### 3.206 $$\int \frac{x^4}{-1+x^4} \, dx$$

Optimal. Leaf size=14 $x-\frac{1}{2} \tan ^{-1}(x)-\frac{1}{2} \tanh ^{-1}(x)$

[Out]

x - ArcTan[x]/2 - ArcTanh[x]/2

________________________________________________________________________________________

Rubi [A]  time = 0.0063513, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.364, Rules used = {321, 212, 206, 203} $x-\frac{1}{2} \tan ^{-1}(x)-\frac{1}{2} \tanh ^{-1}(x)$

Antiderivative was successfully veriﬁed.

[In]

Int[x^4/(-1 + x^4),x]

[Out]

x - ArcTan[x]/2 - ArcTanh[x]/2

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 212

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

Rubi steps

\begin{align*} \int \frac{x^4}{-1+x^4} \, dx &=x+\int \frac{1}{-1+x^4} \, dx\\ &=x-\frac{1}{2} \int \frac{1}{1-x^2} \, dx-\frac{1}{2} \int \frac{1}{1+x^2} \, dx\\ &=x-\frac{1}{2} \tan ^{-1}(x)-\frac{1}{2} \tanh ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0041189, size = 26, normalized size = 1.86 $x+\frac{1}{4} \log (1-x)-\frac{1}{4} \log (x+1)-\frac{1}{2} \tan ^{-1}(x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^4/(-1 + x^4),x]

[Out]

x - ArcTan[x]/2 + Log[1 - x]/4 - Log[1 + x]/4

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 19, normalized size = 1.4 \begin{align*} x+{\frac{\ln \left ( -1+x \right ) }{4}}-{\frac{\ln \left ( 1+x \right ) }{4}}-{\frac{\arctan \left ( x \right ) }{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(x^4-1),x)

[Out]

x+1/4*ln(-1+x)-1/4*ln(1+x)-1/2*arctan(x)

________________________________________________________________________________________

Maxima [A]  time = 1.40145, size = 24, normalized size = 1.71 \begin{align*} x - \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left (x + 1\right ) + \frac{1}{4} \, \log \left (x - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^4-1),x, algorithm="maxima")

[Out]

x - 1/2*arctan(x) - 1/4*log(x + 1) + 1/4*log(x - 1)

________________________________________________________________________________________

Fricas [A]  time = 1.90322, size = 72, normalized size = 5.14 \begin{align*} x - \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left (x + 1\right ) + \frac{1}{4} \, \log \left (x - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^4-1),x, algorithm="fricas")

[Out]

x - 1/2*arctan(x) - 1/4*log(x + 1) + 1/4*log(x - 1)

________________________________________________________________________________________

Sympy [A]  time = 0.113575, size = 19, normalized size = 1.36 \begin{align*} x + \frac{\log{\left (x - 1 \right )}}{4} - \frac{\log{\left (x + 1 \right )}}{4} - \frac{\operatorname{atan}{\left (x \right )}}{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(x**4-1),x)

[Out]

x + log(x - 1)/4 - log(x + 1)/4 - atan(x)/2

________________________________________________________________________________________

Giac [A]  time = 1.05932, size = 27, normalized size = 1.93 \begin{align*} x - \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left ({\left | x + 1 \right |}\right ) + \frac{1}{4} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^4-1),x, algorithm="giac")

[Out]

x - 1/2*arctan(x) - 1/4*log(abs(x + 1)) + 1/4*log(abs(x - 1))