3.6.49 \(\int \frac {1}{x^{19} \sqrt {-1+x^6}} \, dx\)

Optimal. Leaf size=43 \[ \frac {5}{48} \tan ^{-1}\left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1} \left (15 x^{12}+10 x^6+8\right )}{144 x^{18}} \]

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Rubi [A]  time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.47, number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {266, 51, 63, 203} \begin {gather*} \frac {5 \sqrt {x^6-1}}{48 x^6}+\frac {5}{48} \tan ^{-1}\left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1}}{18 x^{18}}+\frac {5 \sqrt {x^6-1}}{72 x^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^19*Sqrt[-1 + x^6]),x]

[Out]

Sqrt[-1 + x^6]/(18*x^18) + (5*Sqrt[-1 + x^6])/(72*x^12) + (5*Sqrt[-1 + x^6])/(48*x^6) + (5*ArcTan[Sqrt[-1 + x^
6]])/48

Rule 51

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] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 {1}{x^{19} \sqrt {-1+x^6}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^4} \, dx,x,x^6\right )\\ &=\frac {\sqrt {-1+x^6}}{18 x^{18}}+\frac {5}{36} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^3} \, dx,x,x^6\right )\\ &=\frac {\sqrt {-1+x^6}}{18 x^{18}}+\frac {5 \sqrt {-1+x^6}}{72 x^{12}}+\frac {5}{48} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^2} \, dx,x,x^6\right )\\ &=\frac {\sqrt {-1+x^6}}{18 x^{18}}+\frac {5 \sqrt {-1+x^6}}{72 x^{12}}+\frac {5 \sqrt {-1+x^6}}{48 x^6}+\frac {5}{96} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )\\ &=\frac {\sqrt {-1+x^6}}{18 x^{18}}+\frac {5 \sqrt {-1+x^6}}{72 x^{12}}+\frac {5 \sqrt {-1+x^6}}{48 x^6}+\frac {5}{48} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )\\ &=\frac {\sqrt {-1+x^6}}{18 x^{18}}+\frac {5 \sqrt {-1+x^6}}{72 x^{12}}+\frac {5 \sqrt {-1+x^6}}{48 x^6}+\frac {5}{48} \tan ^{-1}\left (\sqrt {-1+x^6}\right )\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 28, normalized size = 0.65 \begin {gather*} \frac {1}{3} \sqrt {x^6-1} \, _2F_1\left (\frac {1}{2},4;\frac {3}{2};1-x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^19*Sqrt[-1 + x^6]),x]

[Out]

(Sqrt[-1 + x^6]*Hypergeometric2F1[1/2, 4, 3/2, 1 - x^6])/3

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IntegrateAlgebraic [A]  time = 0.03, size = 43, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6} \left (8+10 x^6+15 x^{12}\right )}{144 x^{18}}+\frac {5}{48} \tan ^{-1}\left (\sqrt {-1+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^19*Sqrt[-1 + x^6]),x]

[Out]

(Sqrt[-1 + x^6]*(8 + 10*x^6 + 15*x^12))/(144*x^18) + (5*ArcTan[Sqrt[-1 + x^6]])/48

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fricas [A]  time = 0.44, size = 39, normalized size = 0.91 \begin {gather*} \frac {15 \, x^{18} \arctan \left (\sqrt {x^{6} - 1}\right ) + {\left (15 \, x^{12} + 10 \, 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(1/x^19/(x^6-1)^(1/2),x, algorithm="fricas")

[Out]

1/144*(15*x^18*arctan(sqrt(x^6 - 1)) + (15*x^12 + 10*x^6 + 8)*sqrt(x^6 - 1))/x^18

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giac [A]  time = 0.36, size = 44, normalized size = 1.02 \begin {gather*} \frac {15 \, {\left (x^{6} - 1\right )}^{\frac {5}{2}} + 40 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {x^{6} - 1}}{144 \, x^{18}} + \frac {5}{48} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*(15*(x^6 - 1)^(5/2) + 40*(x^6 - 1)^(3/2) + 33*sqrt(x^6 - 1))/x^18 + 5/48*arctan(sqrt(x^6 - 1))

