\(\int \frac {\sqrt [3]{-a+x}}{x} \, dx\) [224]

Optimal result

Integrand size = 13, antiderivative size = 88 \[ \int \frac {\sqrt [3]{-a+x}}{x} \, dx=3 \sqrt [3]{-a+x}+\sqrt {3} \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{-a+x}}{\sqrt {3} \sqrt [3]{a}}\right )+\frac {1}{2} \sqrt [3]{a} \log (x)-\frac {3}{2} \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{-a+x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {52, 60, 631, 210, 31} \[ \int \frac {\sqrt [3]{-a+x}}{x} \, dx=\sqrt {3} \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{a}}\right )+3 \sqrt [3]{x-a}+\frac {1}{2} \sqrt [3]{a} \log (x)-\frac {3}{2} \sqrt [3]{a} \log \left (\sqrt [3]{x-a}+\sqrt [3]{a}\right ) \]

[In]

Int[(-a + x)^(1/3)/x,x]

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 60

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-(b*c - a*d)/b, 3]}, Simp[-
Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (Dist[3/(2*b*q), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x
)^(1/3)], x] + Dist[3/(2*b*q^2), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x]
&& NegQ[(b*c - a*d)/b]

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]

Rubi steps \begin{align*} \text {integral}& = 3 \sqrt [3]{-a+x}-a \int \frac {1}{x (-a+x)^{2/3}} \, dx \\ & = 3 \sqrt [3]{-a+x}+\frac {1}{2} \sqrt [3]{a} \log (x)-\frac {1}{2} \left (3 \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+x} \, dx,x,\sqrt [3]{-a+x}\right )-\frac {1}{2} \left (3 a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{-a+x}\right ) \\ & = 3 \sqrt [3]{-a+x}+\frac {1}{2} \sqrt [3]{a} \log (x)-\frac {3}{2} \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{-a+x}\right )-\left (3 \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{-a+x}}{\sqrt [3]{a}}\right ) \\ & = 3 \sqrt [3]{-a+x}+\sqrt {3} \sqrt [3]{a} \arctan \left (\frac {1-\frac {2 \sqrt [3]{-a+x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+\frac {1}{2} \sqrt [3]{a} \log (x)-\frac {3}{2} \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{-a+x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt [3]{-a+x}}{x} \, dx=3 \sqrt [3]{-a+x}+\sqrt {3} \sqrt [3]{a} \arctan \left (\frac {1-\frac {2 \sqrt [3]{-a+x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{-a+x}\right )+\frac {1}{2} \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right ) \]

[In]

Integrate[(-a + x)^(1/3)/x,x]

[Out]

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.01

method	result	size
derivativedivides	\(3 \left (-a +x \right )^{\frac {1}{3}}-3 \left (\frac {\ln \left (a^{\frac {1}{3}}+\left (-a +x \right )^{\frac {1}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {\ln \left (\left (-a +x \right )^{\frac {2}{3}}-a^{\frac {1}{3}} \left (-a +x \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (-a +x \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a^{\frac {2}{3}}}\right ) a\)	\(89\)
default	\(3 \left (-a +x \right )^{\frac {1}{3}}-3 \left (\frac {\ln \left (a^{\frac {1}{3}}+\left (-a +x \right )^{\frac {1}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {\ln \left (\left (-a +x \right )^{\frac {2}{3}}-a^{\frac {1}{3}} \left (-a +x \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (-a +x \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a^{\frac {2}{3}}}\right ) a\)	\(89\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt [3]{-a+x}}{x} \, dx=\sqrt {3} \left (-a\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} a - 2 \, \sqrt {3} \left (-a\right )^{\frac {2}{3}} {\left (-a + x\right )}^{\frac {1}{3}}}{3 \, a}\right ) - \frac {1}{2} \, \left (-a\right )^{\frac {1}{3}} \log \left (\left (-a\right )^{\frac {2}{3}} + \left (-a\right )^{\frac {1}{3}} {\left (-a + x\right )}^{\frac {1}{3}} + {\left (-a + x\right )}^{\frac {2}{3}}\right ) + \left (-a\right )^{\frac {1}{3}} \log \left (-\left (-a\right )^{\frac {1}{3}} + {\left (-a + x\right )}^{\frac {1}{3}}\right ) + 3 \, {\left (-a + x\right )}^{\frac {1}{3}} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.10 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.74 \[ \int \frac {\sqrt [3]{-a+x}}{x} \, dx=\frac {4 \sqrt [3]{a} e^{- \frac {i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{- a + x} e^{\frac {i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {4}{3}\right )}{3 \Gamma \left (\frac {7}{3}\right )} - \frac {4 \sqrt [3]{a} \log {\left (1 - \frac {\sqrt [3]{- a + x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {4}{3}\right )}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {4 \sqrt [3]{a} e^{\frac {i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{- a + x} e^{\frac {5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {4}{3}\right )}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {4 \sqrt [3]{- a + x} \Gamma \left (\frac {4}{3}\right )}{\Gamma \left (\frac {7}{3}\right )} \]

