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

Optimal. Leaf size=22 \[ \frac {x \tanh ^{-1}\left (2 \sqrt [4]{x^4}\right )}{2 \sqrt [4]{x^4}} \]

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Rubi [A]  time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {254, 206} \begin {gather*} \frac {x \tanh ^{-1}\left (2 \sqrt [4]{x^4}\right )}{2 \sqrt [4]{x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 4*Sqrt[x^4])^(-1),x]

[Out]

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

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 254

Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_.), x_Symbol] :> Dist[x/(c*x^q)^(1/q), Subst[Int[(a + b*x^(n*q))
^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, n, p, q}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} \frac {x \tanh ^{-1}\left (2 \sqrt [4]{x^4}\right )}{2 \sqrt [4]{x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 4*Sqrt[x^4])^(-1),x]

[Out]

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

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IntegrateAlgebraic [B]  time = 0.17, size = 67, normalized size = 3.05 \begin {gather*} \frac {1}{4} \tan ^{-1}\left (\frac {\sqrt {x^4}}{2 x^3}\right )+\frac {1}{4} \tanh ^{-1}\left (\frac {\sqrt {x^4}}{2 x^3}\right )-\frac {1}{8} \log (1-2 x)+\frac {1}{8} \log (2 x+1)+\frac {1}{4} \tan ^{-1}(2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 - 4*Sqrt[x^4])^(-1),x]

[Out]

ArcTan[2*x]/4 + ArcTan[Sqrt[x^4]/(2*x^3)]/4 + ArcTanh[Sqrt[x^4]/(2*x^3)]/4 - Log[1 - 2*x]/8 + Log[1 + 2*x]/8

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fricas [A]  time = 1.35, size = 17, normalized size = 0.77 \begin {gather*} \frac {1}{4} \, \log \left (2 \, x + 1\right ) - \frac {1}{4} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(2*x + 1) - 1/4*log(2*x - 1)

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giac [A]  time = 0.15, size = 15, normalized size = 0.68 \begin {gather*} \frac {1}{4} \, \log \left ({\left | x + \frac {1}{2} \right |}\right ) - \frac {1}{4} \, \log \left ({\left | x - \frac {1}{2} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(abs(x + 1/2)) - 1/4*log(abs(x - 1/2))

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maple [A]  time = 0.04, size = 29, normalized size = 1.32 \begin {gather*} \frac {\arctanh \left (2 \sqrt {\frac {\sqrt {x^{4}}}{x^{2}}}\, x \right )}{2 \sqrt {\frac {\sqrt {x^{4}}}{x^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-4*(x^4)^(1/2)),x)

[Out]

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

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maxima [A]  time = 0.54, size = 17, normalized size = 0.77 \begin {gather*} \frac {1}{4} \, \log \left (2 \, x + 1\right ) - \frac {1}{4} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(2*x + 1) - 1/4*log(2*x - 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int -\frac {1}{4\,\sqrt {x^4}-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(4*(x^4)^(1/2) - 1),x)

[Out]

int(-1/(4*(x^4)^(1/2) - 1), x)

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sympy [A]  time = 0.15, size = 15, normalized size = 0.68 \begin {gather*} - \frac {\log {\left (x - \frac {1}{2} \right )}}{4} + \frac {\log {\left (x + \frac {1}{2} \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x - 1/2)/4 + log(x + 1/2)/4

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