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

Optimal. Leaf size=63 \[ \frac {1}{3} \log \left (\sqrt {x^6-1}+x^3\right )+\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {x^6}{\sqrt {6}}+\frac {\sqrt {x^6-1} x^3}{\sqrt {6}}+\sqrt {\frac {2}{3}}\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 49, normalized size of antiderivative = 0.78, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {575, 523, 217, 206, 377} \begin {gather*} \frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x^3}{\sqrt {x^6-1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 575

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x
_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^
q*(e + f*x^(n/k))^r, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IGtQ[n, 0] && Inte
gerQ[m]

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 47, normalized size = 0.75 \begin {gather*} \frac {1}{3} \left (\tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\sqrt {6} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x^3}{\sqrt {x^6-1}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.15, size = 67, normalized size = 1.06 \begin {gather*} -\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\sqrt {\frac {2}{3}}+\frac {x^6}{\sqrt {6}}-\frac {x^3 \sqrt {-1+x^6}}{\sqrt {6}}\right )-\frac {1}{3} \log \left (-x^3+\sqrt {-1+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 82, normalized size = 1.30 \begin {gather*} \frac {1}{6} \, \sqrt {3} \sqrt {2} \log \left (\frac {25 \, x^{6} - 2 \, \sqrt {3} \sqrt {2} {\left (5 \, x^{6} - 2\right )} - 2 \, \sqrt {x^{6} - 1} {\left (5 \, \sqrt {3} \sqrt {2} x^{3} - 12 \, x^{3}\right )} - 10}{x^{6} + 2}\right ) - \frac {1}{3} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3)*sqrt(2)*log((25*x^6 - 2*sqrt(3)*sqrt(2)*(5*x^6 - 2) - 2*sqrt(x^6 - 1)*(5*sqrt(3)*sqrt(2)*x^3 - 12*
x^3) - 10)/(x^6 + 2)) - 1/3*log(-x^3 + sqrt(x^6 - 1))

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giac [A]  time = 0.20, size = 72, normalized size = 1.14 \begin {gather*} \frac {1}{6} \, \sqrt {6} \log \left (\frac {{\left (x^{3} - \sqrt {x^{6} - 1}\right )}^{2} - 2 \, \sqrt {6} + 5}{{\left (x^{3} - \sqrt {x^{6} - 1}\right )}^{2} + 2 \, \sqrt {6} + 5}\right ) - \frac {1}{6} \, \log \left ({\left (x^{3} - \sqrt {x^{6} - 1}\right )}^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(6)*log(((x^3 - sqrt(x^6 - 1))^2 - 2*sqrt(6) + 5)/((x^3 - sqrt(x^6 - 1))^2 + 2*sqrt(6) + 5)) - 1/6*log
((x^3 - sqrt(x^6 - 1))^2)

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maple [C]  time = 0.31, size = 65, normalized size = 1.03

method	result	size
trager	\(\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}+\frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \ln \left (\frac {-5 \RootOf \left (\textit {\_Z}^{2}-6\right ) x^{6}+12 x^{3} \sqrt {x^{6}-1}+2 \RootOf \left (\textit {\_Z}^{2}-6\right )}{x^{6}+2}\right )}{6}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*ln(x^3+(x^6-1)^(1/2))+1/6*RootOf(_Z^2-6)*ln((-5*RootOf(_Z^2-6)*x^6+12*x^3*(x^6-1)^(1/2)+2*RootOf(_Z^2-6))/
(x^6+2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^2\,\left (x^6-4\right )}{\sqrt {x^6-1}\,\left (x^6+2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (x^{3} - 2\right ) \left (x^{3} + 2\right )}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{6} + 2\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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