3.9.49 \(\int \frac {\sqrt {-1+x^6} (-1+2 x^6)^2}{x^7 (-1+4 x^6)} \, dx\)

Optimal. Leaf size=64 \[ \frac {\sqrt {x^6-1} \left (2 x^6+1\right )}{6 x^6}-\frac {1}{6} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt {x^6-1}}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

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Rubi [A]  time = 0.10, antiderivative size = 70, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {573, 180, 47, 63, 203, 50} \begin {gather*} \frac {\sqrt {x^6-1}}{6 x^6}+\frac {\sqrt {x^6-1}}{3}-\frac {1}{6} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt {x^6-1}}{\sqrt {3}}\right )}{2 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[-1 + x^6]*(-1 + 2*x^6)^2)/(x^7*(-1 + 4*x^6)),x]

[Out]

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

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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 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 180

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_))^(q_), x
_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x] /; FreeQ[{a, b, c, d,
e, f, g, h, m, n}, x] && IntegersQ[p, q]

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 573

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x
_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n],
x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\sqrt {-1+x^6} \left (-1+2 x^6\right )^2}{x^7 \left (-1+4 x^6\right )} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x} (-1+2 x)^2}{x^2 (-1+4 x)} \, dx,x,x^6\right )\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \left (-\frac {\sqrt {-1+x}}{x^2}+\frac {4 \sqrt {-1+x}}{-1+4 x}\right ) \, dx,x,x^6\right )\\ &=-\left (\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x^2} \, dx,x,x^6\right )\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{-1+4 x} \, dx,x,x^6\right )\\ &=\frac {1}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} (-1+4 x)} \, dx,x,x^6\right )\\ &=\frac {1}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )-\operatorname {Subst}\left (\int \frac {1}{3+4 x^2} \, dx,x,\sqrt {-1+x^6}\right )\\ &=\frac {1}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {1}{6} \tan ^{-1}\left (\sqrt {-1+x^6}\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt {-1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}}\\ \end {align*}

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Mathematica [B]  time = 0.09, size = 135, normalized size = 2.11 \begin {gather*} \frac {4 x^{12}-2 x^6+2 \sqrt {1-x^6} x^6 \tanh ^{-1}\left (\sqrt {1-x^6}\right )-\sqrt {3} \sqrt {x^6-1} x^6 \tan ^{-1}\left (\frac {x^3-2}{\sqrt {3} \sqrt {x^6-1}}\right )+\sqrt {3} \sqrt {x^6-1} x^6 \tan ^{-1}\left (\frac {x^3+2}{\sqrt {3} \sqrt {x^6-1}}\right )-2}{12 x^6 \sqrt {x^6-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[-1 + x^6]*(-1 + 2*x^6)^2)/(x^7*(-1 + 4*x^6)),x]

[Out]

(-2 - 2*x^6 + 4*x^12 - Sqrt[3]*x^6*Sqrt[-1 + x^6]*ArcTan[(-2 + x^3)/(Sqrt[3]*Sqrt[-1 + x^6])] + Sqrt[3]*x^6*Sq
rt[-1 + x^6]*ArcTan[(2 + x^3)/(Sqrt[3]*Sqrt[-1 + x^6])] + 2*x^6*Sqrt[1 - x^6]*ArcTanh[Sqrt[1 - x^6]])/(12*x^6*
Sqrt[-1 + x^6])

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IntegrateAlgebraic [A]  time = 0.08, size = 64, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6} \left (1+2 x^6\right )}{6 x^6}-\frac {1}{6} \tan ^{-1}\left (\sqrt {-1+x^6}\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt {-1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[-1 + x^6]*(-1 + 2*x^6)^2)/(x^7*(-1 + 4*x^6)),x]

[Out]

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

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fricas [A]  time = 0.47, size = 54, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {3} x^{6} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) + x^{6} \arctan \left (\sqrt {x^{6} - 1}\right ) - {\left (2 \, x^{6} + 1\right )} \sqrt {x^{6} - 1}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)^(1/2)*(2*x^6-1)^2/x^7/(4*x^6-1),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.42, size = 50, normalized size = 0.78 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) + \frac {1}{3} \, \sqrt {x^{6} - 1} + \frac {\sqrt {x^{6} - 1}}{6 \, x^{6}} - \frac {1}{6} \, \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)*(2*x^6-1)^2/x^7/(4*x^6-1),x, algorithm="giac")

[Out]

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

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maple [C]  time = 1.56, size = 105, normalized size = 1.64

method	result	size
trager	\(\frac {\sqrt {x^{6}-1}\, \left (2 x^{6}+1\right )}{6 x^{6}}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{6}+\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {4 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}-7 \RootOf \left (\textit {\_Z}^{2}+3\right )+12 \sqrt {x^{6}-1}}{\left (2 x^{3}-1\right ) \left (2 x^{3}+1\right )}\right )}{12}\)	\(105\)
risch	\(\frac {\sqrt {x^{6}-1}}{6 x^{6}}+\frac {\sqrt {x^{6}-1}}{3}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{6}+\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {4 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}-7 \RootOf \left (\textit {\_Z}^{2}+3\right )+12 \sqrt {x^{6}-1}}{\left (2 x^{3}-1\right ) \left (2 x^{3}+1\right )}\right )}{12}\)	\(107\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6-1)^(1/2)*(2*x^6-1)^2/x^7/(4*x^6-1),x,method=_RETURNVERBOSE)

[Out]

1/6*(x^6-1)^(1/2)*(2*x^6+1)/x^6-1/6*RootOf(_Z^2+1)*ln((RootOf(_Z^2+1)+(x^6-1)^(1/2))/x^3)+1/12*RootOf(_Z^2+3)*
ln((4*RootOf(_Z^2+3)*x^6-7*RootOf(_Z^2+3)+12*(x^6-1)^(1/2))/(2*x^3-1)/(2*x^3+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{6} - 1\right )}^{2} \sqrt {x^{6} - 1}}{{\left (4 \, x^{6} - 1\right )} x^{7}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)^(1/2)*(2*x^6-1)^2/x^7/(4*x^6-1),x, algorithm="maxima")

[Out]

integrate((2*x^6 - 1)^2*sqrt(x^6 - 1)/((4*x^6 - 1)*x^7), x)

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mupad [B]  time = 0.98, size = 50, normalized size = 0.78 \begin {gather*} \frac {\sqrt {x^6-1}}{3}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,\sqrt {x^6-1}}{3}\right )}{6}-\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{6}+\frac {\sqrt {x^6-1}}{6\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^6 - 1)^(1/2)*(2*x^6 - 1)^2)/(x^7*(4*x^6 - 1)),x)

[Out]

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

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sympy [A]  time = 50.60, size = 60, normalized size = 0.94 \begin {gather*} \frac {\sqrt {x^{6} - 1}}{3} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x^{6} - 1}}{3} \right )}}{6} - \frac {\operatorname {atan}{\left (\sqrt {x^{6} - 1} \right )}}{6} + \frac {\sqrt {x^{6} - 1}}{6 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6-1)**(1/2)*(2*x**6-1)**2/x**7/(4*x**6-1),x)

[Out]

sqrt(x**6 - 1)/3 - sqrt(3)*atan(2*sqrt(3)*sqrt(x**6 - 1)/3)/6 - atan(sqrt(x**6 - 1))/6 + sqrt(x**6 - 1)/(6*x**
6)

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