3.1.26 \(\int \frac {x^5}{x-x^3} \, dx\) [26]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 13, 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 = {1598, 308, 212} \begin {gather*} -\frac {x^3}{3}-x+\tanh ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(x - x^3),x]

[Out]

-x - x^3/3 + ArcTanh[x]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 308

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 1598

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^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] Leaf count is larger than twice the leaf count of optimal. \(29\) vs. \(2(13)=26\).
time = 0.00, size = 29, normalized size = 2.23 \begin {gather*} -x-\frac {x^3}{3}-\frac {1}{2} \log (1-x)+\frac {1}{2} \log (1+x) \end {gather*}

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

________________________________________________________________________________________

Maple [A]
time = 0.35, size = 22, normalized size = 1.69

method	result	size
meijerg	\(-\frac {i \left (-\frac {2 i x \left (5 x^{2}+15\right )}{15}+2 i \arctanh \left (x \right )\right )}{2}\)	\(21\)
default	\(-\frac {x^{3}}{3}-x -\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2}\)	\(22\)
norman	\(-\frac {x^{3}}{3}-x -\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2}\)	\(22\)
risch	\(-\frac {x^{3}}{3}-x -\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(-x^3+x),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 21, normalized size = 1.62 \begin {gather*} -\frac {1}{3} \, x^{3} - x + \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^5/(-x^3+x),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 1.60, size = 21, normalized size = 1.62 \begin {gather*} -\frac {1}{3} \, x^{3} - x + \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^5/(-x^3+x),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (8) = 16\).
time = 0.03, size = 19, normalized size = 1.46 \begin {gather*} - \frac {x^{3}}{3} - x - \frac {\log {\left (x - 1 \right )}}{2} + \frac {\log {\left (x + 1 \right )}}{2} \end {gather*}

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (11) = 22\).
time = 1.08, size = 23, normalized size = 1.77 \begin {gather*} -\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 {gather*}

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))

________________________________________________________________________________________

Mupad [B]
time = 4.98, size = 11, normalized size = 0.85 \begin {gather*} \mathrm {atanh}\left (x\right )-x-\frac {x^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(x - x^3),x)

[Out]

atanh(x) - x - x^3/3

________________________________________________________________________________________