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

Optimal. Leaf size=54 \[ \frac {1}{12} \tan ^{-1}\left (\frac {x^3+1}{\sqrt {x^6-1}}\right )+\frac {\sqrt {x^6-1} \left (8 x^9-3 x^6-8 x^3+6\right )}{72 x^{12}} \]

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Rubi [A]  time = 0.05, antiderivative size = 63, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1475, 835, 807, 266, 47, 63, 203} \begin {gather*} \frac {\sqrt {x^6-1}}{24 x^6}-\frac {1}{24} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {\left (x^6-1\right )^{3/2}}{12 x^{12}}+\frac {\left (x^6-1\right )^{3/2}}{9 x^9} \end {gather*}

Antiderivative was successfully verified.

[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 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]]

Rule 807

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 835

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 1475

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*} \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx &=\frac {1}{3} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \tan ^{-1}\left (\sqrt {-1+x^6}\right )\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 37, normalized size = 0.69 \begin {gather*} -\frac {\left (x^6-1\right )^{3/2} \left (x^9 \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};1-x^6\right )-1\right )}{9 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*((-1 + x^6)^(3/2)*(-1 + x^9*Hypergeometric2F1[3/2, 3, 5/2, 1 - x^6]))/x^9

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IntegrateAlgebraic [A]  time = 0.17, size = 56, normalized size = 1.04 \begin {gather*} \frac {\sqrt {-1+x^6} \left (6-8 x^3-3 x^6+8 x^9\right )}{72 x^{12}}+\frac {1}{12} \tan ^{-1}\left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-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

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fricas [A]  time = 0.46, size = 56, normalized size = 1.04 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{6} - 1} {\left (x^{3} - 1\right )}}{x^{13}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.40, size = 47, normalized size = 0.87

method	result	size
risch	\(\frac {8 x^{15}-3 x^{12}-16 x^{9}+9 x^{6}+8 x^{3}-6}{72 x^{12} \sqrt {x^{6}-1}}+\frac {\arcsin \left (\frac {1}{x^{3}}\right )}{24}\)	\(47\)
trager	\(\frac {\sqrt {x^{6}-1}\, \left (8 x^{9}-3 x^{6}-8 x^{3}+6\right )}{72 x^{12}}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{24}\)	\(60\)
meijerg	\(\frac {\sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (-\frac {\sqrt {\pi }\, \left (x^{12}-8 x^{6}+8\right )}{8 x^{12}}+\frac {\sqrt {\pi }\, \left (-4 x^{6}+8\right ) \sqrt {-x^{6}+1}}{8 x^{12}}-\frac {\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )}{2}+\frac {\left (\frac {1}{2}-2 \ln \relax (2)+6 \ln \relax (x )+i \pi \right ) \sqrt {\pi }}{4}+\frac {\sqrt {\pi }}{x^{12}}-\frac {\sqrt {\pi }}{x^{6}}\right )}{12 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}}-\frac {\sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (-x^{6}+1\right )^{\frac {3}{2}}}{9 \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, x^{9}}\)	\(153\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*(8*x^15-3*x^12-16*x^9+9*x^6+8*x^3-6)/x^12/(x^6-1)^(1/2)+1/24*arcsin(1/x^3)

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maxima [A]  time = 0.43, size = 58, normalized size = 1.07 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 1.36, size = 46, normalized size = 0.85 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 4.82, size = 56, normalized size = 1.04 \begin {gather*} \frac {\begin {cases} \frac {\left (x^{6} - 1\right )^{\frac {3}{2}}}{3 x^{9}} & \text {for}\: x > -1 \wedge x < 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 > -1 \wedge x < 1 \end {cases}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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