left (\frac {\sqrt {3}}{6}+\frac {1}{12}\right )-\ln \left (x+\sqrt {3}\right )\,\left (\frac {\sqrt {3}}{6}-\frac {1}{12}\right )-\frac {\ln \relax (x)}{3}-\ln \left (x-\sqrt {3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\ln \left (x+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x^3 + 3)/(9*x - x^5),x)

[Out]

log(x - 3^(1/2))*(3^(1/2)/6 + 1/12) - log(x + 3^(1/2))*(3^(1/2)/6 - 1/12) - log(x)/3 - log(x - 3^(1/2)*1i)*((3
^(1/2)*1i)/6 - 1/12) + log(x + 3^(1/2)*1i)*((3^(1/2)*1i)/6 + 1/12)

________________________________________________________________________________________

sympy [C]  time = 0.59, size = 306, normalized size = 6.38 \[ - \frac {\log {\relax (x )}}{3} + \left (\frac {1}{12} + \frac {\sqrt {3} i}{6}\right ) \log {\left (x + \frac {17413}{11544} - \frac {943 \sqrt {3} i}{5772} + \frac {1368 \left (\frac {1}{12} + \frac {\sqrt {3} i}{6}\right )^{3}}{481} + \frac {4158 \left (\frac {1}{12} + \frac {\sqrt {3} i}{6}\right )^{2}}{481} - \frac {108000 \left (\frac {1}{12} + \frac {\sqrt {3} i}{6}\right )^{4}}{481} \right )} + \left (\frac {1}{12} - \frac {\sqrt {3} i}{6}\right ) \log {\left (x + \frac {17413}{11544} - \frac {108000 \left (\frac {1}{12} - \frac {\sqrt {3} i}{6}\right )^{4}}{481} + \frac {4158 \left (\frac {1}{12} - \frac {\sqrt {3} i}{6}\right )^{2}}{481} + \frac {1368 \left (\frac {1}{12} - \frac {\sqrt {3} i}{6}\right )^{3}}{481} + \frac {943 \sqrt {3} i}{5772} \right )} + \left (\frac {1}{12} - \frac {\sqrt {3}}{6}\right ) \log {\left (x - \frac {108000 \left (\frac {1}{12} - \frac {\sqrt {3}}{6}\right )^{4}}{481} + \frac {1368 \left (\frac {1}{12} - \frac {\sqrt {3}}{6}\right )^{3}}{481} + \frac {943 \sqrt {3}}{5772} + \frac {4158 \left (\frac {1}{12} - \frac {\sqrt {3}}{6}\right )^{2}}{481} + \frac {17413}{11544} \right )} + \left (\frac {1}{12} + \frac {\sqrt {3}}{6}\right ) \log {\left (x - \frac {108000 \left (\frac {1}{12} + \frac {\sqrt {3}}{6}\right )^{4}}{481} - \frac {943 \sqrt {3}}{5772} + \frac {1368 \left (\frac {1}{12} + \frac {\sqrt {3}}{6}\right )^{3}}{481} + \frac {4158 \left (\frac {1}{12} + \frac {\sqrt {3}}{6}\right )^{2}}{481} + \frac {17413}{11544} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**3+3)/(x**5-9*x),x)

[Out]

-log(x)/3 + (1/12 + sqrt(3)*I/6)*log(x + 17413/11544 - 943*sqrt(3)*I/5772 + 1368*(1/12 + sqrt(3)*I/6)**3/481 +
4158*(1/12 + sqrt(3)*I/6)**2/481 - 108000*(1/12 + sqrt(3)*I/6)**4/481) + (1/12 - sqrt(3)*I/6)*log(x + 17413/1
1544 - 108000*(1/12 - sqrt(3)*I/6)**4/481 + 4158*(1/12 - sqrt(3)*I/6)**2/481 + 1368*(1/12 - sqrt(3)*I/6)**3/48
1 + 943*sqrt(3)*I/5772) + (1/12 - sqrt(3)/6)*log(x - 108000*(1/12 - sqrt(3)/6)**4/481 + 1368*(1/12 - sqrt(3)/6
)**3/481 + 943*sqrt(3)/5772 + 4158*(1/12 - sqrt(3)/6)**2/481 + 17413/11544) + (1/12 + sqrt(3)/6)*log(x - 10800
0*(1/12 + sqrt(3)/6)**4/481 - 943*sqrt(3)/5772 + 1368*(1/12 + sqrt(3)/6)**3/481 + 4158*(1/12 + sqrt(3)/6)**2/4
81 + 17413/11544)

________________________________________________________________________________________