3.26 \(\int \frac{x^5}{x-x^3} \, dx\)

Optimal. Leaf size=13 \[ -\frac{x^3}{3}-x+\tanh ^{-1}(x) \]

[Out]

-x - x^3/3 + ArcTanh[x]

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Rubi [A]  time = 0.010937, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1584, 302, 206} \[ -\frac{x^3}{3}-x+\tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[x^5/(x - x^3),x]

[Out]

-x - x^3/3 + ArcTanh[x]

Rule 1584

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

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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])

Rubi steps

\begin{align*} \int \frac{x^5}{x-x^3} \, dx &=\int \frac{x^4}{1-x^2} \, dx\\ &=\int \left (-1-x^2+\frac{1}{1-x^2}\right ) \, dx\\ &=-x-\frac{x^3}{3}+\int \frac{1}{1-x^2} \, dx\\ &=-x-\frac{x^3}{3}+\tanh ^{-1}(x)\\ \end{align*}

Mathematica [B]  time = 0.004208, size = 29, normalized size = 2.23 \[ -\frac{x^3}{3}-x-\frac{1}{2} \log (1-x)+\frac{1}{2} \log (x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(x - x^3),x]

[Out]

-x - x^3/3 - Log[1 - x]/2 + Log[1 + x]/2

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Maple [A]  time = 0.003, size = 22, normalized size = 1.7 \begin{align*} -{\frac{{x}^{3}}{3}}-x-{\frac{\ln \left ( -1+x \right ) }{2}}+{\frac{\ln \left ( 1+x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(-x^3+x),x)

[Out]

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

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Maxima [A]  time = 1.06704, size = 28, normalized size = 2.15 \begin{align*} -\frac{1}{3} \, x^{3} - x + \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.50523, size = 65, normalized size = 5. \begin{align*} -\frac{1}{3} \, x^{3} - x + \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.093696, size = 19, normalized size = 1.46 \begin{align*} - \frac{x^{3}}{3} - x - \frac{\log{\left (x - 1 \right )}}{2} + \frac{\log{\left (x + 1 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.19669, size = 31, normalized size = 2.38 \begin{align*} -\frac{1}{3} \, x^{3} - x + \frac{1}{2} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*x^3 - x + 1/2*log(abs(x + 1)) - 1/2*log(abs(x - 1))