3.195 \(\int \frac {1}{x \sqrt {1+x^4}} \, dx\)

Optimal. Leaf size=14 \[ -\frac {1}{2} \tanh ^{-1}\left (\sqrt {x^4+1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {266, 63, 207} \[ -\frac {1}{2} \tanh ^{-1}\left (\sqrt {x^4+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[1 + x^4]),x]

[Out]

-ArcTanh[Sqrt[1 + x^4]]/2

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 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[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^4}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^4\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^4}\right )\\ &=-\frac {1}{2} \tanh ^{-1}\left (\sqrt {1+x^4}\right )\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \[ -\frac {1}{2} \tanh ^{-1}\left (\sqrt {x^4+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[1 + x^4]),x]

[Out]

-1/2*ArcTanh[Sqrt[1 + x^4]]

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fricas [B]  time = 0.40, size = 25, normalized size = 1.79 \[ -\frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.19, size = 25, normalized size = 1.79 \[ -\frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 11, normalized size = 0.79 \[ -\frac {\arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.44, size = 25, normalized size = 1.79 \[ -\frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.19, size = 10, normalized size = 0.71 \[ -\frac {\mathrm {atanh}\left (\sqrt {x^4+1}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(x^4 + 1)^(1/2)),x)

[Out]

-atanh((x^4 + 1)^(1/2))/2

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sympy [A]  time = 0.92, size = 8, normalized size = 0.57 \[ - \frac {\operatorname {asinh}{\left (\frac {1}{x^{2}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-asinh(x**(-2))/2

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