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

Optimal. Leaf size=51 \[ \frac {\sqrt {x^6-1}}{3}+\frac {1}{3} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {2 \tan ^{-1}\left (\frac {\sqrt {x^6-1}}{\sqrt {3}}\right )}{\sqrt {3}} \]

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Rubi [A]  time = 0.05, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {573, 154, 156, 63, 203} \begin {gather*} \frac {\sqrt {x^6-1}}{3}+\frac {1}{3} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {2 \tan ^{-1}\left (\frac {\sqrt {x^6-1}}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[-1 + x^6]/3 + ArcTan[Sqrt[-1 + x^6]]/3 - (2*ArcTan[Sqrt[-1 + x^6]/Sqrt[3]])/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 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

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

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Mathematica [A]  time = 0.05, size = 47, normalized size = 0.92 \begin {gather*} \frac {1}{3} \left (\sqrt {x^6-1}+\tan ^{-1}\left (\sqrt {x^6-1}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {x^6-1}}{\sqrt {3}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.04, size = 51, normalized size = 1.00 \begin {gather*} \frac {1}{3} \sqrt {-1+x^6}+\frac {1}{3} \tan ^{-1}\left (\sqrt {-1+x^6}\right )-\frac {2 \tan ^{-1}\left (\frac {\sqrt {-1+x^6}}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.05, size = 38, normalized size = 0.75 \begin {gather*} -\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) + \frac {1}{3} \, \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^6-2)*(x^6-1)^(1/2)/x/(x^6+2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.18, size = 38, normalized size = 0.75 \begin {gather*} -\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) + \frac {1}{3} \, \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^6-2)*(x^6-1)^(1/2)/x/(x^6+2),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.86, size = 83, normalized size = 1.63

method	result	size
trager	\(\frac {\sqrt {x^{6}-1}}{3}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}+\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}+6 \sqrt {x^{6}-1}-4 \RootOf \left (\textit {\_Z}^{2}+3\right )}{x^{6}+2}\right )}{3}\)	\(83\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.86, size = 38, normalized size = 0.75 \begin {gather*} \frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{3}-\frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\sqrt {x^6-1}}{3}\right )}{3}+\frac {\sqrt {x^6-1}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 19.83, size = 46, normalized size = 0.90 \begin {gather*} \frac {\sqrt {x^{6} - 1}}{3} - \frac {2 \sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} \sqrt {x^{6} - 1}}{3} \right )}}{3} + \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**6-2)*(x**6-1)**(1/2)/x/(x**6+2),x)

[Out]

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

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