3.75 \(\int \frac{x}{\sqrt{-1+x^3} (8+x^3)} \, dx\)

Optimal. Leaf size=74 \[ \frac{1}{18} \tan ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{x^3-1}}\right )+\frac{1}{18} \tan ^{-1}\left (\frac{\sqrt{x^3-1}}{3}\right )-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} (1-x)}{\sqrt{x^3-1}}\right )}{6 \sqrt{3}} \]

[Out]

ArcTan[(1 - x)^2/(3*Sqrt[-1 + x^3])]/18 + ArcTan[Sqrt[-1 + x^3]/3]/18 - ArcTanh[(Sqrt[3]*(1 - x))/Sqrt[-1 + x^
3]]/(6*Sqrt[3])

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Rubi [A]  time = 0.155523, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {486, 444, 63, 204, 2138, 203, 2145, 206} \[ \frac{1}{18} \tan ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{x^3-1}}\right )+\frac{1}{18} \tan ^{-1}\left (\frac{\sqrt{x^3-1}}{3}\right )-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} (1-x)}{\sqrt{x^3-1}}\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[-1 + x^3]*(8 + x^3)),x]

[Out]

ArcTan[(1 - x)^2/(3*Sqrt[-1 + x^3])]/18 + ArcTan[Sqrt[-1 + x^3]/3]/18 - ArcTanh[(Sqrt[3]*(1 - x))/Sqrt[-1 + x^
3]]/(6*Sqrt[3])

Rule 486

Int[(x_)/(((a_) + (b_.)*(x_)^3)*Sqrt[(c_) + (d_.)*(x_)^3]), x_Symbol] :> With[{q = Rt[d/c, 3]}, Dist[(d*q)/(4*
b), Int[x^2/((8*c - d*x^3)*Sqrt[c + d*x^3]), x], x] + (-Dist[q^2/(12*b), Int[(1 + q*x)/((2 - q*x)*Sqrt[c + d*x
^3]), x], x] + Dist[1/(12*b*c), Int[(2*c*q^2 - 2*d*x - d*q*x^2)/((4 + 2*q*x + q^2*x^2)*Sqrt[c + d*x^3]), x], x
])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[8*b*c + a*d, 0]

Rule 444

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

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 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 2138

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[(-2*e)/d, Subst[Int
[1/(9 - a*x^2), x], x, (1 + (f*x)/e)^2/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f,
0] && EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]

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 2145

Int[((f_) + (g_.)*(x_) + (h_.)*(x_)^2)/(((c_) + (d_.)*(x_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbo
l] :> Dist[-2*g*h, Subst[Int[1/(2*e*h - (b*d*f - 2*a*e*h)*x^2), x], x, (1 + (2*h*x)/g)/Sqrt[a + b*x^3]], x] /;
FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b*d*f - 2*a*e*h, 0] && EqQ[b*g^3 - 8*a*h^3, 0] && EqQ[g^2 + 2*f*h,
0] && EqQ[b*d*f + b*c*g - 4*a*e*h, 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])

Rubi steps

\begin{align*} \int \frac{x}{\sqrt{-1+x^3} \left (8+x^3\right )} \, dx &=-\left (\frac{1}{12} \int \frac{1-x}{(2+x) \sqrt{-1+x^3}} \, dx\right )-\frac{1}{12} \int \frac{-2-2 x+x^2}{\left (4-2 x+x^2\right ) \sqrt{-1+x^3}} \, dx-\frac{1}{4} \int \frac{x^2}{\left (-8-x^3\right ) \sqrt{-1+x^3}} \, dx\\ &=-\left (\frac{1}{12} \operatorname{Subst}\left (\int \frac{1}{(-8-x) \sqrt{-1+x}} \, dx,x,x^3\right )\right )+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{9+x^2} \, dx,x,\frac{(1-x)^2}{\sqrt{-1+x^3}}\right )-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{2-6 x^2} \, dx,x,\frac{1-x}{\sqrt{-1+x^3}}\right )\\ &=\frac{1}{18} \tan ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{-1+x^3}}\right )-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} (1-x)}{\sqrt{-1+x^3}}\right )}{6 \sqrt{3}}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{-9-x^2} \, dx,x,\sqrt{-1+x^3}\right )\\ &=\frac{1}{18} \tan ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{-1+x^3}}\right )+\frac{1}{18} \tan ^{-1}\left (\frac{1}{3} \sqrt{-1+x^3}\right )-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} (1-x)}{\sqrt{-1+x^3}}\right )}{6 \sqrt{3}}\\ \end{align*}

