\(\int \frac {1}{3-19 x^2+32 x^4-16 x^6} \, dx\) [74]

Optimal result

Integrand size = 19, antiderivative size = 31 \[ \int \frac {1}{3-19 x^2+32 x^4-16 x^6} \, dx=\frac {\text {arctanh}(x)}{3}+\frac {1}{3} \text {arctanh}(2 x)-\frac {\text {arctanh}\left (\frac {2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2082, 213} \[ \int \frac {1}{3-19 x^2+32 x^4-16 x^6} \, dx=\frac {\text {arctanh}(x)}{3}+\frac {1}{3} \text {arctanh}(2 x)-\frac {\text {arctanh}\left (\frac {2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]

[In]

Int[(3 - 19*x^2 + 32*x^4 - 16*x^6)^(-1),x]

[Out]

ArcTanh[x]/3 + ArcTanh[2*x]/3 - ArcTanh[(2*x)/Sqrt[3]]/Sqrt[3]

Rule 213

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 2082

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(u /. x -> x^2)^p, x], x
] /;  !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x^2] && ILtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{3 \left (-1+x^2\right )}+\frac {2}{-3+4 x^2}-\frac {2}{3 \left (-1+4 x^2\right )}\right ) \, dx \\ & = -\left (\frac {1}{3} \int \frac {1}{-1+x^2} \, dx\right )-\frac {2}{3} \int \frac {1}{-1+4 x^2} \, dx+2 \int \frac {1}{-3+4 x^2} \, dx \\ & = \frac {1}{3} \tanh ^{-1}(x)+\frac {1}{3} \tanh ^{-1}(2 x)-\frac {\tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )}{\sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.00 \[ \int \frac {1}{3-19 x^2+32 x^4-16 x^6} \, dx=\frac {1}{6} \left (\sqrt {3} \log \left (\sqrt {3}-2 x\right )-\sqrt {3} \log \left (\sqrt {3}+2 x\right )-\log \left (1-3 x+2 x^2\right )+\log \left (1+3 x+2 x^2\right )\right ) \]

[In]

Integrate[(3 - 19*x^2 + 32*x^4 - 16*x^6)^(-1),x]

[Out]

(Sqrt[3]*Log[Sqrt[3] - 2*x] - Sqrt[3]*Log[Sqrt[3] + 2*x] - Log[1 - 3*x + 2*x^2] + Log[1 + 3*x + 2*x^2])/6

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35

method	result	size
default	\(\frac {\ln \left (1+2 x \right )}{6}-\frac {\ln \left (2 x -1\right )}{6}+\frac {\ln \left (x +1\right )}{6}-\frac {\ln \left (x -1\right )}{6}-\frac {\operatorname {arctanh}\left (\frac {2 x \sqrt {3}}{3}\right ) \sqrt {3}}{3}\)	\(42\)
risch	\(\frac {\sqrt {3}\, \ln \left (2 x -\sqrt {3}\right )}{6}-\frac {\sqrt {3}\, \ln \left (2 x +\sqrt {3}\right )}{6}+\frac {\ln \left (2 x^{2}+3 x +1\right )}{6}-\frac {\ln \left (2 x^{2}-3 x +1\right )}{6}\)	\(56\)

[In]

int(1/(-16*x^6+32*x^4-19*x^2+3),x,method=_RETURNVERBOSE)

[Out]

1/6*ln(1+2*x)-1/6*ln(2*x-1)+1/6*ln(x+1)-1/6*ln(x-1)-1/3*arctanh(2/3*x*3^(1/2))*3^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (23) = 46\).

Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81 \[ \int \frac {1}{3-19 x^2+32 x^4-16 x^6} \, dx=\frac {1}{6} \, \sqrt {3} \log \left (\frac {4 \, x^{2} - 4 \, \sqrt {3} x + 3}{4 \, x^{2} - 3}\right ) + \frac {1}{6} \, \log \left (2 \, x^{2} + 3 \, x + 1\right ) - \frac {1}{6} \, \log \left (2 \, x^{2} - 3 \, x + 1\right ) \]

[In]

integrate(1/(-16*x^6+32*x^4-19*x^2+3),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*log((4*x^2 - 4*sqrt(3)*x + 3)/(4*x^2 - 3)) + 1/6*log(2*x^2 + 3*x + 1) - 1/6*log(2*x^2 - 3*x + 1)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (29) = 58\).

Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.03 \[ \int \frac {1}{3-19 x^2+32 x^4-16 x^6} \, dx=\frac {\sqrt {3} \log {\left (x - \frac {\sqrt {3}}{2} \right )}}{6} - \frac {\sqrt {3} \log {\left (x + \frac {\sqrt {3}}{2} \right )}}{6} - \frac {\log {\left (x^{2} - \frac {3 x}{2} + \frac {1}{2} \right )}}{6} + \frac {\log {\left (x^{2} + \frac {3 x}{2} + \frac {1}{2} \right )}}{6} \]

[In]

integrate(1/(-16*x**6+32*x**4-19*x**2+3),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (23) = 46\).

Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \frac {1}{3-19 x^2+32 x^4-16 x^6} \, dx=\frac {1}{6} \, \sqrt {3} \log \left (\frac {2 \, x - \sqrt {3}}{2 \, x + \sqrt {3}}\right ) + \frac {1}{6} \, \log \left (2 \, x + 1\right ) - \frac {1}{6} \, \log \left (2 \, x - 1\right ) + \frac {1}{6} \, \log \left (x + 1\right ) - \frac {1}{6} \, \log \left (x - 1\right ) \]

[In]

integrate(1/(-16*x^6+32*x^4-19*x^2+3),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (23) = 46\).

Time = 0.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.00 \[ \int \frac {1}{3-19 x^2+32 x^4-16 x^6} \, dx=\frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | 8 \, x - 4 \, \sqrt {3} \right |}}{{\left | 8 \, x + 4 \, \sqrt {3} \right |}}\right ) + \frac {1}{6} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) - \frac {1}{6} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) + \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{6} \, \log \left ({\left | x - 1 \right |}\right ) \]

[In]

integrate(1/(-16*x^6+32*x^4-19*x^2+3),x, algorithm="giac")

[Out]

1/6*sqrt(3)*log(abs(8*x - 4*sqrt(3))/abs(8*x + 4*sqrt(3))) + 1/6*log(abs(2*x + 1)) - 1/6*log(abs(2*x - 1)) + 1
/6*log(abs(x + 1)) - 1/6*log(abs(x - 1))

Mupad [B] (verification not implemented)

Time = 10.33 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {1}{3-19 x^2+32 x^4-16 x^6} \, dx=\frac {\mathrm {atanh}\left (\frac {x}{4608\,\left (\frac {x^2}{6912}+\frac {1}{13824}\right )}\right )}{3}-\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {2\,\sqrt {3}\,x}{3}\right )}{3} \]

[In]

int(-1/(19*x^2 - 32*x^4 + 16*x^6 - 3),x)

[Out]

atanh(x/(4608*(x^2/6912 + 1/13824)))/3 - (3^(1/2)*atanh((2*3^(1/2)*x)/3))/3