\(\int \frac {(-1+x^3) \sqrt {-1+x^6}}{x^{10}} \, dx\) [627]

Optimal result

Integrand size = 18, antiderivative size = 49 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{10}} \, dx=\frac {\left (2-3 x^3-2 x^6\right ) \sqrt {-1+x^6}}{18 x^9}-\frac {1}{3} \arctan \left (\frac {1+x^3}{\sqrt {-1+x^6}}\right ) \]

[Out]

1/18*(-2*x^6-3*x^3+2)*(x^6-1)^(1/2)/x^9-1/3*arctan((x^3+1)/(x^6-1)^(1/2))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1489, 821, 272, 43, 65, 209} \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{10}} \, dx=\frac {1}{6} \arctan \left (\sqrt {x^6-1}\right )-\frac {\sqrt {x^6-1}}{6 x^6}-\frac {\left (x^6-1\right )^{3/2}}{9 x^9} \]

[In]

Int[((-1 + x^3)*Sqrt[-1 + x^6])/x^10,x]

[Out]

-1/6*Sqrt[-1 + x^6]/x^6 - (-1 + x^6)^(3/2)/(9*x^9) + ArcTan[Sqrt[-1 + x^6]]/6

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 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]]

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 1489

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x
] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {(-1+x) \sqrt {-1+x^2}}{x^4} \, dx,x,x^3\right ) \\ & = -\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}+\frac {1}{3} \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{x^3} \, dx,x,x^3\right ) \\ & = -\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}+\frac {1}{6} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{x^2} \, dx,x,x^6\right ) \\ & = -\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}+\frac {1}{12} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right ) \\ & = -\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}+\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right ) \\ & = -\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}+\frac {1}{6} \arctan \left (\sqrt {-1+x^6}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{10}} \, dx=\frac {\left (2-3 x^3-2 x^6\right ) \sqrt {-1+x^6}}{18 x^9}-\frac {1}{3} \arctan \left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right ) \]

[In]

Integrate[((-1 + x^3)*Sqrt[-1 + x^6])/x^10,x]

[Out]

((2 - 3*x^3 - 2*x^6)*Sqrt[-1 + x^6])/(18*x^9) - ArcTan[Sqrt[-1 + x^6]/(-1 + x^3)]/3

Maple [A] (verified)

Time = 1.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.82

method	result	size
pseudoelliptic	\(\frac {-3 \arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right ) x^{9}+\left (-2 x^{6}-3 x^{3}+2\right ) \sqrt {x^{6}-1}}{18 x^{9}}\)	\(40\)
trager	\(-\frac {\left (2 x^{6}+3 x^{3}-2\right ) \sqrt {x^{6}-1}}{18 x^{9}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{6}\)	\(53\)
risch	\(-\frac {2 x^{12}+3 x^{9}-4 x^{6}-3 x^{3}+2}{18 x^{9} \sqrt {x^{6}-1}}+\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{12 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\)	\(96\)
meijerg	\(\frac {\sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (-x^{6}+1\right )^{\frac {3}{2}}}{9 \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, x^{9}}+\frac {\sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (\frac {\sqrt {\pi }\, \left (-4 x^{6}+8\right )}{4 x^{6}}-\frac {2 \sqrt {\pi }\, \sqrt {-x^{6}+1}}{x^{6}}+2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )-\left (-2 \ln \left (2\right )-1+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }-\frac {2 \sqrt {\pi }}{x^{6}}\right )}{12 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}\)	\(136\)

[In]

int((x^3-1)*(x^6-1)^(1/2)/x^10,x,method=_RETURNVERBOSE)

[Out]

1/18*(-3*arctan(1/(x^6-1)^(1/2))*x^9+(-2*x^6-3*x^3+2)*(x^6-1)^(1/2))/x^9

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{10}} \, dx=\frac {6 \, x^{9} \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) - 2 \, x^{9} - {\left (2 \, x^{6} + 3 \, x^{3} - 2\right )} \sqrt {x^{6} - 1}}{18 \, x^{9}} \]

[In]

integrate((x^3-1)*(x^6-1)^(1/2)/x^10,x, algorithm="fricas")

[Out]

1/18*(6*x^9*arctan(-x^3 + sqrt(x^6 - 1)) - 2*x^9 - (2*x^6 + 3*x^3 - 2)*sqrt(x^6 - 1))/x^9

Sympy [A] (verification not implemented)

Time = 1.77 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.14 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{10}} \, dx=- \frac {\begin {cases} \frac {\left (x^{6} - 1\right )^{\frac {3}{2}}}{3 x^{9}} & \text {for}\: x^{3} > -1 \wedge x^{3} < 1 \end {cases}}{3} + \frac {\begin {cases} \frac {\operatorname {acos}{\left (\frac {1}{x^{3}} \right )}}{2} - \frac {\sqrt {1 - \frac {1}{x^{6}}}}{2 x^{3}} & \text {for}\: x^{3} > -1 \wedge x^{3} < 1 \end {cases}}{3} \]

[In]

integrate((x**3-1)*(x**6-1)**(1/2)/x**10,x)

[Out]

-Piecewise(((x**6 - 1)**(3/2)/(3*x**9), (x**3 > -1) & (x**3 < 1)))/3 + Piecewise((acos(x**(-3))/2 - sqrt(1 - 1
/x**6)/(2*x**3), (x**3 > -1) & (x**3 < 1)))/3

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.71 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{10}} \, dx=-\frac {\sqrt {x^{6} - 1}}{6 \, x^{6}} - \frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} + \frac {1}{6} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]

[In]

integrate((x^3-1)*(x^6-1)^(1/2)/x^10,x, algorithm="maxima")

[Out]

-1/6*sqrt(x^6 - 1)/x^6 - 1/9*(x^6 - 1)^(3/2)/x^9 + 1/6*arctan(sqrt(x^6 - 1))

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{10}} \, dx=\text {Exception raised: NotImplementedError} \]

[In]

integrate((x^3-1)*(x^6-1)^(1/2)/x^10,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> unable to parse Giac output: 1/9*sign(sageVARx)+2*((1/sageVARx)^3*(1/
18*(1/sageVARx)^3/sign(sageVARx)-1/12/sign(sageVARx))-1/18/sign(sageVARx))*sqrt(-(1/sageVARx)^6+1)+1/2/sign(sa
geVARx)*(-atan(i)/3*s

Mupad [B] (verification not implemented)

Time = 6.32 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.71 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{10}} \, dx=\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{6}-\frac {\sqrt {x^6-1}}{6\,x^6}-\frac {{\left (x^6-1\right )}^{3/2}}{9\,x^9} \]

[In]

int(((x^3 - 1)*(x^6 - 1)^(1/2))/x^10,x)

[Out]

atan((x^6 - 1)^(1/2))/6 - (x^6 - 1)^(1/2)/(6*x^6) - (x^6 - 1)^(3/2)/(9*x^9)