3.4.66 \(\int \frac {1}{-1+a-b x^3} \, dx\) [366]

Optimal. Leaf size=138 \[ \frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{1-a}}}{\sqrt {3}}\right )}{\sqrt {3} (1-a)^{2/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{1-a}+\sqrt [3]{b} x\right )}{3 (1-a)^{2/3} \sqrt [3]{b}}+\frac {\log \left ((1-a)^{2/3}-\sqrt [3]{1-a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 (1-a)^{2/3} \sqrt [3]{b}} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 138, 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 = {206, 31, 648, 631, 210, 642} \begin {gather*} \frac {\text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{1-a}}}{\sqrt {3}}\right )}{\sqrt {3} (1-a)^{2/3} \sqrt [3]{b}}+\frac {\log \left (-\sqrt [3]{1-a} \sqrt [3]{b} x+(1-a)^{2/3}+b^{2/3} x^2\right )}{6 (1-a)^{2/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{1-a}+\sqrt [3]{b} x\right )}{3 (1-a)^{2/3} \sqrt [3]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + a - b*x^3)^(-1),x]

[Out]

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

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Mathematica [A]
time = 0.04, size = 124, normalized size = 0.90 \begin {gather*} \frac {(-1)^{2/3} \left (-2 \sqrt {3} \tan ^{-1}\left (\frac {-1+\frac {2 \sqrt [3]{-1} \sqrt [3]{b} x}{\sqrt [3]{-1+a}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{-1+a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )+\log \left ((-1+a)^{2/3}-\sqrt [3]{-1} \sqrt [3]{-1+a} \sqrt [3]{b} x+(-1)^{2/3} b^{2/3} x^2\right )\right )}{6 (-1+a)^{2/3} \sqrt [3]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + a - b*x^3)^(-1),x]

[Out]

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

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Maple [A]
time = 0.13, size = 106, normalized size = 0.77

method	result	size
risch	\(-\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}-a +1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b}\)	\(30\)
default	\(-\frac {\ln \left (x -\left (\frac {-1+a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {-1+a}{b}\right )^{\frac {2}{3}}}+\frac {\ln \left (x^{2}+\left (\frac {-1+a}{b}\right )^{\frac {1}{3}} x +\left (\frac {-1+a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {-1+a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {-1+a}{b}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 b \left (\frac {-1+a}{b}\right )^{\frac {2}{3}}}\)	\(106\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a-1.0>0)', see `assume?` for m
ore details)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (104) = 208\).
time = 0.38, size = 467, normalized size = 3.38 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (a - 1\right )} b \sqrt {\frac {\left (-{\left (a^{2} - 2 \, a + 1\right )} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, {\left (a - 1\right )} b x^{3} + 3 \, \left (-{\left (a^{2} - 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a - 1\right )} x + a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (a - 1\right )} b x^{2} - \left (-{\left (a^{2} - 2 \, a + 1\right )} b\right )^{\frac {2}{3}} x + \left (-{\left (a^{2} - 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a - 1\right )}\right )} \sqrt {\frac {\left (-{\left (a^{2} - 2 \, a + 1\right )} b\right )^{\frac {1}{3}}}{b}} - 2 \, a + 1}{b x^{3} - a + 1}\right ) + \left (-{\left (a^{2} - 2 \, a + 1\right )} b\right )^{\frac {2}{3}} \log \left ({\left (a - 1\right )} b x^{2} + \left (-{\left (a^{2} - 2 \, a + 1\right )} b\right )^{\frac {2}{3}} x - \left (-{\left (a^{2} - 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a - 1\right )}\right ) - 2 \, \left (-{\left (a^{2} - 2 \, a + 1\right )} b\right )^{\frac {2}{3}} \log \left ({\left (a - 1\right )} b x - \left (-{\left (a^{2} - 2 \, a + 1\right )} b\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{2} - 2 \, a + 1\right )} b}, \frac {6 \, \sqrt {\frac {1}{3}} {\left (a - 1\right )} b \sqrt {-\frac {\left (-{\left (a^{2} - 2 \, a + 1\right )} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (-{\left (a^{2} - 2 \, a + 1\right )} b\right )^{\frac {2}{3}} x - \left (-{\left (a^{2} - 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a - 1\right )}\right )} \sqrt {-\frac {\left (-{\left (a^{2} - 2 \, a + 1\right )} b\right )^{\frac {1}{3}}}{b}}}{a^{2} - 2 \, a + 1}\right ) + \left (-{\left (a^{2} - 2 \, a + 1\right )} b\right )^{\frac {2}{3}} \log \left ({\left (a - 1\right )} b x^{2} + \left (-{\left (a^{2} - 2 \, a + 1\right )} b\right )^{\frac {2}{3}} x - \left (-{\left (a^{2} - 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a - 1\right )}\right ) - 2 \, \left (-{\left (a^{2} - 2 \, a + 1\right )} b\right )^{\frac {2}{3}} \log \left ({\left (a - 1\right )} b x - \left (-{\left (a^{2} - 2 \, a + 1\right )} b\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{2} - 2 \, a + 1\right )} b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.12, size = 34, normalized size = 0.25 \begin {gather*} - \operatorname {RootSum} {\left (t^{3} \cdot \left (27 a^{2} b - 54 a b + 27 b\right ) - 1, \left ( t \mapsto t \log {\left (- 3 t a + 3 t + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-RootSum(_t**3*(27*a**2*b - 54*a*b + 27*b) - 1, Lambda(_t, _t*log(-3*_t*a + 3*_t + x)))