Mathematica [C]  time = 0.0194046, size = 48, normalized size = 0.65 \[ \frac{x^2 \sqrt{1-x^3} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};x^3,-\frac{x^3}{8}\right )}{16 \sqrt{x^3-1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x/(Sqrt[-1 + x^3]*(8 + x^3)),x]

[Out]

(x^2*Sqrt[1 - x^3]*AppellF1[2/3, 1/2, 1, 5/3, x^3, -x^3/8])/(16*Sqrt[-1 + x^3])

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Maple [C]  time = 0.115, size = 286, normalized size = 3.9 \begin{align*} -{\frac{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}{9}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }}{\it EllipticPi} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},{\frac{i}{6}}\sqrt{3}+{\frac{1}{2}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}-1}}}}+{\frac{\sqrt{2}}{36}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{2}-2\,{\it \_Z}+4 \right ) }{ \left ( 2-{\it \_alpha} \right ) \left ({\it \_alpha}-1 \right ) \left ( -i\sqrt{3}-3 \right ) \sqrt{{\frac{-1+x}{-i\sqrt{3}-3}}}\sqrt{{\frac{2\,x+1-i\sqrt{3}}{-i\sqrt{3}+3}}}\sqrt{{\frac{2\,x+1+i\sqrt{3}}{i\sqrt{3}+3}}}{\it EllipticPi} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},{\frac{i}{6}}{\it \_alpha}\,\sqrt{3}+{\frac{{\it \_alpha}}{2}}-{\frac{i}{6}}\sqrt{3}-{\frac{1}{2}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}-1}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^3+8)/(x^3-1)^(1/2),x)

[Out]

-1/9*(-3/2-1/2*I*3^(1/2))*((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2-1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2
)*((x+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(1/2)/(x^3-1)^(1/2)*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/
2),1/6*I*3^(1/2)+1/2,((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))+1/36*2^(1/2)*sum((2-_alpha)*(_alpha-1)*(
-I*3^(1/2)-3)*((-1+x)/(-I*3^(1/2)-3))^(1/2)*((2*x+1-I*3^(1/2))/(-I*3^(1/2)+3))^(1/2)*((2*x+1+I*3^(1/2))/(I*3^(
1/2)+3))^(1/2)/(x^3-1)^(1/2)*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),1/6*I*_alpha*3^(1/2)+1/2*_alpha-1/
6*I*3^(1/2)-1/2,((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)),_alpha=RootOf(_Z^2-2*_Z+4))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (x^{3} + 8\right )} \sqrt{x^{3} - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^3+8)/(x^3-1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x/((x^3 + 8)*sqrt(x^3 - 1)), x)

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Fricas [B]  time = 3.58602, size = 1447, normalized size = 19.55 \begin{align*} \frac{1}{216} \, \sqrt{3} \log \left (\frac{4 \,{\left (x^{6} + 48 \, x^{5} + 186 \, x^{4} - 56 \, x^{3} + 6 \, \sqrt{3}{\left (x^{4} + 12 \, x^{3} + 12 \, x^{2} - 16 \, x\right )} \sqrt{x^{3} - 1} - 120 \, x^{2} - 96 \, x + 64\right )}}{x^{6} - 6 \, x^{5} + 24 \, x^{4} - 56 \, x^{3} + 96 \, x^{2} - 96 \, x + 64}\right ) - \frac{1}{216} \, \sqrt{3} \log \left (\frac{4 \,{\left (x^{6} + 48 \, x^{5} + 186 \, x^{4} - 56 \, x^{3} - 6 \, \sqrt{3}{\left (x^{4} + 12 \, x^{3} + 12 \, x^{2} - 16 \, x\right )} \sqrt{x^{3} - 1} - 120 \, x^{2} - 96 \, x + 64\right )}}{x^{6} - 6 \, x^{5} + 24 \, x^{4} - 56 \, x^{3} + 96 \, x^{2} - 96 \, x + 64}\right ) + \frac{1}{54} \, \arctan \left (\frac{{\left (x^{3} - 12 \, x^{2} - 6 \, x - 10\right )} \sqrt{x^{3} - 1}}{6 \,{\left (x^{4} - x^{3} - x + 1\right )}}\right ) - \frac{1}{54} \, \arctan \left (-\frac{\sqrt{x^{3} - 1}{\left (x^{2} - 8 \, x + 10\right )} +{\left (3 \, \sqrt{3}{\left (x^{3} + x^{2} - 2 \, x\right )} - \sqrt{x^{3} - 1}{\left (x^{2} + 10 \, x - 8\right )}\right )} \sqrt{\frac{x^{6} + 48 \, x^{5} + 186 \, x^{4} - 56 \, x^{3} + 6 \, \sqrt{3}{\left (x^{4} + 12 \, x^{3} + 12 \, x^{2} - 16 \, x\right )} \sqrt{x^{3} - 1} - 120 \, x^{2} - 96 \, x + 64}{x^{6} - 6 \, x^{5} + 24 \, x^{4} - 56 \, x^{3} + 96 \, x^{2} - 96 \, x + 64}}}{3 \,{\left (x^{3} - 3 \, x^{2} + 2\right )}}\right ) - \frac{1}{54} \, \arctan \left (-\frac{\sqrt{x^{3} - 1}{\left (x^{2} - 8 \, x + 10\right )} -{\left (3 \, \sqrt{3}{\left (x^{3} + x^{2} - 2 \, x\right )} + \sqrt{x^{3} - 1}{\left (x^{2} + 10 \, x - 8\right )}\right )} \sqrt{\frac{x^{6} + 48 \, x^{5} + 186 \, x^{4} - 56 \, x^{3} - 6 \, \sqrt{3}{\left (x^{4} + 12 \, x^{3} + 12 \, x^{2} - 16 \, x\right )} \sqrt{x^{3} - 1} - 120 \, x^{2} - 96 \, x + 64}{x^{6} - 6 \, x^{5} + 24 \, x^{4} - 56 \, x^{3} + 96 \, x^{2} - 96 \, x + 64}}}{3 \,{\left (x^{3} - 3 \, x^{2} + 2\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^3+8)/(x^3-1)^(1/2),x, algorithm="fricas")

