3.9.51 \(\int \frac {-x^2+10 x^8}{\sqrt {-1+x^6} (-1+4 x^6)} \, dx\)

Optimal. Leaf size=64 \[ \frac {5}{6} \log \left (\sqrt {x^6-1}+x^3\right )-\frac {\tan ^{-1}\left (-\frac {4 x^6}{\sqrt {3}}-\frac {4 \sqrt {x^6-1} x^3}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

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Rubi [A]  time = 0.09, antiderivative size = 47, normalized size of antiderivative = 0.73, number of steps used = 7, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {1593, 575, 523, 217, 206, 377, 204} \begin {gather*} \frac {5}{6} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^3}{\sqrt {x^6-1}}\right )}{2 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x^2 + 10*x^8)/(Sqrt[-1 + x^6]*(-1 + 4*x^6)),x]

[Out]

-1/2*ArcTan[(Sqrt[3]*x^3)/Sqrt[-1 + x^6]]/Sqrt[3] + (5*ArcTanh[x^3/Sqrt[-1 + x^6]])/6

Rule 204

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

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]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int \frac {-x^2+10 x^8}{\sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx &=\int \frac {x^2 \left (-1+10 x^6\right )}{\sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {-1+10 x^2}{\sqrt {-1+x^2} \left (-1+4 x^2\right )} \, dx,x,x^3\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (-1+4 x^2\right )} \, dx,x,x^3\right )+\frac {5}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-1-3 x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )+\frac {5}{6} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^3}{\sqrt {-1+x^6}}\right )}{2 \sqrt {3}}+\frac {5}{6} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 47, normalized size = 0.73 \begin {gather*} \frac {5}{6} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^3}{\sqrt {x^6-1}}\right )}{2 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^2 + 10*x^8)/(Sqrt[-1 + x^6]*(-1 + 4*x^6)),x]

[Out]

-1/2*ArcTan[(Sqrt[3]*x^3)/Sqrt[-1 + x^6]]/Sqrt[3] + (5*ArcTanh[x^3/Sqrt[-1 + x^6]])/6

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IntegrateAlgebraic [A]  time = 0.10, size = 66, normalized size = 1.03 \begin {gather*} \frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {4 x^6}{\sqrt {3}}+\frac {4 x^3 \sqrt {-1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {5}{6} \log \left (-x^3+\sqrt {-1+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-x^2 + 10*x^8)/(Sqrt[-1 + x^6]*(-1 + 4*x^6)),x]

[Out]

ArcTan[1/Sqrt[3] - (4*x^6)/Sqrt[3] + (4*x^3*Sqrt[-1 + x^6])/Sqrt[3]]/(2*Sqrt[3]) - (5*Log[-x^3 + Sqrt[-1 + x^6
]])/6

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fricas [A]  time = 0.47, size = 51, normalized size = 0.80 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {4}{3} \, \sqrt {3} \sqrt {x^{6} - 1} x^{3} - \frac {1}{3} \, \sqrt {3} {\left (4 \, x^{6} - 1\right )}\right ) - \frac {5}{6} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x^8-x^2)/(x^6-1)^(1/2)/(4*x^6-1),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*arctan(4/3*sqrt(3)*sqrt(x^6 - 1)*x^3 - 1/3*sqrt(3)*(4*x^6 - 1)) - 5/6*log(-x^3 + sqrt(x^6 - 1))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x^8-x^2)/(x^6-1)^(1/2)/(4*x^6-1),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:root
of minimal polynomial must be unitary Error: Bad Argument Valuerootof minimal polynomial must be unitary Error
: Bad Argument Valuerootof minimal polynomial must be unitary Error: Bad Argument Valuerootof minimal polynomi
al must be unitary Error: Bad Argument Valuerootof minimal polynomial must be unitary Error: Bad Argument Valu
erootof minimal polynomial must be unitary Error: Bad Argument Valuerootof minimal polynomial must be unitary
Error: Bad Argument Valuerootof minimal polynomial must be unitary Error: Bad Argument Valuerootof minimal pol
ynomial must be unitary Error: Bad Argument Valuerootof minimal polynomial must be unitary Error: Bad Argument
ValueWarning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real)
:Check [abs(t_nostep)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the a
rgument is real):Check [abs(x)]Unable to cancel step at 0 of -5/12*ln(-sqrt(-x^6+1)+1)+5/12*ln(sqrt(-x^6+1)+1)
+1/2/sqrt(3)*atan(sqrt(-x^6+1)/sqrt(3))-5/12*ln(-sqrt(-x^6+1)+1)-5/12*ln(sqrt(-x^6+1)+1)-1/2/sqrt(3)*atan(sqrt
(-x^6+1)/sqrt(3))Done

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maple [C]  time = 0.33, size = 74, normalized size = 1.16

method	result	size
trager	\(\frac {5 \ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{6}-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}-6 x^{3} \sqrt {x^{6}-1}+\RootOf \left (\textit {\_Z}^{2}+3\right )}{\left (2 x^{3}-1\right ) \left (2 x^{3}+1\right )}\right )}{12}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x^8-x^2)/(x^6-1)^(1/2)/(4*x^6-1),x,method=_RETURNVERBOSE)

[Out]

5/6*ln(x^3+(x^6-1)^(1/2))-1/12*RootOf(_Z^2+3)*ln((2*RootOf(_Z^2+3)*x^6-6*x^3*(x^6-1)^(1/2)+RootOf(_Z^2+3))/(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 {10 \, x^{8} - x^{2}}{{\left (4 \, x^{6} - 1\right )} \sqrt {x^{6} - 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x^8-x^2)/(x^6-1)^(1/2)/(4*x^6-1),x, algorithm="maxima")

[Out]

integrate((10*x^8 - x^2)/((4*x^6 - 1)*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^2-10\,x^8}{\sqrt {x^6-1}\,\left (4\,x^6-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2 - 10*x^8)/((x^6 - 1)^(1/2)*(4*x^6 - 1)),x)

[Out]

int(-(x^2 - 10*x^8)/((x^6 - 1)^(1/2)*(4*x^6 - 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (10 x^{6} - 1\right )}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (2 x^{3} - 1\right ) \left (2 x^{3} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x**8-x**2)/(x**6-1)**(1/2)/(4*x**6-1),x)

[Out]

Integral(x**2*(10*x**6 - 1)/(sqrt((x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1))*(2*x**3 - 1)*(2*x**3 + 1)), x
)

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