[In]

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

[Out]

4*a**(1/3)*exp(-I*pi/3)*log(1 - (-a + x)**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(4/3)/(3*gamma(7/3)) - 4*a**(
1/3)*log(1 - (-a + x)**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(4/3)/(3*gamma(7/3)) + 4*a**(1/3)*exp(I*pi/3)*log(
1 - (-a + x)**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(4/3)/(3*gamma(7/3)) + 4*(-a + x)**(1/3)*gamma(4/3)/gam
ma(7/3)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt [3]{-a+x}}{x} \, dx=-\sqrt {3} a^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} {\left (a^{\frac {1}{3}} - 2 \, {\left (-a + x\right )}^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) + \frac {1}{2} \, a^{\frac {1}{3}} \log \left (a^{\frac {2}{3}} - a^{\frac {1}{3}} {\left (-a + x\right )}^{\frac {1}{3}} + {\left (-a + x\right )}^{\frac {2}{3}}\right ) - a^{\frac {1}{3}} \log \left (a^{\frac {1}{3}} + {\left (-a + x\right )}^{\frac {1}{3}}\right ) + 3 \, {\left (-a + x\right )}^{\frac {1}{3}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt [3]{-a+x}}{x} \, dx=-\sqrt {3} \left (-a\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (-a\right )^{\frac {1}{3}} + 2 \, {\left (-a + x\right )}^{\frac {1}{3}}\right )}}{3 \, \left (-a\right )^{\frac {1}{3}}}\right ) - \frac {1}{2} \, \left (-a\right )^{\frac {1}{3}} \log \left (\left (-a\right )^{\frac {2}{3}} + \left (-a\right )^{\frac {1}{3}} {\left (-a + x\right )}^{\frac {1}{3}} + {\left (-a + x\right )}^{\frac {2}{3}}\right ) + \left (-a\right )^{\frac {1}{3}} \log \left ({\left | -\left (-a\right )^{\frac {1}{3}} + {\left (-a + x\right )}^{\frac {1}{3}} \right |}\right ) + 3 \, {\left (-a + x\right )}^{\frac {1}{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.35 \[ \int \frac {\sqrt [3]{-a+x}}{x} \, dx={\left (-a\right )}^{1/3}\,\ln \left (-9\,{\left (-a\right )}^{4/3}-9\,a\,{\left (x-a\right )}^{1/3}\right )+3\,{\left (x-a\right )}^{1/3}+\frac {{\left (-a\right )}^{1/3}\,\ln \left (\frac {9\,{\left (-a\right )}^{4/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}+9\,a\,{\left (x-a\right )}^{1/3}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {{\left (-a\right )}^{1/3}\,\ln \left (\frac {9\,{\left (-a\right )}^{4/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-9\,a\,{\left (x-a\right )}^{1/3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2} \]

[In]

int((x - a)^(1/3)/x,x)

[Out]

(-a)^(1/3)*log(- 9*(-a)^(4/3) - 9*a*(x - a)^(1/3)) + 3*(x - a)^(1/3) + ((-a)^(1/3)*log((9*(-a)^(4/3)*(3^(1/2)*
1i - 1))/2 + 9*a*(x - a)^(1/3))*(3^(1/2)*1i - 1))/2 - ((-a)^(1/3)*log((9*(-a)^(4/3)*(3^(1/2)*1i + 1))/2 - 9*a*
(x - a)^(1/3))*(3^(1/2)*1i + 1))/2