3.5 \(\int \frac {1}{x \sqrt {1-x^3}} \, dx\)

Optimal. Leaf size=16 \[ -\frac {2}{3} \tanh ^{-1}\left (\sqrt {1-x^3}\right ) \]

[Out]

-2/3*arctanh((-x^3+1)^(1/2))

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {266, 63, 206} \[ -\frac {2}{3} \tanh ^{-1}\left (\sqrt {1-x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[1 - x^3]),x]

[Out]

(-2*ArcTanh[Sqrt[1 - x^3]])/3

Rule 63

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 206

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

Rule 266

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*} \int \frac {1}{x \sqrt {1-x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^3\right )\\ &=-\left (\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^3}\right )\right )\\ &=-\frac {2}{3} \tanh ^{-1}\left (\sqrt {1-x^3}\right )\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ -\frac {2}{3} \tanh ^{-1}\left (\sqrt {1-x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[1 - x^3]),x]

[Out]

(-2*ArcTanh[Sqrt[1 - x^3]])/3

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fricas [B]  time = 0.41, size = 29, normalized size = 1.81 \[ -\frac {1}{3} \, \log \left (\sqrt {-x^{3} + 1} + 1\right ) + \frac {1}{3} \, \log \left (\sqrt {-x^{3} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.92, size = 30, normalized size = 1.88 \[ -\frac {1}{3} \, \log \left (\sqrt {-x^{3} + 1} + 1\right ) + \frac {1}{3} \, \log \left ({\left | \sqrt {-x^{3} + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 13, normalized size = 0.81 \[ -\frac {2 \arctanh \left (\sqrt {-x^{3}+1}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(-x^3+1)^(1/2),x)

[Out]

-2/3*arctanh((-x^3+1)^(1/2))

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maxima [B]  time = 0.43, size = 29, normalized size = 1.81 \[ -\frac {1}{3} \, \log \left (\sqrt {-x^{3} + 1} + 1\right ) + \frac {1}{3} \, \log \left (\sqrt {-x^{3} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.15, size = 180, normalized size = 11.25 \[ -\frac {2\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3-1}\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {1-x^3}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(1 - x^3)^(1/2)),x)

[Out]

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

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sympy [A]  time = 1.02, size = 31, normalized size = 1.94 \[ \begin {cases} - \frac {2 \operatorname {acosh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\\frac {2 i \operatorname {asin}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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