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

Optimal. Leaf size=31 \[ \frac {1}{6} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {\sqrt {x^6-1}}{6 x^6} \]

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Rubi [A]  time = 0.01, antiderivative size = 31, 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, 47, 63, 203} \begin {gather*} \frac {1}{6} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {\sqrt {x^6-1}}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + x^6]/x^7,x]

[Out]

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 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^7} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x^2} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {-1+x^6}}{6 x^6}+\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {-1+x^6}}{6 x^6}+\frac {1}{6} \operatorname {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}{6} \tan ^{-1}\left (\sqrt {-1+x^6}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 48, normalized size = 1.55 \begin {gather*} -\frac {x^6+\sqrt {1-x^6} x^6 \tanh ^{-1}\left (\sqrt {1-x^6}\right )-1}{6 x^6 \sqrt {x^6-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 + x^6]/x^7,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.02, size = 31, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {-1+x^6}}{6 x^6}+\frac {1}{6} \tan ^{-1}\left (\sqrt {-1+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[-1 + x^6]/x^7,x]

[Out]

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

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fricas [A]  time = 0.45, size = 27, normalized size = 0.87 \begin {gather*} \frac {x^{6} \arctan \left (\sqrt {x^{6} - 1}\right ) - \sqrt {x^{6} - 1}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.28, size = 23, normalized size = 0.74 \begin {gather*} -\frac {\sqrt {x^{6} - 1}}{6 \, x^{6}} + \frac {1}{6} \, \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^7,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.34, size = 20, normalized size = 0.65

method	result	size
risch	\(-\frac {\sqrt {x^{6}-1}}{6 x^{6}}-\frac {\arcsin \left (\frac {1}{x^{3}}\right )}{6}\)	\(20\)
trager	\(-\frac {\sqrt {x^{6}-1}}{6 x^{6}}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{6}\)	\(41\)
meijerg	\(\frac {\sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (-\frac {2 \sqrt {\pi }}{x^{6}}-\left (-2 \ln \relax (2)-1+6 \ln \relax (x )+i \pi \right ) \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 x^{6}+8\right )}{4 x^{6}}-\frac {2 \sqrt {\pi }\, \sqrt {-x^{6}+1}}{x^{6}}+2 \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right ) \sqrt {\pi }\right )}{12 \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {\pi }}\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(x^6-1)^(1/2)/x^6-1/6*arcsin(1/x^3)

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maxima [A]  time = 0.49, size = 23, normalized size = 0.74 \begin {gather*} -\frac {\sqrt {x^{6} - 1}}{6 \, x^{6}} + \frac {1}{6} \, \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^7,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.34, size = 23, normalized size = 0.74 \begin {gather*} \frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{6}-\frac {\sqrt {x^6-1}}{6\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6 - 1)^(1/2)/x^7,x)

[Out]

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

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sympy [A]  time = 1.38, size = 73, normalized size = 2.35 \begin {gather*} \begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{6} + \frac {i}{6 x^{3} \sqrt {-1 + \frac {1}{x^{6}}}} - \frac {i}{6 x^{9} \sqrt {-1 + \frac {1}{x^{6}}}} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {1}{x^{3}} \right )}}{6} - \frac {\sqrt {1 - \frac {1}{x^{6}}}}{6 x^{3}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((I*acosh(x**(-3))/6 + I/(6*x**3*sqrt(-1 + x**(-6))) - I/(6*x**9*sqrt(-1 + x**(-6))), 1/Abs(x**6) > 1
), (-asin(x**(-3))/6 - sqrt(1 - 1/x**6)/(6*x**3), True))

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