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maple [A]  time = 0.37, size = 37, normalized size = 0.86

method	result	size
risch	\(\frac {15 x^{18}-5 x^{12}-2 x^{6}-8}{144 x^{18} \sqrt {x^{6}-1}}-\frac {5 \arcsin \left (\frac {1}{x^{3}}\right )}{48}\)	\(37\)
trager	\(\frac {\sqrt {x^{6}-1}\, \left (15 x^{12}+10 x^{6}+8\right )}{144 x^{18}}+\frac {5 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{48}\)	\(53\)
meijerg	\(-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (\frac {\sqrt {\pi }}{3 x^{18}}+\frac {\sqrt {\pi }}{4 x^{12}}+\frac {3 \sqrt {\pi }}{8 x^{6}}-\frac {5 \left (\frac {37}{30}-2 \ln \relax (2)+6 \ln \relax (x )+i \pi \right ) \sqrt {\pi }}{16}-\frac {\sqrt {\pi }\, \left (-148 x^{18}+144 x^{12}+96 x^{6}+128\right )}{384 x^{18}}+\frac {\sqrt {\pi }\, \left (240 x^{12}+160 x^{6}+128\right ) \sqrt {-x^{6}+1}}{384 x^{18}}+\frac {5 \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right ) \sqrt {\pi }}{8}\right )}{6 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {\pi }}\)	\(141\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*(15*x^18-5*x^12-2*x^6-8)/x^18/(x^6-1)^(1/2)-5/48*arcsin(1/x^3)

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maxima [A]  time = 0.42, size = 66, normalized size = 1.53 \begin {gather*} \frac {15 \, {\left (x^{6} - 1\right )}^{\frac {5}{2}} + 40 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} + 33 \, \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 {5}{48} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*(15*(x^6 - 1)^(5/2) + 40*(x^6 - 1)^(3/2) + 33*sqrt(x^6 - 1))/(3*x^6 + (x^6 - 1)^3 + 3*(x^6 - 1)^2 - 2) +
5/48*arctan(sqrt(x^6 - 1))

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mupad [B]  time = 0.59, size = 47, normalized size = 1.09 \begin {gather*} \frac {5\,\mathrm {atan}\left (\sqrt {x^6-1}\right )}{48}+\frac {11\,\sqrt {x^6-1}}{48\,x^{18}}+\frac {5\,{\left (x^6-1\right )}^{3/2}}{18\,x^{18}}+\frac {5\,{\left (x^6-1\right )}^{5/2}}{48\,x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^19*(x^6 - 1)^(1/2)),x)

[Out]

(5*atan((x^6 - 1)^(1/2)))/48 + (11*(x^6 - 1)^(1/2))/(48*x^18) + (5*(x^6 - 1)^(3/2))/(18*x^18) + (5*(x^6 - 1)^(
5/2))/(48*x^18)

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sympy [B]  time = 4.77, size = 165, normalized size = 3.84 \begin {gather*} \begin {cases} \frac {5 i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{48} - \frac {5 i}{48 x^{3} \sqrt {-1 + \frac {1}{x^{6}}}} + \frac {5 i}{144 x^{9} \sqrt {-1 + \frac {1}{x^{6}}}} + \frac {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 {5 \operatorname {asin}{\left (\frac {1}{x^{3}} \right )}}{48} + \frac {5}{48 x^{3} \sqrt {1 - \frac {1}{x^{6}}}} - \frac {5}{144 x^{9} \sqrt {1 - \frac {1}{x^{6}}}} - \frac {1}{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(1/x**19/(x**6-1)**(1/2),x)

[Out]

Piecewise((5*I*acosh(x**(-3))/48 - 5*I/(48*x**3*sqrt(-1 + x**(-6))) + 5*I/(144*x**9*sqrt(-1 + x**(-6))) + I/(7
2*x**15*sqrt(-1 + x**(-6))) + I/(18*x**21*sqrt(-1 + x**(-6))), 1/Abs(x**6) > 1), (-5*asin(x**(-3))/48 + 5/(48*
x**3*sqrt(1 - 1/x**6)) - 5/(144*x**9*sqrt(1 - 1/x**6)) - 1/(72*x**15*sqrt(1 - 1/x**6)) - 1/(18*x**21*sqrt(1 -
1/x**6)), True))

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