\(\int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx\) [332]

Optimal result

Integrand size = 18, antiderivative size = 28 \[ \int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx=\frac {\sqrt {1+x^6}}{3}+\frac {1}{3} \text {arctanh}\left (\sqrt {1+x^6}\right ) \]

[Out]

1/3*(x^6+1)^(1/2)+1/3*arctanh((x^6+1)^(1/2))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {457, 81, 65, 213} \[ \int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx=\frac {1}{3} \text {arctanh}\left (\sqrt {x^6+1}\right )+\frac {\sqrt {x^6+1}}{3} \]

[In]

Int[(-1 + x^6)/(x*Sqrt[1 + x^6]),x]

[Out]

Sqrt[1 + x^6]/3 + ArcTanh[Sqrt[1 + x^6]]/3

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx=\frac {1}{3} \left (\sqrt {1+x^6}+\text {arctanh}\left (\sqrt {1+x^6}\right )\right ) \]

[In]

Integrate[(-1 + x^6)/(x*Sqrt[1 + x^6]),x]

[Out]

(Sqrt[1 + x^6] + ArcTanh[Sqrt[1 + x^6]])/3

Maple [A] (verified)

Time = 0.85 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75

method	result	size
pseudoelliptic	\(\frac {\sqrt {x^{6}+1}}{3}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{6}+1}}\right )}{3}\)	\(21\)
trager	\(\frac {\sqrt {x^{6}+1}}{3}+\frac {\ln \left (\frac {\sqrt {x^{6}+1}+1}{x^{3}}\right )}{3}\)	\(27\)
meijerg	\(\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {x^{6}+1}}{6 \sqrt {\pi }}-\frac {-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }}{6 \sqrt {\pi }}\)	\(61\)

[In]

int((x^6-1)/x/(x^6+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(x^6+1)^(1/2)+1/3*arctanh(1/(x^6+1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx=\frac {1}{3} \, \sqrt {x^{6} + 1} + \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) - \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \]

[In]

integrate((x^6-1)/x/(x^6+1)^(1/2),x, algorithm="fricas")

[Out]

1/3*sqrt(x^6 + 1) + 1/6*log(sqrt(x^6 + 1) + 1) - 1/6*log(sqrt(x^6 + 1) - 1)

Sympy [A] (verification not implemented)

Time = 4.21 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx=\frac {\sqrt {x^{6} + 1}}{3} - \frac {\log {\left (\sqrt {x^{6} + 1} - 1 \right )}}{6} + \frac {\log {\left (\sqrt {x^{6} + 1} + 1 \right )}}{6} \]

[In]

integrate((x**6-1)/x/(x**6+1)**(1/2),x)

[Out]

sqrt(x**6 + 1)/3 - log(sqrt(x**6 + 1) - 1)/6 + log(sqrt(x**6 + 1) + 1)/6

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx=\frac {1}{3} \, \sqrt {x^{6} + 1} + \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) - \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \]

[In]

integrate((x^6-1)/x/(x^6+1)^(1/2),x, algorithm="maxima")

[Out]

1/3*sqrt(x^6 + 1) + 1/6*log(sqrt(x^6 + 1) + 1) - 1/6*log(sqrt(x^6 + 1) - 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx=\frac {1}{3} \, \sqrt {x^{6} + 1} + \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) - \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \]

[In]

integrate((x^6-1)/x/(x^6+1)^(1/2),x, algorithm="giac")

[Out]

1/3*sqrt(x^6 + 1) + 1/6*log(sqrt(x^6 + 1) + 1) - 1/6*log(sqrt(x^6 + 1) - 1)

Mupad [B] (verification not implemented)

Time = 5.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx=\frac {\mathrm {atanh}\left (\sqrt {x^6+1}\right )}{3}+\frac {\sqrt {x^6+1}}{3} \]

[In]

int((x^6 - 1)/(x*(x^6 + 1)^(1/2)),x)

[Out]

atanh((x^6 + 1)^(1/2))/3 + (x^6 + 1)^(1/2)/3