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

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, 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 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])

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 \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*}

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Mathematica [C]  time = 0.01, 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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fricas [B]  time = 0.99, size = 58, normalized size = 1.81 \[ -\frac {{\left (x^{4} - 1\right )} \log \left (\sqrt {-x^{4} + 1} + 1\right ) - {\left (x^{4} - 1\right )} \log \left (\sqrt {-x^{4} + 1} - 1\right ) + 2 \, \sqrt {-x^{4} + 1}}{4 \, {\left (x^{4} - 1\right )}} \]

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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giac [A]  time = 0.15, size = 42, normalized size = 1.31 \[ \frac {1}{2 \, \sqrt {-x^{4} + 1}} - \frac {1}{4} \, \log \left (\sqrt {-x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (-\sqrt {-x^{4} + 1} + 1\right ) \]

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)

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maple [B]  time = 0.02, size = 68, normalized size = 2.12 \[ -\frac {\arctanh \left (\frac {1}{\sqrt {-x^{4}+1}}\right )}{2}-\frac {\sqrt {-2 x^{2}-\left (x^{2}-1\right )^{2}+2}}{4 \left (x^{2}-1\right )}+\frac {\sqrt {2 x^{2}-\left (x^{2}+1\right )^{2}+2}}{4 x^{2}+4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.31, size = 40, normalized size = 1.25 \[ \frac {1}{2 \, \sqrt {-x^{4} + 1}} - \frac {1}{4} \, \log \left (\sqrt {-x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {-x^{4} + 1} - 1\right ) \]

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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mupad [B]  time = 1.22, size = 24, normalized size = 0.75 \[ \frac {1}{2\,\sqrt {1-x^4}}-\frac {\mathrm {atanh}\left (\sqrt {1-x^4}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(1 - x^4)^(3/2)),x)

[Out]

1/(2*(1 - x^4)^(1/2)) - atanh((1 - x^4)^(1/2))/2

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sympy [C]  time = 2.70, size = 228, normalized size = 7.12 \[ \begin {cases} \frac {2 x^{4} \log {\left (x^{2} \right )}}{4 - 4 x^{4}} - \frac {x^{4} \log {\left (x^{4} \right )}}{4 - 4 x^{4}} - \frac {2 i x^{4} \operatorname {asin}{\left (\frac {1}{x^{2}} \right )}}{4 - 4 x^{4}} + \frac {2 i \sqrt {x^{4} - 1}}{4 - 4 x^{4}} - \frac {2 \log {\left (x^{2} \right )}}{4 - 4 x^{4}} + \frac {\log {\left (x^{4} \right )}}{4 - 4 x^{4}} + \frac {2 i \operatorname {asin}{\left (\frac {1}{x^{2}} \right )}}{4 - 4 x^{4}} & \text {for}\: \left |{x^{4}}\right | > 1 \\- \frac {x^{4} \log {\left (x^{4} \right )}}{4 - 4 x^{4}} + \frac {2 x^{4} \log {\left (\sqrt {1 - x^{4}} + 1 \right )}}{4 - 4 x^{4}} - \frac {i \pi x^{4}}{4 - 4 x^{4}} + \frac {2 \sqrt {1 - x^{4}}}{4 - 4 x^{4}} + \frac {\log {\left (x^{4} \right )}}{4 - 4 x^{4}} - \frac {2 \log {\left (\sqrt {1 - x^{4}} + 1 \right )}}{4 - 4 x^{4}} + \frac {i \pi }{4 - 4 x^{4}} & \text {otherwise} \end {cases} \]

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

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