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

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

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {x^2}{2}-\frac {1}{2} \log \left (1-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(x - x^3),x]

[Out]

-x^2/2 - Log[1 - x^2]/2

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])

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 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]

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*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 0.90 \begin {gather*} -\frac {x^2}{2}-\frac {1}{2} \log \left (x^2-1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/(x - x^3),x]

[Out]

-1/2*x^2 - Log[-1 + x^2]/2

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4}{x-x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^4/(x - x^3),x]

[Out]

IntegrateAlgebraic[x^4/(x - x^3), x]

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fricas [A]  time = 0.39, size = 14, normalized size = 0.70 \begin {gather*} -\frac {1}{2} \, x^{2} - \frac {1}{2} \, \log \left (x^{2} - 1\right ) \end {gather*}

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)

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giac [A]  time = 0.15, size = 15, normalized size = 0.75 \begin {gather*} -\frac {1}{2} \, x^{2} - \frac {1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end {gather*}

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))

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maple [A]  time = 0.04, size = 19, normalized size = 0.95 \begin {gather*} -\frac {x^{2}}{2}-\frac {\ln \left (x -1\right )}{2}-\frac {\ln \left (x +1\right )}{2} \end {gather*}

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

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maxima [A]  time = 1.32, size = 18, normalized size = 0.90 \begin {gather*} -\frac {1}{2} \, x^{2} - \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \end {gather*}

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)

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mupad [B]  time = 0.04, size = 14, normalized size = 0.70 \begin {gather*} -\frac {\ln \left (x^2-1\right )}{2}-\frac {x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(x - x^3),x)

[Out]

- log(x^2 - 1)/2 - x^2/2

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sympy [A]  time = 0.08, size = 14, normalized size = 0.70 \begin {gather*} - \frac {x^{2}}{2} - \frac {\log {\left (x^{2} - 1 \right )}}{2} \end {gather*}

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

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