3.6.34 \(\int \frac {-4+13 x^6}{x \sqrt {-1+x^6} (-1+4 x^6)} \, dx\)

Optimal. Leaf size=41 \[ \frac {4}{3} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt {x^6-1}}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

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Rubi [A]  time = 0.06, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {573, 156, 63, 203} \begin {gather*} \frac {4}{3} \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[(-4 + 13*x^6)/(x*Sqrt[-1 + x^6]*(-1 + 4*x^6)),x]

[Out]

(4*ArcTan[Sqrt[-1 + x^6]])/3 - ArcTan[(2*Sqrt[-1 + x^6])/Sqrt[3]]/(2*Sqrt[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 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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 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 {-4+13 x^6}{x \sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {-4+13 x}{\sqrt {-1+x} x (-1+4 x)} \, dx,x,x^6\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} (-1+4 x)} \, dx,x,x^6\right )\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )\\ &=\frac {4}{3} \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 {4}{3} \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 [A]  time = 0.07, size = 73, normalized size = 1.78 \begin {gather*} \frac {1}{12} \left (16 \tan ^{-1}\left (\sqrt {x^6-1}\right )+\sqrt {3} \tan ^{-1}\left (\frac {2-x^3}{\sqrt {3} \sqrt {x^6-1}}\right )+\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[(-4 + 13*x^6)/(x*Sqrt[-1 + x^6]*(-1 + 4*x^6)),x]

[Out]

(Sqrt[3]*ArcTan[(2 - x^3)/(Sqrt[3]*Sqrt[-1 + x^6])] + Sqrt[3]*ArcTan[(2 + x^3)/(Sqrt[3]*Sqrt[-1 + x^6])] + 16*
ArcTan[Sqrt[-1 + x^6]])/12

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IntegrateAlgebraic [A]  time = 0.04, size = 41, normalized size = 1.00 \begin {gather*} \frac {4}{3} \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[(-4 + 13*x^6)/(x*Sqrt[-1 + x^6]*(-1 + 4*x^6)),x]

[Out]

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

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fricas [A]  time = 0.65, size = 29, normalized size = 0.71 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) + \frac {4}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((13*x^6-4)/x/(x^6-1)^(1/2)/(4*x^6-1),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.32, size = 29, normalized size = 0.71 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) + \frac {4}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((13*x^6-4)/x/(x^6-1)^(1/2)/(4*x^6-1),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.88, size = 87, normalized size = 2.12

method	result	size
trager	\(\frac {4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}-7 \RootOf \left (\textit {\_Z}^{2}+3\right )-12 \sqrt {x^{6}-1}}{\left (2 x^{3}-1\right ) \left (2 x^{3}+1\right )}\right )}{12}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((13*x^6-4)/x/(x^6-1)^(1/2)/(4*x^6-1),x,method=_RETURNVERBOSE)

[Out]

4/3*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*Root
Of(_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 {13 \, x^{6} - 4}{{\left (4 \, x^{6} - 1\right )} \sqrt {x^{6} - 1} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((13*x^6-4)/x/(x^6-1)^(1/2)/(4*x^6-1),x, algorithm="maxima")

[Out]

integrate((13*x^6 - 4)/((4*x^6 - 1)*sqrt(x^6 - 1)*x), x)

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mupad [B]  time = 0.54, size = 29, normalized size = 0.71 \begin {gather*} \frac {4\,\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 )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((13*x^6 - 4)/(x*(x^6 - 1)^(1/2)*(4*x^6 - 1)),x)

[Out]

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

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sympy [A]  time = 22.45, size = 37, normalized size = 0.90 \begin {gather*} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x^{6} - 1}}{3} \right )}}{6} + \frac {4 \operatorname {atan}{\left (\sqrt {x^{6} - 1} \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((13*x**6-4)/x/(x**6-1)**(1/2)/(4*x**6-1),x)

[Out]

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

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