3.11.23 \(\int \frac {1+x^6}{x (1-x^6)} \, dx\) [1023]

Optimal. Leaf size=15 \[ \log (x)-\frac {1}{3} \log \left (1-x^6\right ) \]

[Out]

ln(x)-1/3*ln(-x^6+1)

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {457, 78} \begin {gather*} \log (x)-\frac {1}{3} \log \left (1-x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^6)/(x*(1 - x^6)),x]

[Out]

Log[x] - Log[1 - x^6]/3

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1+x^6}{x \left (1-x^6\right )} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {1+x}{(1-x) x} \, dx,x,x^6\right )\\ &=\frac {1}{6} \text {Subst}\left (\int \left (-\frac {2}{-1+x}+\frac {1}{x}\right ) \, dx,x,x^6\right )\\ &=\log (x)-\frac {1}{3} \log \left (1-x^6\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \log (x)-\frac {1}{3} \log \left (1-x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^6)/(x*(1 - x^6)),x]

[Out]

Log[x] - Log[1 - x^6]/3

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(35\) vs. \(2(13)=26\).
time = 0.32, size = 36, normalized size = 2.40

method	result	size
risch	\(\ln \left (x \right )-\frac {\ln \left (x^{6}-1\right )}{3}\)	\(12\)
meijerg	\(-\frac {\ln \left (-x^{6}+1\right )}{3}+\ln \left (x \right )+\frac {i \pi }{6}\)	\(18\)
default	\(-\frac {\ln \left (x +1\right )}{3}-\frac {\ln \left (x^{2}+x +1\right )}{3}-\frac {\ln \left (x -1\right )}{3}+\ln \left (x \right )-\frac {\ln \left (x^{2}-x +1\right )}{3}\)	\(36\)
norman	\(-\frac {\ln \left (x +1\right )}{3}-\frac {\ln \left (x^{2}+x +1\right )}{3}-\frac {\ln \left (x -1\right )}{3}+\ln \left (x \right )-\frac {\ln \left (x^{2}-x +1\right )}{3}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 15, normalized size = 1.00 \begin {gather*} -\frac {1}{3} \, \log \left (x^{6} - 1\right ) + \frac {1}{6} \, \log \left (x^{6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*log(x^6 - 1) + 1/6*log(x^6)

________________________________________________________________________________________

Fricas [A]
time = 2.63, size = 11, normalized size = 0.73 \begin {gather*} -\frac {1}{3} \, \log \left (x^{6} - 1\right ) + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*log(x^6 - 1) + log(x)

________________________________________________________________________________________

Sympy [A]
time = 0.04, size = 10, normalized size = 0.67 \begin {gather*} \log {\left (x \right )} - \frac {\log {\left (x^{6} - 1 \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) - log(x**6 - 1)/3

________________________________________________________________________________________

Giac [A]
time = 0.76, size = 16, normalized size = 1.07 \begin {gather*} \frac {1}{6} \, \log \left (x^{6}\right ) - \frac {1}{3} \, \log \left ({\left | x^{6} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*log(x^6) - 1/3*log(abs(x^6 - 1))

________________________________________________________________________________________

Mupad [B]
time = 0.09, size = 11, normalized size = 0.73 \begin {gather*} \ln \left (x\right )-\frac {\ln \left (x^6-1\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^6 + 1)/(x*(x^6 - 1)),x)

[Out]

log(x) - log(x^6 - 1)/3

________________________________________________________________________________________