3.4.60 \(\int \frac {1}{a-b x^3} \, dx\) [360]

Optimal. Leaf size=114 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {206, 31, 648, 631, 210, 642} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b}}+\frac {\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a - b*x^3)^(-1),x]

[Out]

ArcTan[(a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))]/(Sqrt[3]*a^(2/3)*b^(1/3)) - Log[a^(1/3) - b^(1/3)*x]/(3*a^(2
/3)*b^(1/3)) + Log[a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(6*a^(2/3)*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 206

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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 631

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 642

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 648

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

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 89, normalized size = 0.78 \begin {gather*} \frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )+\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^3)^(-1),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.12, size = 92, normalized size = 0.81

method	result	size
risch	\(-\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}-a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b}\)	\(29\)
default	\(-\frac {\ln \left (x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\)	\(92\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.51, size = 97, normalized size = 0.85 \begin {gather*} \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x^{2} + x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.38, size = 320, normalized size = 2.81 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} a b \sqrt {\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} + 3 \, \left (-a^{2} b\right )^{\frac {1}{3}} a x + a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} - \left (-a^{2} b\right )^{\frac {2}{3}} x + \left (-a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} - a}\right ) + \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} + \left (-a^{2} b\right )^{\frac {2}{3}} x - \left (-a^{2} b\right )^{\frac {1}{3}} a\right ) - 2 \, \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b}, \frac {6 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (-a^{2} b\right )^{\frac {2}{3}} x - \left (-a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) + \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} + \left (-a^{2} b\right )^{\frac {2}{3}} x - \left (-a^{2} b\right )^{\frac {1}{3}} a\right ) - 2 \, \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.06, size = 22, normalized size = 0.19 \begin {gather*} - \operatorname {RootSum} {\left (27 t^{3} a^{2} b - 1, \left ( t \mapsto t \log {\left (- 3 t a + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 2.02, size = 104, normalized size = 0.91 \begin {gather*} -\frac {\left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a} + \frac {\sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b} + \frac {\left (a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 1.23, size = 115, normalized size = 1.01 \begin {gather*} \frac {\ln \left (a^{1/3}\,{\left (-b\right )}^{5/3}+b^2\,x\right )}{3\,a^{2/3}\,{\left (-b\right )}^{1/3}}+\frac {\ln \left (3\,b^2\,x+\frac {3\,a^{1/3}\,{\left (-b\right )}^{5/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,{\left (-b\right )}^{1/3}}-\frac {\ln \left (3\,b^2\,x-\frac {3\,a^{1/3}\,{\left (-b\right )}^{5/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,{\left (-b\right )}^{1/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a - b*x^3),x)

[Out]

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

________________________________________________________________________________________