3.3034 \(\int \frac{1}{1+4 \sqrt{x^4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0060316, 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, 203} \[ \frac{x \tan ^{-1}\left (2 \sqrt [4]{x^4}\right )}{2 \sqrt [4]{x^4}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + 4*Sqrt[x^4])^(-1),x]

[Out]

(x*ArcTan[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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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 \tan ^{-1}\left (2 \sqrt [4]{x^4}\right )}{2 \sqrt [4]{x^4}}\\ \end{align*}

Mathematica [A]  time = 0.009221, size = 22, normalized size = 1. \[ \frac{x \tan ^{-1}\left (2 \sqrt [4]{x^4}\right )}{2 \sqrt [4]{x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 4*Sqrt[x^4])^(-1),x]

[Out]

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

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Maple [A]  time = 0.037, size = 29, normalized size = 1.3 \begin{align*}{\frac{1}{2}\arctan \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)*arctan(2*((x^4)^(1/2)/x^2)^(1/2)*x)

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Maxima [A]  time = 1.41167, size = 8, normalized size = 0.36 \begin{align*} \frac{1}{2} \, \arctan \left (2 \, x\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]

1/2*arctan(2*x)

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Fricas [A]  time = 1.22391, size = 23, normalized size = 1.05 \begin{align*} \frac{1}{2} \, \arctan \left (2 \, x\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]

1/2*arctan(2*x)

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Sympy [A]  time = 0.097322, size = 5, normalized size = 0.23 \begin{align*} \frac{\operatorname{atan}{\left (2 x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(2*x)/2

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Giac [A]  time = 1.11448, size = 8, normalized size = 0.36 \begin{align*} \frac{1}{2} \, \arctan \left (2 \, x\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]

1/2*arctan(2*x)