\(\int \frac {\sqrt {-1+x^3}}{x} \, dx\) [305]

Optimal result

Integrand size = 13, antiderivative size = 28 \[ \int \frac {\sqrt {-1+x^3}}{x} \, dx=\frac {2}{3} \sqrt {-1+x^3}-\frac {2}{3} \arctan \left (\sqrt {-1+x^3}\right ) \]

[Out]

2/3*(x^3-1)^(1/2)-2/3*arctan((x^3-1)^(1/2))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 52, 65, 209} \[ \int \frac {\sqrt {-1+x^3}}{x} \, dx=\frac {2 \sqrt {x^3-1}}{3}-\frac {2}{3} \arctan \left (\sqrt {x^3-1}\right ) \]

[In]

Int[Sqrt[-1 + x^3]/x,x]

[Out]

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

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 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^3}}{x} \, dx=\frac {2}{3} \sqrt {-1+x^3}-\frac {2}{3} \arctan \left (\sqrt {-1+x^3}\right ) \]

[In]

Integrate[Sqrt[-1 + x^3]/x,x]

[Out]

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

Maple [A] (verified)

Time = 2.53 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75

method	result	size
default	\(\frac {2 \sqrt {x^{3}-1}}{3}-\frac {2 \arctan \left (\sqrt {x^{3}-1}\right )}{3}\)	\(21\)
elliptic	\(\frac {2 \sqrt {x^{3}-1}}{3}-\frac {2 \arctan \left (\sqrt {x^{3}-1}\right )}{3}\)	\(21\)
pseudoelliptic	\(\frac {2 \sqrt {x^{3}-1}}{3}-\frac {2 \arctan \left (\sqrt {x^{3}-1}\right )}{3}\)	\(21\)
trager	\(\frac {2 \sqrt {x^{3}-1}}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {x^{3}-1}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{3}}\right )}{3}\)	\(53\)
meijerg	\(-\frac {\sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, \left (4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {-x^{3}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )-2 \left (2-2 \ln \left (2\right )+3 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{6 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}}\)	\(82\)

[In]

int((x^3-1)^(1/2)/x,x,method=_RETURNVERBOSE)

[Out]

2/3*(x^3-1)^(1/2)-2/3*arctan((x^3-1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {-1+x^3}}{x} \, dx=\frac {2}{3} \, \sqrt {x^{3} - 1} - \frac {2}{3} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \]

[In]

integrate((x^3-1)^(1/2)/x,x, algorithm="fricas")

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.71 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.04 \[ \int \frac {\sqrt {-1+x^3}}{x} \, dx=\begin {cases} \frac {2 \sqrt {x^{3} - 1}}{3} - \frac {2 i \log {\left (x^{\frac {3}{2}} \right )}}{3} + \frac {i \log {\left (x^{3} \right )}}{3} + \frac {2 \operatorname {asin}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} & \text {for}\: \left |{x^{3}}\right | > 1 \\\frac {2 i \sqrt {1 - x^{3}}}{3} + \frac {i \log {\left (x^{3} \right )}}{3} - \frac {2 i \log {\left (\sqrt {1 - x^{3}} + 1 \right )}}{3} & \text {otherwise} \end {cases} \]

[In]

integrate((x**3-1)**(1/2)/x,x)

[Out]

Piecewise((2*sqrt(x**3 - 1)/3 - 2*I*log(x**(3/2))/3 + I*log(x**3)/3 + 2*asin(x**(-3/2))/3, Abs(x**3) > 1), (2*
I*sqrt(1 - x**3)/3 + I*log(x**3)/3 - 2*I*log(sqrt(1 - x**3) + 1)/3, True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {-1+x^3}}{x} \, dx=\frac {2}{3} \, \sqrt {x^{3} - 1} - \frac {2}{3} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \]

[In]

integrate((x^3-1)^(1/2)/x,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {-1+x^3}}{x} \, dx=\frac {2}{3} \, \sqrt {x^{3} - 1} - \frac {2}{3} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \]

[In]

integrate((x^3-1)^(1/2)/x,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.36 (sec) , antiderivative size = 174, normalized size of antiderivative = 6.21 \[ \int \frac {\sqrt {-1+x^3}}{x} \, dx=\frac {2\,\sqrt {x^3-1}}{3}+\frac {2\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]

[In]

int((x^3 - 1)^(1/2)/x,x)

[Out]

(2*(x^3 - 1)^(1/2))/3 + (2*((3^(1/2)*1i)/2 + 3/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*(
(x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi((3
^(1/2)*1i)/2 + 3/2, asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3
/2)))/(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) +
x^3)^(1/2)