\(\int \frac {x^5}{1+x^6} \, dx\) [1364]

Optimal result

Integrand size = 11, antiderivative size = 10 \[ \int \frac {x^5}{1+x^6} \, dx=\frac {1}{6} \log \left (1+x^6\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {266} \[ \int \frac {x^5}{1+x^6} \, dx=\frac {1}{6} \log \left (x^6+1\right ) \]

[In]

Int[x^5/(1 + x^6),x]

[Out]

Log[1 + x^6]/6

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{1+x^6} \, dx=\frac {1}{6} \log \left (1+x^6\right ) \]

[In]

Integrate[x^5/(1 + x^6),x]

[Out]

Log[1 + x^6]/6

Maple [A] (verified)

Time = 4.38 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {\ln \left (x^{6}+1\right )}{6}\)	\(9\)
default	\(\frac {\ln \left (x^{6}+1\right )}{6}\)	\(9\)
meijerg	\(\frac {\ln \left (x^{6}+1\right )}{6}\)	\(9\)
risch	\(\frac {\ln \left (x^{6}+1\right )}{6}\)	\(9\)
norman	\(\frac {\ln \left (x^{2}+1\right )}{6}+\frac {\ln \left (x^{4}-x^{2}+1\right )}{6}\)	\(23\)
parallelrisch	\(\frac {\ln \left (x^{2}+1\right )}{6}+\frac {\ln \left (x^{4}-x^{2}+1\right )}{6}\)	\(23\)

[In]

int(x^5/(x^6+1),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {x^5}{1+x^6} \, dx=\frac {1}{6} \, \log \left (x^{6} + 1\right ) \]

[In]

integrate(x^5/(x^6+1),x, algorithm="fricas")

[Out]

1/6*log(x^6 + 1)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \frac {x^5}{1+x^6} \, dx=\frac {\log {\left (x^{6} + 1 \right )}}{6} \]

[In]

integrate(x**5/(x**6+1),x)

[Out]

log(x**6 + 1)/6

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {x^5}{1+x^6} \, dx=\frac {1}{6} \, \log \left (x^{6} + 1\right ) \]

[In]

integrate(x^5/(x^6+1),x, algorithm="maxima")

[Out]

1/6*log(x^6 + 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {x^5}{1+x^6} \, dx=\frac {1}{6} \, \log \left (x^{6} + 1\right ) \]

[In]

integrate(x^5/(x^6+1),x, algorithm="giac")

[Out]

1/6*log(x^6 + 1)

Mupad [B] (verification not implemented)

Time = 5.75 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {x^5}{1+x^6} \, dx=\frac {\ln \left (x^6+1\right )}{6} \]

[In]

int(x^5/(x^6 + 1),x)

[Out]

log(x^6 + 1)/6