\(\int \frac {-1+x^6}{x^7 \sqrt {1+x^6}} \, dx\) [385]

Optimal result

Integrand size = 18, antiderivative size = 31 \[ \int \frac {-1+x^6}{x^7 \sqrt {1+x^6}} \, dx=\frac {\sqrt {1+x^6}}{6 x^6}-\frac {1}{2} \text {arctanh}\left (\sqrt {1+x^6}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, 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, 79, 65, 213} \[ \int \frac {-1+x^6}{x^7 \sqrt {1+x^6}} \, dx=\frac {\sqrt {x^6+1}}{6 x^6}-\frac {1}{2} \text {arctanh}\left (\sqrt {x^6+1}\right ) \]

[In]

Int[(-1 + x^6)/(x^7*Sqrt[1 + x^6]),x]

[Out]

Sqrt[1 + x^6]/(6*x^6) - ArcTanh[Sqrt[1 + x^6]]/2

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 79

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

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^2 \sqrt {1+x}} \, dx,x,x^6\right ) \\ & = \frac {\sqrt {1+x^6}}{6 x^6}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^6\right ) \\ & = \frac {\sqrt {1+x^6}}{6 x^6}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^6}\right ) \\ & = \frac {\sqrt {1+x^6}}{6 x^6}-\frac {1}{2} \text {arctanh}\left (\sqrt {1+x^6}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^6}{x^7 \sqrt {1+x^6}} \, dx=\frac {\sqrt {1+x^6}}{6 x^6}-\frac {1}{2} \text {arctanh}\left (\sqrt {1+x^6}\right ) \]

[In]

Integrate[(-1 + x^6)/(x^7*Sqrt[1 + x^6]),x]

[Out]

Sqrt[1 + x^6]/(6*x^6) - ArcTanh[Sqrt[1 + x^6]]/2

Maple [A] (verified)

Time = 1.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(\frac {-3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{6}+1}}\right ) x^{6}+\sqrt {x^{6}+1}}{6 x^{6}}\)	\(27\)
trager	\(\frac {\sqrt {x^{6}+1}}{6 x^{6}}+\frac {\ln \left (\frac {\sqrt {x^{6}+1}-1}{x^{3}}\right )}{2}\)	\(30\)
risch	\(\frac {\sqrt {x^{6}+1}}{6 x^{6}}+\frac {-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }}{4 \sqrt {\pi }}\)	\(50\)
meijerg	\(\frac {-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }}{6 \sqrt {\pi }}-\frac {\frac {\sqrt {\pi }\, \left (4 x^{6}+8\right )}{8 x^{6}}-\frac {\sqrt {\pi }\, \sqrt {x^{6}+1}}{x^{6}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{x^{6}}}{6 \sqrt {\pi }}\)	\(113\)

[In]

int((x^6-1)/x^7/(x^6+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45 \[ \int \frac {-1+x^6}{x^7 \sqrt {1+x^6}} \, dx=-\frac {3 \, x^{6} \log \left (\sqrt {x^{6} + 1} + 1\right ) - 3 \, x^{6} \log \left (\sqrt {x^{6} + 1} - 1\right ) - 2 \, \sqrt {x^{6} + 1}}{12 \, x^{6}} \]

[In]

integrate((x^6-1)/x^7/(x^6+1)^(1/2),x, algorithm="fricas")

[Out]

-1/12*(3*x^6*log(sqrt(x^6 + 1) + 1) - 3*x^6*log(sqrt(x^6 + 1) - 1) - 2*sqrt(x^6 + 1))/x^6

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).

Time = 16.62 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71 \[ \int \frac {-1+x^6}{x^7 \sqrt {1+x^6}} \, dx=\frac {\log {\left (\sqrt {x^{6} + 1} - 1 \right )}}{4} - \frac {\log {\left (\sqrt {x^{6} + 1} + 1 \right )}}{4} + \frac {1}{12 \left (\sqrt {x^{6} + 1} + 1\right )} + \frac {1}{12 \left (\sqrt {x^{6} + 1} - 1\right )} \]

[In]

integrate((x**6-1)/x**7/(x**6+1)**(1/2),x)

[Out]

log(sqrt(x**6 + 1) - 1)/4 - log(sqrt(x**6 + 1) + 1)/4 + 1/(12*(sqrt(x**6 + 1) + 1)) + 1/(12*(sqrt(x**6 + 1) -
1))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {-1+x^6}{x^7 \sqrt {1+x^6}} \, dx=\frac {\sqrt {x^{6} + 1}}{6 \, x^{6}} - \frac {1}{4} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \]

[In]

integrate((x^6-1)/x^7/(x^6+1)^(1/2),x, algorithm="maxima")

[Out]

1/6*sqrt(x^6 + 1)/x^6 - 1/4*log(sqrt(x^6 + 1) + 1) + 1/4*log(sqrt(x^6 + 1) - 1)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {-1+x^6}{x^7 \sqrt {1+x^6}} \, dx=\frac {\sqrt {x^{6} + 1}}{6 \, x^{6}} - \frac {1}{4} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \]

[In]

integrate((x^6-1)/x^7/(x^6+1)^(1/2),x, algorithm="giac")

[Out]

1/6*sqrt(x^6 + 1)/x^6 - 1/4*log(sqrt(x^6 + 1) + 1) + 1/4*log(sqrt(x^6 + 1) - 1)

Mupad [B] (verification not implemented)

Time = 5.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {-1+x^6}{x^7 \sqrt {1+x^6}} \, dx=\frac {\sqrt {x^6+1}}{6\,x^6}-\frac {\mathrm {atanh}\left (\sqrt {x^6+1}\right )}{2} \]

[In]

int((x^6 - 1)/(x^7*(x^6 + 1)^(1/2)),x)

[Out]

(x^6 + 1)^(1/2)/(6*x^6) - atanh((x^6 + 1)^(1/2))/2