3.6.51 \(\int \frac {\sqrt {-1+x^6}}{x^{19}} \, dx\) [551]

Optimal. Leaf size=43 \[ \frac {\sqrt {-1+x^6} \left (-8+2 x^6+3 x^{12}\right )}{144 x^{18}}+\frac {1}{48} \text {ArcTan}\left (\sqrt {-1+x^6}\right ) \]

[Out]

1/144*(x^6-1)^(1/2)*(3*x^12+2*x^6-8)/x^18+1/48*arctan((x^6-1)^(1/2))

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Rubi [A]
time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.47, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {272, 43, 44, 65, 209} \begin {gather*} \frac {1}{48} \text {ArcTan}\left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1}}{48 x^6}-\frac {\sqrt {x^6-1}}{18 x^{18}}+\frac {\sqrt {x^6-1}}{72 x^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + x^6]/x^19,x]

[Out]

-1/18*Sqrt[-1 + x^6]/x^18 + Sqrt[-1 + x^6]/(72*x^12) + Sqrt[-1 + x^6]/(48*x^6) + ArcTan[Sqrt[-1 + x^6]]/48

Rule 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 44

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

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*} \int \frac {\sqrt {-1+x^6}}{x^{19}} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{x^4} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {-1+x^6}}{18 x^{18}}+\frac {1}{36} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^3} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {-1+x^6}}{18 x^{18}}+\frac {\sqrt {-1+x^6}}{72 x^{12}}+\frac {1}{48} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^2} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {-1+x^6}}{18 x^{18}}+\frac {\sqrt {-1+x^6}}{72 x^{12}}+\frac {\sqrt {-1+x^6}}{48 x^6}+\frac {1}{96} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {-1+x^6}}{18 x^{18}}+\frac {\sqrt {-1+x^6}}{72 x^{12}}+\frac {\sqrt {-1+x^6}}{48 x^6}+\frac {1}{48} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )\\ &=-\frac {\sqrt {-1+x^6}}{18 x^{18}}+\frac {\sqrt {-1+x^6}}{72 x^{12}}+\frac {\sqrt {-1+x^6}}{48 x^6}+\frac {1}{48} \tan ^{-1}\left (\sqrt {-1+x^6}\right )\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 43, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6} \left (-8+2 x^6+3 x^{12}\right )}{144 x^{18}}+\frac {1}{48} \text {ArcTan}\left (\sqrt {-1+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 + x^6]/x^19,x]

[Out]

(Sqrt[-1 + x^6]*(-8 + 2*x^6 + 3*x^12))/(144*x^18) + ArcTan[Sqrt[-1 + x^6]]/48

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Maple [A]
time = 0.64, size = 37, normalized size = 0.86

method	result	size
risch	\(\frac {3 x^{18}-x^{12}-10 x^{6}+8}{144 x^{18} \sqrt {x^{6}-1}}-\frac {\arcsin \left (\frac {1}{x^{3}}\right )}{48}\)	\(37\)
trager	\(\frac {\sqrt {x^{6}-1}\, \left (3 x^{12}+2 x^{6}-8\right )}{144 x^{18}}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{48}\)	\(55\)
meijerg	\(\frac {\sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (\frac {\sqrt {\pi }\, \left (20 x^{18}-48 x^{12}-96 x^{6}+128\right )}{192 x^{18}}-\frac {\sqrt {\pi }\, \left (-48 x^{12}-32 x^{6}+128\right ) \sqrt {-x^{6}+1}}{192 x^{18}}+\frac {\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )}{4}-\frac {\left (\frac {5}{6}-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{8}-\frac {2 \sqrt {\pi }}{3 x^{18}}+\frac {\sqrt {\pi }}{2 x^{12}}+\frac {\sqrt {\pi }}{4 x^{6}}\right )}{12 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}}\)	\(141\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6-1)^(1/2)/x^19,x,method=_RETURNVERBOSE)

[Out]

1/144*(3*x^18-x^12-10*x^6+8)/x^18/(x^6-1)^(1/2)-1/48*arcsin(1/x^3)

