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

Optimal. Leaf size=58 \[ \frac{1}{3} x \sqrt{x^4+1}-\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}(x),\frac{1}{2}\right )}{6 \sqrt{x^4+1}} \]

[Out]

(x*Sqrt[1 + x^4])/3 - ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(6*Sqrt[1 + x^4])

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Rubi [A]  time = 0.0090074, antiderivative size = 58, 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 = {321, 220} \[ \frac{1}{3} x \sqrt{x^4+1}-\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{6 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/Sqrt[1 + x^4],x]

[Out]

(x*Sqrt[1 + x^4])/3 - ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(6*Sqrt[1 + x^4])

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rubi steps

\begin{align*} \int \frac{x^4}{\sqrt{1+x^4}} \, dx &=\frac{1}{3} x \sqrt{1+x^4}-\frac{1}{3} \int \frac{1}{\sqrt{1+x^4}} \, dx\\ &=\frac{1}{3} x \sqrt{1+x^4}-\frac{\left (1+x^2\right ) \sqrt{\frac{1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{6 \sqrt{1+x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0059533, size = 32, normalized size = 0.55 \[ \frac{1}{3} x \left (\sqrt{x^4+1}-\, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-x^4\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/Sqrt[1 + x^4],x]

[Out]

(x*(Sqrt[1 + x^4] - Hypergeometric2F1[1/4, 1/2, 5/4, -x^4]))/3

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Maple [C]  time = 0.008, size = 72, normalized size = 1.2 \begin{align*}{\frac{x}{3}\sqrt{{x}^{4}+1}}-{\frac{{\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) }{{\frac{3\,\sqrt{2}}{2}}+{\frac{3\,i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x*(x^4+1)^(1/2)-1/3/(1/2*2^(1/2)+1/2*I*2^(1/2))*(1-I*x^2)^(1/2)*(1+I*x^2)^(1/2)/(x^4+1)^(1/2)*EllipticF(x*
(1/2*2^(1/2)+1/2*I*2^(1/2)),I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\sqrt{x^{4} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4/sqrt(x^4 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{4}}{\sqrt{x^{4} + 1}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^4/sqrt(x^4 + 1), x)

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Sympy [C]  time = 0.770995, size = 29, normalized size = 0.5 \begin{align*} \frac{x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**5*gamma(5/4)*hyper((1/2, 5/4), (9/4,), x**4*exp_polar(I*pi))/(4*gamma(9/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\sqrt{x^{4} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4/sqrt(x^4 + 1), x)