3.279 \(\int \frac{1+x^6}{x-x^7} \, dx\)

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

[Out]

Log[x] - Log[1 - x^6]/3

________________________________________________________________________________________

Rubi [A]  time = 0.0219739, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1593, 446, 72} \[ \log (x)-\frac{1}{3} \log \left (1-x^6\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^6)/(x - x^7),x]

[Out]

Log[x] - Log[1 - x^6]/3

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 446

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]]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int \frac{1+x^6}{x-x^7} \, dx &=\int \frac{1+x^6}{x \left (1-x^6\right )} \, dx\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1+x}{(1-x) x} \, dx,x,x^6\right )\\ &=\frac{1}{6} \operatorname{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 = 0.0050982, size = 15, normalized size = 1. \[ \log (x)-\frac{1}{3} \log \left (1-x^6\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^6)/(x - x^7),x]

[Out]

Log[x] - Log[1 - x^6]/3

________________________________________________________________________________________

Maple [B]  time = 0.055, size = 36, normalized size = 2.4 \begin{align*} -{\frac{\ln \left ({x}^{2}-x+1 \right ) }{3}}+\ln \left ( x \right ) -{\frac{\ln \left ( 1+x \right ) }{3}}-{\frac{\ln \left ( -1+x \right ) }{3}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.76386, size = 47, normalized size = 3.13 \begin{align*} -\frac{1}{3} \, \log \left (x^{2} + x + 1\right ) - \frac{1}{3} \, \log \left (x^{2} - x + 1\right ) - \frac{1}{3} \, \log \left (x + 1\right ) - \frac{1}{3} \, \log \left (x - 1\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.11852, size = 38, normalized size = 2.53 \begin{align*} -\frac{1}{3} \, \log \left (x^{6} - 1\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*log(x^6 - 1) + log(x)

________________________________________________________________________________________

Sympy [A]  time = 0.133748, size = 10, normalized size = 0.67 \begin{align*} \log{\left (x \right )} - \frac{\log{\left (x^{6} - 1 \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) - log(x**6 - 1)/3

________________________________________________________________________________________

Giac [A]  time = 1.24728, size = 22, normalized size = 1.47 \begin{align*} \frac{1}{6} \, \log \left (x^{6}\right ) - \frac{1}{3} \, \log \left ({\left | x^{6} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*log(x^6) - 1/3*log(abs(x^6 - 1))