3.1.35 \(\int \frac {1}{x^4 (x-x^3)} \, dx\) [35]

Optimal. Leaf size=29 \[ -\frac {1}{4 x^4}-\frac {1}{2 x^2}+\log (x)-\frac {1}{2} \log \left (1-x^2\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 29, 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, 272, 46} \begin {gather*} -\frac {1}{4 x^4}-\frac {1}{2 x^2}-\frac {1}{2} \log \left (1-x^2\right )+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(x - x^3)),x]

[Out]

-1/4*1/x^4 - 1/(2*x^2) + Log[x] - Log[1 - x^2]/2

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 272

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

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Mathematica [A]
time = 0.00, size = 29, normalized size = 1.00 \begin {gather*} -\frac {1}{4 x^4}-\frac {1}{2 x^2}+\log (x)-\frac {1}{2} \log \left (1-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(x - x^3)),x]

[Out]

-1/4*1/x^4 - 1/(2*x^2) + Log[x] - Log[1 - x^2]/2

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Maple [A]
time = 0.36, size = 26, normalized size = 0.90

method	result	size
risch	\(\frac {-\frac {1}{4}-\frac {x^{2}}{2}}{x^{4}}+\ln \left (x \right )-\frac {\ln \left (x^{2}-1\right )}{2}\)	\(23\)
default	\(-\frac {\ln \left (x +1\right )}{2}-\frac {\ln \left (x -1\right )}{2}-\frac {1}{4 x^{4}}-\frac {1}{2 x^{2}}+\ln \left (x \right )\)	\(26\)
norman	\(\frac {-\frac {1}{4}-\frac {x^{2}}{2}}{x^{4}}-\frac {\ln \left (x +1\right )}{2}-\frac {\ln \left (x -1\right )}{2}+\ln \left (x \right )\)	\(27\)
meijerg	\(-\frac {\ln \left (-x^{2}+1\right )}{2}+\ln \left (x \right )+\frac {i \pi }{2}-\frac {1}{4 x^{4}}-\frac {1}{2 x^{2}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(-x^3+x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 27, normalized size = 0.93 \begin {gather*} -\frac {2 \, x^{2} + 1}{4 \, x^{4}} - \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(-x^3+x),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 2.70, size = 30, normalized size = 1.03 \begin {gather*} -\frac {2 \, x^{4} \log \left (x^{2} - 1\right ) - 4 \, x^{4} \log \left (x\right ) + 2 \, x^{2} + 1}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(-x^3+x),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.04, size = 22, normalized size = 0.76 \begin {gather*} \log {\left (x \right )} - \frac {\log {\left (x^{2} - 1 \right )}}{2} - \frac {2 x^{2} + 1}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(-x**3+x),x)

[Out]

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

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Giac [A]
time = 1.55, size = 33, normalized size = 1.14 \begin {gather*} -\frac {3 \, x^{4} + 2 \, x^{2} + 1}{4 \, x^{4}} + \frac {1}{2} \, \log \left (x^{2}\right ) - \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(1/x^4/(-x^3+x),x, algorithm="giac")

[Out]

-1/4*(3*x^4 + 2*x^2 + 1)/x^4 + 1/2*log(x^2) - 1/2*log(abs(x^2 - 1))

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Mupad [B]
time = 0.03, size = 23, normalized size = 0.79 \begin {gather*} \ln \left (x\right )-\frac {\ln \left (x^2-1\right )}{2}-\frac {\frac {x^2}{2}+\frac {1}{4}}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(x - x^3)),x)

[Out]

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

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