3.897 \(\int \frac{1}{x (1-x^4)^{3/2}} \, dx\)

Optimal. Leaf size=32 \[ \frac{1}{2 \sqrt{1-x^4}}-\frac{1}{2} \tanh ^{-1}\left (\sqrt{1-x^4}\right ) \]

[Out]

1/(2*Sqrt[1 - x^4]) - ArcTanh[Sqrt[1 - x^4]]/2

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Rubi [A]  time = 0.0143036, antiderivative size = 32, 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} \[ \frac{1}{2 \sqrt{1-x^4}}-\frac{1}{2} \tanh ^{-1}\left (\sqrt{1-x^4}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(1 - x^4)^(3/2)),x]

[Out]

1/(2*Sqrt[1 - x^4]) - ArcTanh[Sqrt[1 - x^4]]/2

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

Mathematica [C]  time = 0.0048648, size = 30, normalized size = 0.94 \[ \frac{\, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};1-x^4\right )}{2 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 - x^4)^(3/2)),x]

[Out]

Hypergeometric2F1[-1/2, 1, 1/2, 1 - x^4]/(2*Sqrt[1 - x^4])

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Maple [B]  time = 0.016, size = 68, normalized size = 2.1 \begin{align*} -{\frac{1}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{4}+1}}} \right ) }+{\frac{1}{4\,{x}^{2}+4}\sqrt{- \left ({x}^{2}+1 \right ) ^{2}+2+2\,{x}^{2}}}-{\frac{1}{4\,{x}^{2}-4}\sqrt{- \left ({x}^{2}-1 \right ) ^{2}+2-2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arctanh(1/(-x^4+1)^(1/2))+1/4/(x^2+1)*(-(x^2+1)^2+2+2*x^2)^(1/2)-1/4/(x^2-1)*(-(x^2-1)^2+2-2*x^2)^(1/2)

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Maxima [A]  time = 1.03724, size = 54, normalized size = 1.69 \begin{align*} \frac{1}{2 \, \sqrt{-x^{4} + 1}} - \frac{1}{4} \, \log \left (\sqrt{-x^{4} + 1} + 1\right ) + \frac{1}{4} \, \log \left (\sqrt{-x^{4} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/sqrt(-x^4 + 1) - 1/4*log(sqrt(-x^4 + 1) + 1) + 1/4*log(sqrt(-x^4 + 1) - 1)

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Fricas [B]  time = 1.57523, size = 144, normalized size = 4.5 \begin{align*} -\frac{{\left (x^{4} - 1\right )} \log \left (\sqrt{-x^{4} + 1} + 1\right ) -{\left (x^{4} - 1\right )} \log \left (\sqrt{-x^{4} + 1} - 1\right ) + 2 \, \sqrt{-x^{4} + 1}}{4 \,{\left (x^{4} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((x^4 - 1)*log(sqrt(-x^4 + 1) + 1) - (x^4 - 1)*log(sqrt(-x^4 + 1) - 1) + 2*sqrt(-x^4 + 1))/(x^4 - 1)

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Sympy [C]  time = 1.41487, size = 228, normalized size = 7.12 \begin{align*} \begin{cases} - \frac{2 x^{4} \log{\left (x^{2} \right )}}{4 x^{4} - 4} + \frac{x^{4} \log{\left (x^{4} \right )}}{4 x^{4} - 4} + \frac{2 i x^{4} \operatorname{asin}{\left (\frac{1}{x^{2}} \right )}}{4 x^{4} - 4} - \frac{2 i \sqrt{x^{4} - 1}}{4 x^{4} - 4} + \frac{2 \log{\left (x^{2} \right )}}{4 x^{4} - 4} - \frac{\log{\left (x^{4} \right )}}{4 x^{4} - 4} - \frac{2 i \operatorname{asin}{\left (\frac{1}{x^{2}} \right )}}{4 x^{4} - 4} & \text{for}\: \left |{x^{4}}\right | > 1 \\\frac{x^{4} \log{\left (x^{4} \right )}}{4 x^{4} - 4} - \frac{2 x^{4} \log{\left (\sqrt{1 - x^{4}} + 1 \right )}}{4 x^{4} - 4} + \frac{i \pi x^{4}}{4 x^{4} - 4} - \frac{2 \sqrt{1 - x^{4}}}{4 x^{4} - 4} - \frac{\log{\left (x^{4} \right )}}{4 x^{4} - 4} + \frac{2 \log{\left (\sqrt{1 - x^{4}} + 1 \right )}}{4 x^{4} - 4} - \frac{i \pi }{4 x^{4} - 4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*x**4*log(x**2)/(4*x**4 - 4) + x**4*log(x**4)/(4*x**4 - 4) + 2*I*x**4*asin(x**(-2))/(4*x**4 - 4)
- 2*I*sqrt(x**4 - 1)/(4*x**4 - 4) + 2*log(x**2)/(4*x**4 - 4) - log(x**4)/(4*x**4 - 4) - 2*I*asin(x**(-2))/(4*x
**4 - 4), Abs(x**4) > 1), (x**4*log(x**4)/(4*x**4 - 4) - 2*x**4*log(sqrt(1 - x**4) + 1)/(4*x**4 - 4) + I*pi*x*
*4/(4*x**4 - 4) - 2*sqrt(1 - x**4)/(4*x**4 - 4) - log(x**4)/(4*x**4 - 4) + 2*log(sqrt(1 - x**4) + 1)/(4*x**4 -
4) - I*pi/(4*x**4 - 4), True))

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Giac [A]  time = 1.46939, size = 57, normalized size = 1.78 \begin{align*} \frac{1}{2 \, \sqrt{-x^{4} + 1}} - \frac{1}{4} \, \log \left (\sqrt{-x^{4} + 1} + 1\right ) + \frac{1}{4} \, \log \left (-\sqrt{-x^{4} + 1} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/sqrt(-x^4 + 1) - 1/4*log(sqrt(-x^4 + 1) + 1) + 1/4*log(-sqrt(-x^4 + 1) + 1)