3.94 \(\int \frac{1}{(1-x^3) \sqrt [3]{a+b x^3}} \, dx\)

Optimal. Leaf size=98 \[ \frac{\log \left (1-x^3\right )}{6 \sqrt [3]{a+b}}-\frac{\log \left (x \sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}}+\frac{\tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{a+b}}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{a+b}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.093337, antiderivative size = 135, normalized size of antiderivative = 1.38, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {377, 200, 31, 634, 617, 204, 628} \[ -\frac{\log \left (1-\frac{x \sqrt [3]{a+b}}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{a+b}}+\frac{\log \left (\frac{x^2 (a+b)^{2/3}}{\left (a+b x^3\right )^{2/3}}+\frac{x \sqrt [3]{a+b}}{\sqrt [3]{a+b x^3}}+1\right )}{6 \sqrt [3]{a+b}}+\frac{\tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{a+b}}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{a+b}} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - x^3)*(a + b*x^3)^(1/3)),x]

[Out]

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

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 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

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 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1-(a+b) x^3} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt [3]{a+b} x} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )+\frac{1}{3} \operatorname{Subst}\left (\int \frac{2+\sqrt [3]{a+b} x}{1+\sqrt [3]{a+b} x+(a+b)^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )\\ &=-\frac{\log \left (1-\frac{\sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{a+b}}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt [3]{a+b} x+(a+b)^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )+\frac{\operatorname{Subst}\left (\int \frac{\sqrt [3]{a+b}+2 (a+b)^{2/3} x}{1+\sqrt [3]{a+b} x+(a+b)^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{6 \sqrt [3]{a+b}}\\ &=-\frac{\log \left (1-\frac{\sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{a+b}}+\frac{\log \left (1+\frac{(a+b)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}\right )}{6 \sqrt [3]{a+b}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}\right )}{\sqrt [3]{a+b}}\\ &=\frac{\tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{a+b}}-\frac{\log \left (1-\frac{\sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{a+b}}+\frac{\log \left (1+\frac{(a+b)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}\right )}{6 \sqrt [3]{a+b}}\\ \end{align*}

Mathematica [A]  time = 0.155155, size = 120, normalized size = 1.22 \[ \frac{-2 \log \left (1-\frac{x \sqrt [3]{a+b}}{\sqrt [3]{a+b x^3}}\right )+\log \left (\frac{x^2 (a+b)^{2/3}}{\left (a+b x^3\right )^{2/3}}+\frac{x \sqrt [3]{a+b}}{\sqrt [3]{a+b x^3}}+1\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{a+b}}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{6 \sqrt [3]{a+b}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x^3)*(a + b*x^3)^(1/3)),x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{x}^{3}+1}{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^3+1)/(b*x^3+a)^(1/3),x)

[Out]

int(1/(-x^3+1)/(b*x^3+a)^(1/3),x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^3+1)/(b*x^3+a)^(1/3),x, algorithm="maxima")

[Out]

-integrate(1/((b*x^3 + a)^(1/3)*(x^3 - 1)), x)

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^3+1)/(b*x^3+a)^(1/3),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{x^{3} \sqrt [3]{a + b x^{3}} - \sqrt [3]{a + b x^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**3+1)/(b*x**3+a)**(1/3),x)

[Out]

-Integral(1/(x**3*(a + b*x**3)**(1/3) - (a + b*x**3)**(1/3)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^3+1)/(b*x^3+a)^(1/3),x, algorithm="giac")

[Out]

integrate(-1/((b*x^3 + a)^(1/3)*(x^3 - 1)), x)