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]

1/6*ln(-x^3+1)/(a+b)^(1/3)-1/2*ln((a+b)^(1/3)*x-(b*x^3+a)^(1/3))/(a+b)^(1/3)+1/3*arctan(1/3*(1+2*(a+b)^(1/3)*x
/(b*x^3+a)^(1/3))*3^(1/2))/(a+b)^(1/3)*3^(1/2)

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Rubi [A]  time = 0.09, 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 31

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

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 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 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 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 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]

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]

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*}

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Mathematica [A]  time = 0.09, size = 120, normalized size = 1.22 \[ \frac {-2 \log \left (1-\frac {x \sqrt [3]{a+b}}{\sqrt [3]{a+b x^3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{a+b}}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )+\log \left (\frac {x \sqrt [3]{a+b}}{\sqrt [3]{a+b x^3}}+\frac {x^2 (a+b)^{2/3}}{\left (a+b x^3\right )^{2/3}}+1\right )}{6 \sqrt [3]{a+b}} \]

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))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )}}\,{d x} \]

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)

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maple [F]  time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-x^{3}+1\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}\, dx \]

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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )}}\,{d x} \]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {1}{\left (x^3-1\right )\,{\left (b\,x^3+a\right )}^{1/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{x^{3} \sqrt [3]{a + b x^{3}} - \sqrt [3]{a + b x^{3}}}\, dx \]

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)

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