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

Optimal. Leaf size=61 \[ \frac {1}{36} \sqrt {x^6-1} \left (4 x^6-13\right )+\frac {1}{3} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt {x^6-1}}{\sqrt {3}}\right )}{8 \sqrt {3}} \]

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Rubi [A]  time = 0.10, antiderivative size = 67, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {573, 178, 88, 63, 203} \begin {gather*} \frac {1}{9} \left (x^6-1\right )^{3/2}-\frac {\sqrt {x^6-1}}{4}+\frac {1}{3} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt {x^6-1}}{\sqrt {3}}\right )}{8 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*Sqrt[-1 + x^6] + (-1 + x^6)^(3/2)/9 + ArcTan[Sqrt[-1 + x^6]]/3 - ArcTan[(2*Sqrt[-1 + x^6])/Sqrt[3]]/(8*Sq
rt[3])

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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 178

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_))^(q_))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbo
l] :> Dist[(b*e - a*f)/(b*c - a*d), Int[((e + f*x)^(p - 1)*(g + h*x)^q)/(a + b*x), x], x] - Dist[(d*e - c*f)/(
b*c - a*d), Int[((e + f*x)^(p - 1)*(g + h*x)^q)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, q}, x] &&
LtQ[0, p, 1]

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 \left (-1+4 x^6\right )} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x} (-1+2 x)^2}{x (-1+4 x)} \, dx,x,x^6\right )\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {(-1+2 x)^2}{\sqrt {-1+x} x} \, dx,x,x^6\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {(-1+2 x)^2}{\sqrt {-1+x} (-1+4 x)} \, dx,x,x^6\right )\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \left (4 \sqrt {-1+x}+\frac {1}{\sqrt {-1+x} x}\right ) \, dx,x,x^6\right )-\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{4 \sqrt {-1+x}}+\sqrt {-1+x}+\frac {1}{4 \sqrt {-1+x} (-1+4 x)}\right ) \, dx,x,x^6\right )\\ &=-\frac {1}{4} \sqrt {-1+x^6}+\frac {1}{9} \left (-1+x^6\right )^{3/2}-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} (-1+4 x)} \, dx,x,x^6\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )\\ &=-\frac {1}{4} \sqrt {-1+x^6}+\frac {1}{9} \left (-1+x^6\right )^{3/2}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{3+4 x^2} \, dx,x,\sqrt {-1+x^6}\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )\\ &=-\frac {1}{4} \sqrt {-1+x^6}+\frac {1}{9} \left (-1+x^6\right )^{3/2}+\frac {1}{3} \tan ^{-1}\left (\sqrt {-1+x^6}\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt {-1+x^6}}{\sqrt {3}}\right )}{8 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 100, normalized size = 1.64 \begin {gather*} \frac {1}{144} \left (16 \sqrt {x^6-1} x^6-52 \sqrt {x^6-1}+48 \tan ^{-1}\left (\sqrt {x^6-1}\right )+3 \sqrt {3} \tan ^{-1}\left (\frac {2-x^3}{\sqrt {3} \sqrt {x^6-1}}\right )+3 \sqrt {3} \tan ^{-1}\left (\frac {x^3+2}{\sqrt {3} \sqrt {x^6-1}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-52*Sqrt[-1 + x^6] + 16*x^6*Sqrt[-1 + x^6] + 3*Sqrt[3]*ArcTan[(2 - x^3)/(Sqrt[3]*Sqrt[-1 + x^6])] + 3*Sqrt[3]
*ArcTan[(2 + x^3)/(Sqrt[3]*Sqrt[-1 + x^6])] + 48*ArcTan[Sqrt[-1 + x^6]])/144

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IntegrateAlgebraic [A]  time = 0.05, size = 61, normalized size = 1.00 \begin {gather*} \frac {1}{36} \sqrt {-1+x^6} \left (-13+4 x^6\right )+\frac {1}{3} \tan ^{-1}\left (\sqrt {-1+x^6}\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt {-1+x^6}}{\sqrt {3}}\right )}{8 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-1 + x^6]*(-13 + 4*x^6))/36 + ArcTan[Sqrt[-1 + x^6]]/3 - ArcTan[(2*Sqrt[-1 + x^6])/Sqrt[3]]/(8*Sqrt[3])

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fricas [A]  time = 0.46, size = 45, normalized size = 0.74 \begin {gather*} \frac {1}{36} \, {\left (4 \, x^{6} - 13\right )} \sqrt {x^{6} - 1} - \frac {1}{24} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) + \frac {1}{3} \, \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/(4*x^6-1),x, algorithm="fricas")

[Out]

1/36*(4*x^6 - 13)*sqrt(x^6 - 1) - 1/24*sqrt(3)*arctan(2/3*sqrt(3)*sqrt(x^6 - 1)) + 1/3*arctan(sqrt(x^6 - 1))

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giac [A]  time = 0.28, size = 47, normalized size = 0.77 \begin {gather*} \frac {1}{9} \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} - \frac {1}{24} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) - \frac {1}{4} \, \sqrt {x^{6} - 1} + \frac {1}{3} \, \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/(4*x^6-1),x, algorithm="giac")

[Out]

1/9*(x^6 - 1)^(3/2) - 1/24*sqrt(3)*arctan(2/3*sqrt(3)*sqrt(x^6 - 1)) - 1/4*sqrt(x^6 - 1) + 1/3*arctan(sqrt(x^6
- 1))

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maple [C]  time = 1.04, size = 101, normalized size = 1.66

method	result	size
trager	\(\left (\frac {x^{6}}{9}-\frac {13}{36}\right ) \sqrt {x^{6}-1}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}+\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {4 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}-7 \RootOf \left (\textit {\_Z}^{2}+3\right )+12 \sqrt {x^{6}-1}}{\left (2 x^{3}-1\right ) \left (2 x^{3}+1\right )}\right )}{48}\)	\(101\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/9*x^6-13/36)*(x^6-1)^(1/2)+1/3*RootOf(_Z^2+1)*ln((RootOf(_Z^2+1)+(x^6-1)^(1/2))/x^3)+1/48*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}\,{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/(4*x^6-1),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.78, size = 47, normalized size = 0.77 \begin {gather*} \frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{3}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,\sqrt {x^6-1}}{3}\right )}{24}-\frac {\sqrt {x^6-1}}{4}+\frac {{\left (x^6-1\right )}^{3/2}}{9} \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*(4*x^6 - 1)),x)

[Out]

atan((x^6 - 1)^(1/2))/3 - (3^(1/2)*atan((2*3^(1/2)*(x^6 - 1)^(1/2))/3))/24 - (x^6 - 1)^(1/2)/4 + (x^6 - 1)^(3/
2)/9

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sympy [A]  time = 43.42, size = 56, normalized size = 0.92 \begin {gather*} \frac {\left (x^{6} - 1\right )^{\frac {3}{2}}}{9} - \frac {\sqrt {x^{6} - 1}}{4} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x^{6} - 1}}{3} \right )}}{24} + \frac {\operatorname {atan}{\left (\sqrt {x^{6} - 1} \right )}}{3} \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/(4*x**6-1),x)

[Out]

(x**6 - 1)**(3/2)/9 - sqrt(x**6 - 1)/4 - sqrt(3)*atan(2*sqrt(3)*sqrt(x**6 - 1)/3)/24 + atan(sqrt(x**6 - 1))/3

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