3.9.97 \(\int \frac {1}{x (1-x^4)^{3/2}} \, dx\) [897]

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 = {272, 53, 65, 212} \begin {gather*} \frac {1}{2 \sqrt {1-x^4}}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {1-x^4}\right ) \end {gather*}

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 53

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 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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[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*} \int \frac {1}{x \left (1-x^4\right )^{3/2}} \, dx &=\frac {1}{4} \text {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} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^4\right )\\ &=\frac {1}{2 \sqrt {1-x^4}}-\frac {1}{2} \text {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 [A]
time = 0.03, size = 30, normalized size = 0.94 \begin {gather*} \frac {1}{2} \left (\frac {1}{\sqrt {1-x^4}}-\tanh ^{-1}\left (\sqrt {1-x^4}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 - x^4)^(3/2)),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(67\) vs. \(2(24)=48\).
time = 0.23, size = 68, normalized size = 2.12

method	result	size
risch	\(\frac {1}{2 \sqrt {-x^{4}+1}}-\frac {\arctanh \left (\frac {1}{\sqrt {-x^{4}+1}}\right )}{2}\)	\(25\)
trager	\(-\frac {\sqrt {-x^{4}+1}}{2 \left (x^{4}-1\right )}-\frac {\ln \left (\frac {1+\sqrt {-x^{4}+1}}{x^{2}}\right )}{2}\)	\(38\)
meijerg	\(\frac {-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {-x^{4}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{4}+1}}{2}\right )+\frac {\left (2-2 \ln \left (2\right )+4 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{2}}{2 \sqrt {\pi }}\)	\(63\)
default	\(\frac {\sqrt {-\left (x^{2}+1\right )^{2}+2 x^{2}+2}}{4 x^{2}+4}-\frac {\arctanh \left (\frac {1}{\sqrt {-x^{4}+1}}\right )}{2}-\frac {\sqrt {-\left (x^{2}-1\right )^{2}-2 x^{2}+2}}{4 \left (x^{2}-1\right )}\)	\(68\)
elliptic	\(\frac {\sqrt {-\left (x^{2}+1\right )^{2}+2 x^{2}+2}}{4 x^{2}+4}-\frac {\arctanh \left (\frac {1}{\sqrt {-x^{4}+1}}\right )}{2}-\frac {\sqrt {-\left (x^{2}-1\right )^{2}-2 x^{2}+2}}{4 \left (x^{2}-1\right )}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 40, normalized size = 1.25 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (24) = 48\).
time = 0.36, size = 58, normalized size = 1.81 \begin {gather*} -\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 {gather*}

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] Result contains complex when optimal does not.
time = 0.67, size = 228, normalized size = 7.12 \begin {gather*} \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 {gather*}

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 = 1.27, size = 42, normalized size = 1.31 \begin {gather*} \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 {gather*}

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

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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