3.3035 \(\int \frac{1}{1-4 \sqrt{x^4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.004652, antiderivative size = 22, normalized size of antiderivative = 1., 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} \[ \frac{x \tanh ^{-1}\left (2 \sqrt [4]{x^4}\right )}{2 \sqrt [4]{x^4}} \]

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 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)]

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}{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*}

Mathematica [A]  time = 0.0036248, size = 22, normalized size = 1. \[ \frac{x \tanh ^{-1}\left (2 \sqrt [4]{x^4}\right )}{2 \sqrt [4]{x^4}} \]

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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Maple [A]  time = 0.019, size = 29, normalized size = 1.3 \begin{align*}{\frac{1}{2}{\it Artanh} \left ( 2\,\sqrt{{\frac{\sqrt{{x}^{4}}}{{x}^{2}}}}x \right ){\frac{1}{\sqrt{{\frac{1}{{x}^{2}}\sqrt{{x}^{4}}}}}}} \end{align*}

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 = 0.941213, size = 23, normalized size = 1.05 \begin{align*} \frac{1}{4} \, \log \left (2 \, x + 1\right ) - \frac{1}{4} \, \log \left (2 \, x - 1\right ) \end{align*}

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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Fricas [A]  time = 1.26415, size = 50, normalized size = 2.27 \begin{align*} \frac{1}{4} \, \log \left (2 \, x + 1\right ) - \frac{1}{4} \, \log \left (2 \, x - 1\right ) \end{align*}

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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Sympy [A]  time = 0.099246, size = 15, normalized size = 0.68 \begin{align*} - \frac{\log{\left (x - \frac{1}{2} \right )}}{4} + \frac{\log{\left (x + \frac{1}{2} \right )}}{4} \end{align*}

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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Giac [A]  time = 1.10384, size = 20, normalized size = 0.91 \begin{align*} \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{align*}

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))