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

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

[Out]

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

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Rubi [A]  time = 0.0073908, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {195, 220} \[ \frac{1}{3} \sqrt{x^4+1} x+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{3 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]

Int[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])/(3*Sqrt[1 + x^4])

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 \sqrt{1+x^4} \, dx &=\frac{1}{3} x \sqrt{1+x^4}+\frac{2}{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 )}{3 \sqrt{1+x^4}}\\ \end{align*}

Mathematica [C]  time = 0.019558, size = 48, normalized size = 0.83 \[ \frac{-2 \sqrt [4]{-1} \sqrt{x^4+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt [4]{-1} x\right ),-1\right )+x^5+x}{3 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x^4],x]

[Out]

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

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Maple [C]  time = 0.005, size = 72, normalized size = 1.2 \begin{align*}{\frac{x}{3}\sqrt{{x}^{4}+1}}+{\frac{2\,{\it EllipticF} \left ( x \left ( 1/2\,\sqrt{2}+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+1)^(1/2),x)

[Out]

1/3*x*(x^4+1)^(1/2)+2/3/(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 \sqrt{x^{4} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x^4 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(x^4 + 1), x)

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Sympy [C]  time = 0.743514, size = 29, normalized size = 0.5 \begin{align*} \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x^4 + 1), x)