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

Optimal. Leaf size=124 \[ \frac {\log \left (\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}+b^{2/3} x^2\right )}{6 (a+1)^{2/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+1}-\sqrt [3]{b} x\right )}{3 (a+1)^{2/3} \sqrt [3]{b}}+\frac {\tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+1}}+1}{\sqrt {3}}\right )}{\sqrt {3} (a+1)^{2/3} \sqrt [3]{b}} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 124, 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} \[ \frac {\log \left (\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}+b^{2/3} x^2\right )}{6 (a+1)^{2/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+1}-\sqrt [3]{b} x\right )}{3 (a+1)^{2/3} \sqrt [3]{b}}+\frac {\tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+1}}+1}{\sqrt {3}}\right )}{\sqrt {3} (a+1)^{2/3} \sqrt [3]{b}} \]

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 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}{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 {\operatorname {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*}

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 124, normalized size = 1.00 \[ \frac {(-1)^{2/3} \left (\log \left (-\sqrt [3]{-1} \sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}+(-1)^{2/3} b^{2/3} x^2\right )-2 \log \left (\sqrt [3]{a+1}+\sqrt [3]{-1} \sqrt [3]{b} x\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{-1} \sqrt [3]{b} x}{\sqrt [3]{a+1}}-1}{\sqrt {3}}\right )\right )}{6 (a+1)^{2/3} \sqrt [3]{b}} \]

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))

________________________________________________________________________________________

fricas [B]  time = 0.70, size = 467, normalized size = 3.77 \[ \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 ] \]

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)]

________________________________________________________________________________________

giac [A]  time = 0.16, size = 131, normalized size = 1.06 \[ \frac {{\left (a b^{2} + b^{2}\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a + 1}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a + 1}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} a b + \sqrt {3} b} + \frac {{\left (a b^{2} + b^{2}\right )}^{\frac {1}{3}} \log \left (x^{2} + x \left (\frac {a + 1}{b}\right )^{\frac {1}{3}} + \left (\frac {a + 1}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a b + b\right )}} - \frac {\left (\frac {a + 1}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (\frac {a + 1}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a + 1\right )}} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 106, normalized size = 0.85 \[ \frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a +1}{b}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 \left (\frac {a +1}{b}\right )^{\frac {2}{3}} b}-\frac {\ln \left (x -\left (\frac {a +1}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a +1}{b}\right )^{\frac {2}{3}} b}+\frac {\ln \left (x^{2}+\left (\frac {a +1}{b}\right )^{\frac {1}{3}} x +\left (\frac {a +1}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a +1}{b}\right )^{\frac {2}{3}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 3.00, size = 113, normalized size = 0.91 \[ \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a + 1}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a + 1}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b \left (\frac {a + 1}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x^{2} + x \left (\frac {a + 1}{b}\right )^{\frac {1}{3}} + \left (\frac {a + 1}{b}\right )^{\frac {2}{3}}\right )}{6 \, b \left (\frac {a + 1}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x - \left (\frac {a + 1}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a + 1}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.32, size = 165, normalized size = 1.33 \[ \frac {\ln \left (3\,b^2\,x+\frac {9\,a\,b^2+9\,b^2}{3\,{\left (-b\right )}^{1/3}\,{\left (a+1\right )}^{2/3}}\right )}{3\,{\left (-b\right )}^{1/3}\,{\left (a+1\right )}^{2/3}}+\frac {\ln \left (3\,b^2\,x+\frac {\left (9\,a\,b^2+9\,b^2\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-b\right )}^{1/3}\,{\left (a+1\right )}^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-b\right )}^{1/3}\,{\left (a+1\right )}^{2/3}}-\frac {\ln \left (3\,b^2\,x-\frac {\left (9\,a\,b^2+9\,b^2\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-b\right )}^{1/3}\,{\left (a+1\right )}^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-b\right )}^{1/3}\,{\left (a+1\right )}^{2/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a - b*x^3 + 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))

________________________________________________________________________________________

sympy [A]  time = 0.42, size = 34, normalized size = 0.27 \[ - \operatorname {RootSum} {\left (t^{3} \left (27 a^{2} b + 54 a b + 27 b\right ) - 1, \left (t \mapsto t \log {\left (- 3 t a - 3 t + x \right )} \right )\right )} \]

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)))

________________________________________________________________________________________