\(\int \frac {\sqrt {-1+x^6}}{x} \, dx\) [331]

Optimal result

Integrand size = 13, antiderivative size = 28 \[ \int \frac {\sqrt {-1+x^6}}{x} \, dx=\frac {1}{3} \sqrt {-1+x^6}-\frac {1}{3} \arctan \left (\sqrt {-1+x^6}\right ) \]

[Out]

1/3*(x^6-1)^(1/2)-1/3*arctan((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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 52, 65, 209} \[ \int \frac {\sqrt {-1+x^6}}{x} \, dx=\frac {\sqrt {x^6-1}}{3}-\frac {1}{3} \arctan \left (\sqrt {x^6-1}\right ) \]

[In]

Int[Sqrt[-1 + x^6]/x,x]

[Out]

Sqrt[-1 + x^6]/3 - ArcTan[Sqrt[-1 + x^6]]/3

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 209

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

Rule 272

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^6}}{x} \, dx=\frac {1}{3} \sqrt {-1+x^6}-\frac {1}{3} \arctan \left (\sqrt {-1+x^6}\right ) \]

[In]

Integrate[Sqrt[-1 + x^6]/x,x]

[Out]

Sqrt[-1 + x^6]/3 - ArcTan[Sqrt[-1 + x^6]]/3

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75

method	result	size
pseudoelliptic	\(\frac {\sqrt {x^{6}-1}}{3}+\frac {\arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right )}{3}\)	\(21\)
trager	\(\frac {\sqrt {x^{6}-1}}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}\)	\(40\)
meijerg	\(-\frac {\sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {-x^{6}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )-2 \left (2-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{12 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}\)	\(82\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {-1+x^6}}{x} \, dx=\frac {1}{3} \, \sqrt {x^{6} - 1} - \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]

[In]

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

[Out]

1/3*sqrt(x^6 - 1) - 1/3*arctan(sqrt(x^6 - 1))

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.71 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.14 \[ \int \frac {\sqrt {-1+x^6}}{x} \, dx=\begin {cases} - \frac {i x^{3}}{3 \sqrt {-1 + \frac {1}{x^{6}}}} - \frac {i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{3} + \frac {i}{3 x^{3} \sqrt {-1 + \frac {1}{x^{6}}}} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\\frac {x^{3}}{3 \sqrt {1 - \frac {1}{x^{6}}}} + \frac {\operatorname {asin}{\left (\frac {1}{x^{3}} \right )}}{3} - \frac {1}{3 x^{3} \sqrt {1 - \frac {1}{x^{6}}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-I*x**3/(3*sqrt(-1 + x**(-6))) - I*acosh(x**(-3))/3 + I/(3*x**3*sqrt(-1 + x**(-6))), 1/Abs(x**6) >
1), (x**3/(3*sqrt(1 - 1/x**6)) + asin(x**(-3))/3 - 1/(3*x**3*sqrt(1 - 1/x**6)), True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {-1+x^6}}{x} \, dx=\frac {1}{3} \, \sqrt {x^{6} - 1} - \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]

[In]

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

[Out]

1/3*sqrt(x^6 - 1) - 1/3*arctan(sqrt(x^6 - 1))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {-1+x^6}}{x} \, dx=\frac {1}{3} \, \sqrt {x^{6} - 1} - \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]

[In]

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

[Out]

1/3*sqrt(x^6 - 1) - 1/3*arctan(sqrt(x^6 - 1))

Mupad [B] (verification not implemented)

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

[In]

int((x^6 - 1)^(1/2)/x,x)

[Out]

(x^6 - 1)^(1/2)/3 - atan((x^6 - 1)^(1/2))/3