\(\int \frac {1}{x^7 (1-x^6)} \, dx\) [1359]

Optimal result

Integrand size = 13, antiderivative size = 22 \[ \int \frac {1}{x^7 \left (1-x^6\right )} \, dx=-\frac {1}{6 x^6}+\log (x)-\frac {1}{6} \log \left (1-x^6\right ) \]

[Out]

-1/6/x^6+ln(x)-1/6*ln(-x^6+1)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 46} \[ \int \frac {1}{x^7 \left (1-x^6\right )} \, dx=-\frac {1}{6 x^6}-\frac {1}{6} \log \left (1-x^6\right )+\log (x) \]

[In]

Int[1/(x^7*(1 - x^6)),x]

[Out]

-1/6*1/x^6 + Log[x] - Log[1 - x^6]/6

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]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \text {Subst}\left (\int \frac {1}{(1-x) x^2} \, dx,x,x^6\right ) \\ & = \frac {1}{6} \text {Subst}\left (\int \left (\frac {1}{1-x}+\frac {1}{x^2}+\frac {1}{x}\right ) \, dx,x,x^6\right ) \\ & = -\frac {1}{6 x^6}+\log (x)-\frac {1}{6} \log \left (1-x^6\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^7 \left (1-x^6\right )} \, dx=-\frac {1}{6 x^6}+\log (x)-\frac {1}{6} \log \left (1-x^6\right ) \]

[In]

Integrate[1/(x^7*(1 - x^6)),x]

[Out]

-1/6*1/x^6 + Log[x] - Log[1 - x^6]/6

Maple [A] (verified)

Time = 4.40 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77

method	result	size
risch	\(-\frac {1}{6 x^{6}}+\ln \left (x \right )-\frac {\ln \left (x^{6}-1\right )}{6}\)	\(17\)
meijerg	\(-\frac {1}{6 x^{6}}+\ln \left (x \right )+\frac {i \pi }{6}-\frac {\ln \left (-x^{6}+1\right )}{6}\)	\(23\)
default	\(-\frac {1}{6 x^{6}}+\ln \left (x \right )-\frac {\ln \left (-1+x \right )}{6}-\frac {\ln \left (1+x \right )}{6}-\frac {\ln \left (x^{2}+x +1\right )}{6}-\frac {\ln \left (x^{2}-x +1\right )}{6}\)	\(41\)
norman	\(-\frac {1}{6 x^{6}}+\ln \left (x \right )-\frac {\ln \left (-1+x \right )}{6}-\frac {\ln \left (1+x \right )}{6}-\frac {\ln \left (x^{2}+x +1\right )}{6}-\frac {\ln \left (x^{2}-x +1\right )}{6}\)	\(41\)
parallelrisch	\(\frac {6 \ln \left (x \right ) x^{6}-\ln \left (1+x \right ) x^{6}-\ln \left (-1+x \right ) x^{6}-\ln \left (x^{2}-x +1\right ) x^{6}-\ln \left (x^{2}+x +1\right ) x^{6}-1}{6 x^{6}}\)	\(59\)

[In]

int(1/x^7/(-x^6+1),x,method=_RETURNVERBOSE)

[Out]

-1/6/x^6+ln(x)-1/6*ln(x^6-1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^7 \left (1-x^6\right )} \, dx=-\frac {x^{6} \log \left (x^{6} - 1\right ) - 6 \, x^{6} \log \left (x\right ) + 1}{6 \, x^{6}} \]

[In]

integrate(1/x^7/(-x^6+1),x, algorithm="fricas")

[Out]

-1/6*(x^6*log(x^6 - 1) - 6*x^6*log(x) + 1)/x^6

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^7 \left (1-x^6\right )} \, dx=\log {\left (x \right )} - \frac {\log {\left (x^{6} - 1 \right )}}{6} - \frac {1}{6 x^{6}} \]

[In]

integrate(1/x**7/(-x**6+1),x)

[Out]

log(x) - log(x**6 - 1)/6 - 1/(6*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^7 \left (1-x^6\right )} \, dx=-\frac {1}{6 \, x^{6}} - \frac {1}{6} \, \log \left (x^{6} - 1\right ) + \frac {1}{6} \, \log \left (x^{6}\right ) \]

[In]

integrate(1/x^7/(-x^6+1),x, algorithm="maxima")

[Out]

-1/6/x^6 - 1/6*log(x^6 - 1) + 1/6*log(x^6)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x^7 \left (1-x^6\right )} \, dx=-\frac {x^{6} + 1}{6 \, x^{6}} + \frac {1}{6} \, \log \left (x^{6}\right ) - \frac {1}{6} \, \log \left ({\left | x^{6} - 1 \right |}\right ) \]

[In]

integrate(1/x^7/(-x^6+1),x, algorithm="giac")

[Out]

-1/6*(x^6 + 1)/x^6 + 1/6*log(x^6) - 1/6*log(abs(x^6 - 1))

Mupad [B] (verification not implemented)

Time = 5.64 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x^7 \left (1-x^6\right )} \, dx=\ln \left (x\right )-\frac {\ln \left (x^6-1\right )}{6}-\frac {1}{6\,x^6} \]

[In]

int(-1/(x^7*(x^6 - 1)),x)

[Out]

log(x) - log(x^6 - 1)/6 - 1/(6*x^6)