[Out]

1/216*sqrt(3)*log(4*(x^6 + 48*x^5 + 186*x^4 - 56*x^3 + 6*sqrt(3)*(x^4 + 12*x^3 + 12*x^2 - 16*x)*sqrt(x^3 - 1)
- 120*x^2 - 96*x + 64)/(x^6 - 6*x^5 + 24*x^4 - 56*x^3 + 96*x^2 - 96*x + 64)) - 1/216*sqrt(3)*log(4*(x^6 + 48*x
^5 + 186*x^4 - 56*x^3 - 6*sqrt(3)*(x^4 + 12*x^3 + 12*x^2 - 16*x)*sqrt(x^3 - 1) - 120*x^2 - 96*x + 64)/(x^6 - 6
*x^5 + 24*x^4 - 56*x^3 + 96*x^2 - 96*x + 64)) + 1/54*arctan(1/6*(x^3 - 12*x^2 - 6*x - 10)*sqrt(x^3 - 1)/(x^4 -
x^3 - x + 1)) - 1/54*arctan(-1/3*(sqrt(x^3 - 1)*(x^2 - 8*x + 10) + (3*sqrt(3)*(x^3 + x^2 - 2*x) - sqrt(x^3 -
1)*(x^2 + 10*x - 8))*sqrt((x^6 + 48*x^5 + 186*x^4 - 56*x^3 + 6*sqrt(3)*(x^4 + 12*x^3 + 12*x^2 - 16*x)*sqrt(x^3
- 1) - 120*x^2 - 96*x + 64)/(x^6 - 6*x^5 + 24*x^4 - 56*x^3 + 96*x^2 - 96*x + 64)))/(x^3 - 3*x^2 + 2)) - 1/54*
arctan(-1/3*(sqrt(x^3 - 1)*(x^2 - 8*x + 10) - (3*sqrt(3)*(x^3 + x^2 - 2*x) + sqrt(x^3 - 1)*(x^2 + 10*x - 8))*s
qrt((x^6 + 48*x^5 + 186*x^4 - 56*x^3 - 6*sqrt(3)*(x^4 + 12*x^3 + 12*x^2 - 16*x)*sqrt(x^3 - 1) - 120*x^2 - 96*x
+ 64)/(x^6 - 6*x^5 + 24*x^4 - 56*x^3 + 96*x^2 - 96*x + 64)))/(x^3 - 3*x^2 + 2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 2\right ) \left (x^{2} - 2 x + 4\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**3+8)/(x**3-1)**(1/2),x)

[Out]

Integral(x/(sqrt((x - 1)*(x**2 + x + 1))*(x + 2)*(x**2 - 2*x + 4)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (x^{3} + 8\right )} \sqrt{x^{3} - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^3+8)/(x^3-1)^(1/2),x, algorithm="giac")

[Out]

integrate(x/((x^3 + 8)*sqrt(x^3 - 1)), x)