\(\int \frac {(-1+x^3) \sqrt {-1+x^6}}{x^{13}} \, dx\) [694]

Optimal result

Integrand size = 18, antiderivative size = 54 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=\frac {\sqrt {-1+x^6} \left (6-8 x^3-3 x^6+8 x^9\right )}{72 x^{12}}+\frac {1}{12} \arctan \left (\frac {1+x^3}{\sqrt {-1+x^6}}\right ) \]

[Out]

1/72*(x^6-1)^(1/2)*(8*x^9-3*x^6-8*x^3+6)/x^12+1/12*arctan((x^3+1)/(x^6-1)^(1/2))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1489, 849, 821, 272, 43, 65, 209} \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=-\frac {1}{24} \arctan \left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1}}{24 x^6}-\frac {\left (x^6-1\right )^{3/2}}{12 x^{12}}+\frac {\left (x^6-1\right )^{3/2}}{9 x^9} \]

[In]

Int[((-1 + x^3)*Sqrt[-1 + x^6])/x^13,x]

[Out]

Sqrt[-1 + x^6]/(24*x^6) - (-1 + x^6)^(3/2)/(12*x^12) + (-1 + x^6)^(3/2)/(9*x^9) - ArcTan[Sqrt[-1 + x^6]]/24

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 849

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)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

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^5} \, dx,x,x^3\right ) \\ & = -\frac {\left (-1+x^6\right )^{3/2}}{12 x^{12}}+\frac {1}{12} \text {Subst}\left (\int \frac {(4-x) \sqrt {-1+x^2}}{x^4} \, dx,x,x^3\right ) \\ & = -\frac {\left (-1+x^6\right )^{3/2}}{12 x^{12}}+\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{12} \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{x^3} \, dx,x,x^3\right ) \\ & = -\frac {\left (-1+x^6\right )^{3/2}}{12 x^{12}}+\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{24} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{x^2} \, dx,x,x^6\right ) \\ & = \frac {\sqrt {-1+x^6}}{24 x^6}-\frac {\left (-1+x^6\right )^{3/2}}{12 x^{12}}+\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{48} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right ) \\ & = \frac {\sqrt {-1+x^6}}{24 x^6}-\frac {\left (-1+x^6\right )^{3/2}}{12 x^{12}}+\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{24} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right ) \\ & = \frac {\sqrt {-1+x^6}}{24 x^6}-\frac {\left (-1+x^6\right )^{3/2}}{12 x^{12}}+\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{24} \arctan \left (\sqrt {-1+x^6}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=\frac {\sqrt {-1+x^6} \left (6-8 x^3-3 x^6+8 x^9\right )}{72 x^{12}}+\frac {1}{12} \arctan \left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right ) \]

[In]

Integrate[((-1 + x^3)*Sqrt[-1 + x^6])/x^13,x]

[Out]

(Sqrt[-1 + x^6]*(6 - 8*x^3 - 3*x^6 + 8*x^9))/(72*x^12) + ArcTan[Sqrt[-1 + x^6]/(-1 + x^3)]/12

Maple [A] (verified)

Time = 1.99 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81

method	result	size
pseudoelliptic	\(\frac {3 \arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right ) x^{12}+8 \left (x^{9}-\frac {3}{8} x^{6}-x^{3}+\frac {3}{4}\right ) \sqrt {x^{6}-1}}{72 x^{12}}\)	\(44\)
trager	\(\frac {\sqrt {x^{6}-1}\, \left (8 x^{9}-3 x^{6}-8 x^{3}+6\right )}{72 x^{12}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{24}\)	\(60\)
risch	\(\frac {8 x^{15}-3 x^{12}-16 x^{9}+9 x^{6}+8 x^{3}-6}{72 x^{12} \sqrt {x^{6}-1}}-\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{48 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\)	\(101\)
meijerg	\(\frac {\sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (-\frac {\sqrt {\pi }\, \left (x^{12}-8 x^{6}+8\right )}{8 x^{12}}+\frac {\sqrt {\pi }\, \left (-4 x^{6}+8\right ) \sqrt {-x^{6}+1}}{8 x^{12}}-\frac {\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )}{2}+\frac {\left (\frac {1}{2}-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{4}+\frac {\sqrt {\pi }}{x^{12}}-\frac {\sqrt {\pi }}{x^{6}}\right )}{12 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}-\frac {\sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (-x^{6}+1\right )^{\frac {3}{2}}}{9 \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, x^{9}}\)	\(153\)

