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

Optimal. Leaf size=28 \[ \frac {\sqrt {x^6-1}}{3}-\frac {1}{3} \tan ^{-1}\left (\sqrt {x^6-1}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {266, 50, 63, 203} \begin {gather*} \frac {\sqrt {x^6-1}}{3}-\frac {1}{3} \tan ^{-1}\left (\sqrt {x^6-1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + x^6]/x,x]

[Out]

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

Rule 50

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 63

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 203

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

Rule 266

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

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Mathematica [A]  time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^6-1}}{3}-\frac {1}{3} \tan ^{-1}\left (\sqrt {x^6-1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 + x^6]/x,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.02, size = 28, normalized size = 1.00 \begin {gather*} \frac {1}{3} \sqrt {-1+x^6}-\frac {1}{3} \tan ^{-1}\left (\sqrt {-1+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[-1 + x^6]/x,x]

[Out]

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

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fricas [A]  time = 0.47, size = 20, normalized size = 0.71 \begin {gather*} \frac {1}{3} \, \sqrt {x^{6} - 1} - \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.36, size = 20, normalized size = 0.71 \begin {gather*} \frac {1}{3} \, \sqrt {x^{6} - 1} - \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [C]  time = 0.35, size = 40, normalized size = 1.43

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x^6-1)^(1/2)+1/3*RootOf(_Z^2+1)*ln((-RootOf(_Z^2+1)+(x^6-1)^(1/2))/x^3)

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maxima [A]  time = 0.51, size = 20, normalized size = 0.71 \begin {gather*} \frac {1}{3} \, \sqrt {x^{6} - 1} - \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 0.29, size = 20, normalized size = 0.71 \begin {gather*} \frac {\sqrt {x^6-1}}{3}-\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6 - 1)^(1/2)/x,x)

[Out]

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

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sympy [A]  time = 1.03, size = 88, normalized size = 3.14 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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