3.461 \(\int \frac{1}{x^4 \sqrt{1-x^3}} \, dx\)

Optimal. Leaf size=35 \[ -\frac{\sqrt{1-x^3}}{3 x^3}-\frac{1}{3} \tanh ^{-1}\left (\sqrt{1-x^3}\right ) \]

[Out]

-Sqrt[1 - x^3]/(3*x^3) - ArcTanh[Sqrt[1 - x^3]]/3

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Rubi [A]  time = 0.0143049, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 206} \[ -\frac{\sqrt{1-x^3}}{3 x^3}-\frac{1}{3} \tanh ^{-1}\left (\sqrt{1-x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[1 - x^3]),x]

[Out]

-Sqrt[1 - x^3]/(3*x^3) - ArcTanh[Sqrt[1 - x^3]]/3

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]]

Rule 51

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

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])

Rubi steps

\begin{align*} \int \frac{1}{x^4 \sqrt{1-x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x^2} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{1-x^3}}{3 x^3}+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{1-x^3}}{3 x^3}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x^3}\right )\\ &=-\frac{\sqrt{1-x^3}}{3 x^3}-\frac{1}{3} \tanh ^{-1}\left (\sqrt{1-x^3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0087565, size = 35, normalized size = 1. \[ -\frac{\sqrt{1-x^3}}{3 x^3}-\frac{1}{3} \tanh ^{-1}\left (\sqrt{1-x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[1 - x^3]),x]

[Out]

-Sqrt[1 - x^3]/(3*x^3) - ArcTanh[Sqrt[1 - x^3]]/3

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Maple [A]  time = 0.025, size = 28, normalized size = 0.8 \begin{align*} -{\frac{1}{3}{\it Artanh} \left ( \sqrt{-{x}^{3}+1} \right ) }-{\frac{1}{3\,{x}^{3}}\sqrt{-{x}^{3}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(-x^3+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.01428, size = 58, normalized size = 1.66 \begin{align*} -\frac{\sqrt{-x^{3} + 1}}{3 \, x^{3}} - \frac{1}{6} \, \log \left (\sqrt{-x^{3} + 1} + 1\right ) + \frac{1}{6} \, \log \left (\sqrt{-x^{3} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(-x^3+1)^(1/2),x, algorithm="maxima")

[Out]

-1/3*sqrt(-x^3 + 1)/x^3 - 1/6*log(sqrt(-x^3 + 1) + 1) + 1/6*log(sqrt(-x^3 + 1) - 1)

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Fricas [A]  time = 1.4613, size = 120, normalized size = 3.43 \begin{align*} -\frac{x^{3} \log \left (\sqrt{-x^{3} + 1} + 1\right ) - x^{3} \log \left (\sqrt{-x^{3} + 1} - 1\right ) + 2 \, \sqrt{-x^{3} + 1}}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(-x^3+1)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.43877, size = 82, normalized size = 2.34 \begin{align*} \begin{cases} - \frac{\operatorname{acosh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} - \frac{\sqrt{-1 + \frac{1}{x^{3}}}}{3 x^{\frac{3}{2}}} & \text{for}\: \frac{1}{\left |{x^{3}}\right |} > 1 \\\frac{i \operatorname{asin}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} - \frac{i}{3 x^{\frac{3}{2}} \sqrt{1 - \frac{1}{x^{3}}}} + \frac{i}{3 x^{\frac{9}{2}} \sqrt{1 - \frac{1}{x^{3}}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(-x**3+1)**(1/2),x)

[Out]

Piecewise((-acosh(x**(-3/2))/3 - sqrt(-1 + x**(-3))/(3*x**(3/2)), 1/Abs(x**3) > 1), (I*asin(x**(-3/2))/3 - I/(
3*x**(3/2)*sqrt(1 - 1/x**3)) + I/(3*x**(9/2)*sqrt(1 - 1/x**3)), True))

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Giac [A]  time = 1.11861, size = 59, normalized size = 1.69 \begin{align*} -\frac{\sqrt{-x^{3} + 1}}{3 \, x^{3}} - \frac{1}{6} \, \log \left (\sqrt{-x^{3} + 1} + 1\right ) + \frac{1}{6} \, \log \left ({\left | \sqrt{-x^{3} + 1} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(-x^3+1)^(1/2),x, algorithm="giac")

[Out]

-1/3*sqrt(-x^3 + 1)/x^3 - 1/6*log(sqrt(-x^3 + 1) + 1) + 1/6*log(abs(sqrt(-x^3 + 1) - 1))