[In]

int((x^3-1)*(x^6-1)^(1/2)/x^13,x,method=_RETURNVERBOSE)

[Out]

1/72*(3*arctan(1/(x^6-1)^(1/2))*x^12+8*(x^9-3/8*x^6-x^3+3/4)*(x^6-1)^(1/2))/x^12

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=-\frac {6 \, x^{12} \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) - 8 \, x^{12} - {\left (8 \, x^{9} - 3 \, x^{6} - 8 \, x^{3} + 6\right )} \sqrt {x^{6} - 1}}{72 \, x^{12}} \]

[In]

integrate((x^3-1)*(x^6-1)^(1/2)/x^13,x, algorithm="fricas")

[Out]

-1/72*(6*x^12*arctan(-x^3 + sqrt(x^6 - 1)) - 8*x^12 - (8*x^9 - 3*x^6 - 8*x^3 + 6)*sqrt(x^6 - 1))/x^12

Sympy [A] (verification not implemented)

Time = 2.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.17 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=\frac {\begin {cases} \frac {\left (x^{6} - 1\right )^{\frac {3}{2}}}{3 x^{9}} & \text {for}\: x^{3} > -1 \wedge x^{3} < 1 \end {cases}}{3} - \frac {\begin {cases} \frac {\operatorname {acos}{\left (\frac {1}{x^{3}} \right )}}{8} - \frac {\left (-1 + \frac {2}{x^{6}}\right ) \sqrt {1 - \frac {1}{x^{6}}}}{8 x^{3}} & \text {for}\: x^{3} > -1 \wedge x^{3} < 1 \end {cases}}{3} \]

[In]

integrate((x**3-1)*(x**6-1)**(1/2)/x**13,x)

[Out]

Piecewise(((x**6 - 1)**(3/2)/(3*x**9), (x**3 > -1) & (x**3 < 1)))/3 - Piecewise((acos(x**(-3))/8 - (-1 + 2/x**
6)*sqrt(1 - 1/x**6)/(8*x**3), (x**3 > -1) & (x**3 < 1)))/3

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.07 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=-\frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}} - \sqrt {x^{6} - 1}}{24 \, {\left (2 \, x^{6} + {\left (x^{6} - 1\right )}^{2} - 1\right )}} + \frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} - \frac {1}{24} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]

[In]

integrate((x^3-1)*(x^6-1)^(1/2)/x^13,x, algorithm="maxima")

[Out]

-1/24*((x^6 - 1)^(3/2) - sqrt(x^6 - 1))/(2*x^6 + (x^6 - 1)^2 - 1) + 1/9*(x^6 - 1)^(3/2)/x^9 - 1/24*arctan(sqrt
(x^6 - 1))

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=\text {Exception raised: NotImplementedError} \]

[In]

integrate((x^3-1)*(x^6-1)^(1/2)/x^13,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> unable to parse Giac output: -1/9*sign(sageVARx)+2*((1/sageVARx)^3*((
1/sageVARx)^3*(1/24*(1/sageVARx)^3/sign(sageVARx)-1/18/sign(sageVARx))-1/48/sign(sageVARx))+1/18/sign(sageVARx
))*sqrt(-(1/sageVARx)

Mupad [B] (verification not implemented)

Time = 6.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=\frac {\frac {\sqrt {x^6-1}}{24}-\frac {{\left (x^6-1\right )}^{3/2}}{24}}{x^{12}}-\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{24}+\frac {{\left (x^6-1\right )}^{3/2}}{9\,x^9} \]

[In]

int(((x^3 - 1)*(x^6 - 1)^(1/2))/x^13,x)

[Out]

((x^6 - 1)^(1/2)/24 - (x^6 - 1)^(3/2)/24)/x^12 - atan((x^6 - 1)^(1/2))/24 + (x^6 - 1)^(3/2)/(9*x^9)