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Giac [A]
time = 1.46, size = 138, normalized size = 1.00 \begin {gather*} \frac {{\left (a b^{2} - b^{2}\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a - 1}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a - 1}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} a b - \sqrt {3} b} + \frac {{\left (a b^{2} - b^{2}\right )}^{\frac {1}{3}} \log \left (x^{2} + x \left (\frac {a - 1}{b}\right )^{\frac {1}{3}} + \left (\frac {a - 1}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a b - b\right )}} - \frac {\left (\frac {a - 1}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (\frac {a - 1}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.34, size = 165, normalized size = 1.20 \begin {gather*} \frac {\ln \left (3\,b^2\,x+\frac {9\,a\,b^2-9\,b^2}{3\,{\left (-b\right )}^{1/3}\,{\left (a-1\right )}^{2/3}}\right )}{3\,{\left (-b\right )}^{1/3}\,{\left (a-1\right )}^{2/3}}+\frac {\ln \left (3\,b^2\,x+\frac {\left (9\,a\,b^2-9\,b^2\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-b\right )}^{1/3}\,{\left (a-1\right )}^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-b\right )}^{1/3}\,{\left (a-1\right )}^{2/3}}-\frac {\ln \left (3\,b^2\,x-\frac {\left (9\,a\,b^2-9\,b^2\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-b\right )}^{1/3}\,{\left (a-1\right )}^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-b\right )}^{1/3}\,{\left (a-1\right )}^{2/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(b*x^3 - a + 1),x)

[Out]

log(3*b^2*x + (9*a*b^2 - 9*b^2)/(3*(-b)^(1/3)*(a - 1)^(2/3)))/(3*(-b)^(1/3)*(a - 1)^(2/3)) + (log(3*b^2*x + ((
9*a*b^2 - 9*b^2)*(3^(1/2)*1i - 1))/(6*(-b)^(1/3)*(a - 1)^(2/3)))*(3^(1/2)*1i - 1))/(6*(-b)^(1/3)*(a - 1)^(2/3)
) - (log(3*b^2*x - ((9*a*b^2 - 9*b^2)*(3^(1/2)*1i + 1))/(6*(-b)^(1/3)*(a - 1)^(2/3)))*(3^(1/2)*1i + 1))/(6*(-b
)^(1/3)*(a - 1)^(2/3))

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