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Maxima [A]
time = 0.46, size = 66, normalized size = 1.53 \begin {gather*} \frac {3 \, {\left (x^{6} - 1\right )}^{\frac {5}{2}} + 8 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} - 3 \, \sqrt {x^{6} - 1}}{144 \, {\left (3 \, x^{6} + {\left (x^{6} - 1\right )}^{3} + 3 \, {\left (x^{6} - 1\right )}^{2} - 2\right )}} + \frac {1}{48} \, \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^19,x, algorithm="maxima")

[Out]

1/144*(3*(x^6 - 1)^(5/2) + 8*(x^6 - 1)^(3/2) - 3*sqrt(x^6 - 1))/(3*x^6 + (x^6 - 1)^3 + 3*(x^6 - 1)^2 - 2) + 1/
48*arctan(sqrt(x^6 - 1))

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Fricas [A]
time = 0.36, size = 39, normalized size = 0.91 \begin {gather*} \frac {3 \, x^{18} \arctan \left (\sqrt {x^{6} - 1}\right ) + {\left (3 \, x^{12} + 2 \, x^{6} - 8\right )} \sqrt {x^{6} - 1}}{144 \, x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)^(1/2)/x^19,x, algorithm="fricas")

[Out]

1/144*(3*x^18*arctan(sqrt(x^6 - 1)) + (3*x^12 + 2*x^6 - 8)*sqrt(x^6 - 1))/x^18

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Sympy [C] Result contains complex when optimal does not.
time = 5.43, size = 160, normalized size = 3.72 \begin {gather*} \begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{48} - \frac {i}{48 x^{3} \sqrt {-1 + \frac {1}{x^{6}}}} + \frac {i}{144 x^{9} \sqrt {-1 + \frac {1}{x^{6}}}} + \frac {5 i}{72 x^{15} \sqrt {-1 + \frac {1}{x^{6}}}} - \frac {i}{18 x^{21} \sqrt {-1 + \frac {1}{x^{6}}}} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {1}{x^{3}} \right )}}{48} + \frac {1}{48 x^{3} \sqrt {1 - \frac {1}{x^{6}}}} - \frac {1}{144 x^{9} \sqrt {1 - \frac {1}{x^{6}}}} - \frac {5}{72 x^{15} \sqrt {1 - \frac {1}{x^{6}}}} + \frac {1}{18 x^{21} \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**19,x)

[Out]

Piecewise((I*acosh(x**(-3))/48 - I/(48*x**3*sqrt(-1 + x**(-6))) + I/(144*x**9*sqrt(-1 + x**(-6))) + 5*I/(72*x*
*15*sqrt(-1 + x**(-6))) - I/(18*x**21*sqrt(-1 + x**(-6))), 1/Abs(x**6) > 1), (-asin(x**(-3))/48 + 1/(48*x**3*s
qrt(1 - 1/x**6)) - 1/(144*x**9*sqrt(1 - 1/x**6)) - 5/(72*x**15*sqrt(1 - 1/x**6)) + 1/(18*x**21*sqrt(1 - 1/x**6
)), True))

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Giac [A]
time = 0.39, size = 44, normalized size = 1.02 \begin {gather*} \frac {3 \, {\left (x^{6} - 1\right )}^{\frac {5}{2}} + 8 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} - 3 \, \sqrt {x^{6} - 1}}{144 \, x^{18}} + \frac {1}{48} \, \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^19,x, algorithm="giac")

[Out]

1/144*(3*(x^6 - 1)^(5/2) + 8*(x^6 - 1)^(3/2) - 3*sqrt(x^6 - 1))/x^18 + 1/48*arctan(sqrt(x^6 - 1))

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Mupad [B]
time = 0.55, size = 47, normalized size = 1.09 \begin {gather*} \frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{48}-\frac {\sqrt {x^6-1}}{48\,x^{18}}+\frac {{\left (x^6-1\right )}^{3/2}}{18\,x^{18}}+\frac {{\left (x^6-1\right )}^{5/2}}{48\,x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6 - 1)^(1/2)/x^19,x)

[Out]

atan((x^6 - 1)^(1/2))/48 - (x^6 - 1)^(1/2)/(48*x^18) + (x^6 - 1)^(3/2)/(18*x^18) + (x^6 - 1)^(5/2)/(48*x^18)

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