Integrand size = 12, antiderivative size = 115 \[ \int \frac {x}{a-b x^3} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} b^{2/3}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}+\frac {\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3}} \] Output:
-1/3*arctan(1/3*(a^(1/3)+2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(1/3)/b^( 2/3)-1/3*ln(a^(1/3)-b^(1/3)*x)/a^(1/3)/b^(2/3)+1/6*ln(a^(2/3)+a^(1/3)*b^(1 /3)*x+b^(2/3)*x^2)/a^(1/3)/b^(2/3)
Time = 0.01 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.77 \[ \int \frac {x}{a-b x^3} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )+\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3}} \] Input:
Integrate[x/(a - b*x^3),x]
Output:
(-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^(1/3)*b^ (2/3))
Time = 0.40 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {821, 16, 1142, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x}{a-b x^3} \, dx\) |
\(\Big \downarrow \) 821 |
\(\displaystyle \frac {\int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b} x}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {\sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle -\frac {\int \frac {\sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle -\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int \frac {\sqrt [3]{b} \left (2 \sqrt [3]{b} x+\sqrt [3]{a}\right )}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \int \frac {2 \sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle -\frac {-\frac {1}{2} \int \frac {2 \sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {3 \int \frac {1}{-\left (\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )^2-3}d\left (\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {2 \sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle -\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b}}-\frac {\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\) |
Input:
Int[x/(a - b*x^3),x]
Output:
-1/3*Log[a^(1/3) - b^(1/3)*x]/(a^(1/3)*b^(2/3)) - ((Sqrt[3]*ArcTan[(1 + (2 *b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) - Log[a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(2*b^(1/3)))/(3*a^(1/3)*b^(1/3))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(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]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.45 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.25
method | result | size |
risch | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}}{3 b}\) | \(29\) |
default | \(-\frac {\ln \left (x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\) | \(92\) |
Input:
int(x/(-b*x^3+a),x,method=_RETURNVERBOSE)
Output:
-1/3/b*sum(1/_R*ln(x-_R),_R=RootOf(_Z^3*b-a))
Time = 0.08 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.67 \[ \int \frac {x}{a-b x^3} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} + a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x - 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{2} - \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} - a}\right ) + \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} - \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x + \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{6 \, a b^{2}}, \frac {6 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) + \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} - \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x + \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{6 \, a b^{2}}\right ] \] Input:
integrate(x/(-b*x^3+a),x, algorithm="fricas")
Output:
[1/6*(3*sqrt(1/3)*a*b*sqrt((-a*b^2)^(1/3)/a)*log((2*b^2*x^3 + a*b + 3*sqrt (1/3)*(a*b*x - 2*(-a*b^2)^(2/3)*x^2 - (-a*b^2)^(1/3)*a)*sqrt((-a*b^2)^(1/3 )/a) - 3*(-a*b^2)^(2/3)*x)/(b*x^3 - a)) + (-a*b^2)^(2/3)*log(b^2*x^2 - (-a *b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 2*(-a*b^2)^(2/3)*log(b*x + (-a*b^2)^(1 /3)))/(a*b^2), 1/6*(6*sqrt(1/3)*a*b*sqrt(-(-a*b^2)^(1/3)/a)*arctan(-sqrt(1 /3)*(2*b*x - (-a*b^2)^(1/3))*sqrt(-(-a*b^2)^(1/3)/a)/b) + (-a*b^2)^(2/3)*l og(b^2*x^2 - (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 2*(-a*b^2)^(2/3)*log(b *x + (-a*b^2)^(1/3)))/(a*b^2)]
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.23 \[ \int \frac {x}{a-b x^3} \, dx=- \operatorname {RootSum} {\left (27 t^{3} a b^{2} - 1, \left ( t \mapsto t \log {\left (- 9 t^{2} a b + x \right )} \right )\right )} \] Input:
integrate(x/(-b*x**3+a),x)
Output:
-RootSum(27*_t**3*a*b**2 - 1, Lambda(_t, _t*log(-9*_t**2*a*b + x)))
Time = 0.12 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.84 \[ \int \frac {x}{a-b x^3} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} + x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\log \left (x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {1}{3}}} \] Input:
integrate(x/(-b*x^3+a),x, algorithm="maxima")
Output:
-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + (a/b)^(1/3))/(a/b)^(1/3))/(b*(a/b)^ (1/3)) + 1/6*log(x^2 + x*(a/b)^(1/3) + (a/b)^(2/3))/(b*(a/b)^(1/3)) - 1/3* log(x - (a/b)^(1/3))/(b*(a/b)^(1/3))
Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.90 \[ \int \frac {x}{a-b x^3} \, dx=-\frac {\left (\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a} - \frac {\sqrt {3} \left (a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b^{2}} + \frac {\left (a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a b^{2}} \] Input:
integrate(x/(-b*x^3+a),x, algorithm="giac")
Output:
-1/3*(a/b)^(2/3)*log(abs(x - (a/b)^(1/3)))/a - 1/3*sqrt(3)*(a*b^2)^(2/3)*a rctan(1/3*sqrt(3)*(2*x + (a/b)^(1/3))/(a/b)^(1/3))/(a*b^2) + 1/6*(a*b^2)^( 2/3)*log(x^2 + x*(a/b)^(1/3) + (a/b)^(2/3))/(a*b^2)
Time = 0.30 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95 \[ \int \frac {x}{a-b x^3} \, dx=\frac {\ln \left ({\left (-a\right )}^{1/3}+b^{1/3}\,x\right )}{3\,{\left (-a\right )}^{1/3}\,b^{2/3}}+\frac {\ln \left (b\,x+\frac {{\left (-a\right )}^{1/3}\,b^{2/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-a\right )}^{1/3}\,b^{2/3}}-\frac {\ln \left (b\,x+\frac {{\left (-a\right )}^{1/3}\,b^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-a\right )}^{1/3}\,b^{2/3}} \] Input:
int(x/(a - b*x^3),x)
Output:
log((-a)^(1/3) + b^(1/3)*x)/(3*(-a)^(1/3)*b^(2/3)) + (log(b*x + ((-a)^(1/3 )*b^(2/3)*(3^(1/2)*1i - 1)^2)/4)*(3^(1/2)*1i - 1))/(6*(-a)^(1/3)*b^(2/3)) - (log(b*x + ((-a)^(1/3)*b^(2/3)*(3^(1/2)*1i + 1)^2)/4)*(3^(1/2)*1i + 1))/ (6*(-a)^(1/3)*b^(2/3))
Time = 0.22 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.57 \[ \int \frac {x}{a-b x^3} \, dx=\frac {-2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}+2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right )+\mathrm {log}\left (a^{\frac {2}{3}}+b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )-2 \,\mathrm {log}\left (a^{\frac {1}{3}}-b^{\frac {1}{3}} x \right )}{6 b^{\frac {2}{3}} a^{\frac {1}{3}}} \] Input:
int(x/(-b*x^3+a),x)
Output:
( - 2*sqrt(3)*atan((a**(1/3) + 2*b**(1/3)*x)/(a**(1/3)*sqrt(3))) + log(a** (2/3) + b**(1/3)*a**(1/3)*x + b**(2/3)*x**2) - 2*log(a**(1/3) - b**(1/3)*x ))/(6*b**(2/3)*a**(1/3))