3.35 \(\int \frac {1}{-b+a x^3} \, dx\)

Optimal. Leaf size=115 \[ -\frac {\log \left (a^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 \sqrt [3]{a} b^{2/3}}+\frac {\log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{a} x+\sqrt [3]{b}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{a} b^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {200, 31, 634, 617, 204, 628} \[ -\frac {\log \left (a^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 \sqrt [3]{a} b^{2/3}}+\frac {\log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{a} x+\sqrt [3]{b}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{a} b^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[(-b + a*x^3)^(-1),x]

[Out]

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

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

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Mathematica [A]  time = 0.03, size = 89, normalized size = 0.77 \[ -\frac {\log \left (a^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )-2 \log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}+1}{\sqrt {3}}\right )}{6 \sqrt [3]{a} b^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-b + a*x^3)^(-1),x]

[Out]

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

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fricas [A]  time = 0.43, size = 300, normalized size = 2.61 \[ \left [\frac {3 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a b^{2}\right )^{\frac {1}{3}} b x + b^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} - \left (a b^{2}\right )^{\frac {2}{3}} x - \left (a b^{2}\right )^{\frac {1}{3}} b\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{a x^{3} - b}\right ) - \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x^{2} + \left (a b^{2}\right )^{\frac {2}{3}} x + \left (a b^{2}\right )^{\frac {1}{3}} b\right ) + 2 \, \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x - \left (a b^{2}\right )^{\frac {2}{3}}\right )}{6 \, a b^{2}}, -\frac {6 \, \sqrt {\frac {1}{3}} a b \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a b^{2}\right )^{\frac {2}{3}} x + \left (a b^{2}\right )^{\frac {1}{3}} b\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{b^{2}}\right ) + \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x^{2} + \left (a b^{2}\right )^{\frac {2}{3}} x + \left (a b^{2}\right )^{\frac {1}{3}} b\right ) - 2 \, \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x - \left (a b^{2}\right )^{\frac {2}{3}}\right )}{6 \, a b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*sqrt(1/3)*a*b*sqrt(-(a*b^2)^(1/3)/a)*log((2*a*b*x^3 - 3*(a*b^2)^(1/3)*b*x + b^2 - 3*sqrt(1/3)*(2*a*b*x
^2 - (a*b^2)^(2/3)*x - (a*b^2)^(1/3)*b)*sqrt(-(a*b^2)^(1/3)/a))/(a*x^3 - b)) - (a*b^2)^(2/3)*log(a*b*x^2 + (a*
b^2)^(2/3)*x + (a*b^2)^(1/3)*b) + 2*(a*b^2)^(2/3)*log(a*b*x - (a*b^2)^(2/3)))/(a*b^2), -1/6*(6*sqrt(1/3)*a*b*s
qrt((a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2*(a*b^2)^(2/3)*x + (a*b^2)^(1/3)*b)*sqrt((a*b^2)^(1/3)/a)/b^2) + (a*b^
2)^(2/3)*log(a*b*x^2 + (a*b^2)^(2/3)*x + (a*b^2)^(1/3)*b) - 2*(a*b^2)^(2/3)*log(a*b*x - (a*b^2)^(2/3)))/(a*b^2
)]

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giac [A]  time = 1.27, size = 104, normalized size = 0.90 \[ \frac {\left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{3 \, b} - \frac {\sqrt {3} \left (a^{2} b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a b} - \frac {\left (a^{2} b\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b/a)^(1/3)*log(abs(x - (b/a)^(1/3)))/b - 1/3*sqrt(3)*(a^2*b)^(1/3)*arctan(1/3*sqrt(3)*(2*x + (b/a)^(1/3))
/(b/a)^(1/3))/(a*b) - 1/6*(a^2*b)^(1/3)*log(x^2 + x*(b/a)^(1/3) + (b/a)^(2/3))/(a*b)

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maple [A]  time = 0.00, size = 92, normalized size = 0.80 \[ -\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 \left (\frac {b}{a}\right )^{\frac {2}{3}} a}+\frac {\ln \left (x -\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {b}{a}\right )^{\frac {2}{3}} a}-\frac {\ln \left (x^{2}+\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {b}{a}\right )^{\frac {2}{3}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.96, size = 97, normalized size = 0.84 \[ -\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a \left (\frac {b}{a}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{2} + x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {\log \left (x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \, a \left (\frac {b}{a}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.31, size = 101, normalized size = 0.88 \[ \frac {\ln \left (a^{1/3}\,x-b^{1/3}\right )}{3\,a^{1/3}\,b^{2/3}}+\frac {\ln \left (3\,a^2\,x-\frac {3\,a^{5/3}\,b^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{1/3}\,b^{2/3}}-\frac {\ln \left (3\,a^2\,x+\frac {3\,a^{5/3}\,b^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{1/3}\,b^{2/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(b - a*x^3),x)

[Out]

log(a^(1/3)*x - b^(1/3))/(3*a^(1/3)*b^(2/3)) + (log(3*a^2*x - (3*a^(5/3)*b^(1/3)*(3^(1/2)*1i - 1))/2)*(3^(1/2)
*1i - 1))/(6*a^(1/3)*b^(2/3)) - (log(3*a^2*x + (3*a^(5/3)*b^(1/3)*(3^(1/2)*1i + 1))/2)*(3^(1/2)*1i + 1))/(6*a^
(1/3)*b^(2/3))

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sympy [A]  time = 0.15, size = 20, normalized size = 0.17 \[ \operatorname {RootSum} {\left (27 t^{3} a b^{2} - 1, \left (t \mapsto t \log {\left (- 3 t b + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(27*_t**3*a*b**2 - 1, Lambda(_t, _t*log(-3*_t*b + x)))

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