3.27 \(\int \frac{x^4}{x-x^3} \, dx\)

Optimal. Leaf size=20 \[ -\frac{x^2}{2}-\frac{1}{2} \log \left (1-x^2\right ) \]

[Out]

-x^2/2 - Log[1 - x^2]/2

________________________________________________________________________________________

Rubi [A]  time = 0.0151658, antiderivative size = 20, 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, 266, 43} \[ -\frac{x^2}{2}-\frac{1}{2} \log \left (1-x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^4/(x - x^3),x]

[Out]

-x^2/2 - Log[1 - x^2]/2

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 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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0027918, size = 18, normalized size = 0.9 \[ -\frac{x^2}{2}-\frac{1}{2} \log \left (x^2-1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(x - x^3),x]

[Out]

-x^2/2 - Log[-1 + x^2]/2

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 19, normalized size = 1. \begin{align*} -{\frac{{x}^{2}}{2}}-{\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^4/(-x^3+x),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.03579, size = 24, normalized size = 1.2 \begin{align*} -\frac{1}{2} \, x^{2} - \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^4/(-x^3+x),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.56051, size = 39, normalized size = 1.95 \begin{align*} -\frac{1}{2} \, x^{2} - \frac{1}{2} \, \log \left (x^{2} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(-x^3+x),x, algorithm="fricas")

[Out]

-1/2*x^2 - 1/2*log(x^2 - 1)

________________________________________________________________________________________

Sympy [A]  time = 0.075149, size = 14, normalized size = 0.7 \begin{align*} - \frac{x^{2}}{2} - \frac{\log{\left (x^{2} - 1 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(-x**3+x),x)

[Out]

-x**2/2 - log(x**2 - 1)/2

________________________________________________________________________________________

Giac [A]  time = 1.32408, size = 20, normalized size = 1. \begin{align*} -\frac{1}{2} \, x^{2} - \frac{1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(-x^3+x),x, algorithm="giac")

[Out]

-1/2*x^2 - 1/2*log(abs(x^2 - 1))