3.10.30 \(\int \frac {1}{x^4 \sqrt {1+x^4}} \, dx\) [930]

Optimal. Leaf size=60 \[ -\frac {\sqrt {1+x^4}}{3 x^3}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}} \]

[Out]

-1/3*(x^4+1)^(1/2)/x^3-1/6*(x^2+1)*(cos(2*arctan(x))^2)^(1/2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),1/2*
2^(1/2))*((x^4+1)/(x^2+1)^2)^(1/2)/(x^4+1)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 60, 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 = {331, 226} \begin {gather*} -\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \text {ArcTan}(x)\left |\frac {1}{2}\right .\right )}{6 \sqrt {x^4+1}}-\frac {\sqrt {x^4+1}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[1 + x^4]),x]

[Out]

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

Rule 226

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)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 331

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

Rubi steps

\begin {align*} \int \frac {1}{x^4 \sqrt {1+x^4}} \, dx &=-\frac {\sqrt {1+x^4}}{3 x^3}-\frac {1}{3} \int \frac {1}{\sqrt {1+x^4}} \, dx\\ &=-\frac {\sqrt {1+x^4}}{3 x^3}-\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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.01, size = 22, normalized size = 0.37 \begin {gather*} -\frac {\, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {1}{4};-x^4\right )}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[1 + x^4]),x]

[Out]

-1/3*Hypergeometric2F1[-3/4, 1/2, 1/4, -x^4]/x^3

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Maple [C] Result contains complex when optimal does not.
time = 0.15, size = 74, normalized size = 1.23

method	result	size
meijerg	\(-\frac {\hypergeom \left (\left [-\frac {3}{4}, \frac {1}{2}\right ], \left [\frac {1}{4}\right ], -x^{4}\right )}{3 x^{3}}\)	\(17\)
default	\(-\frac {\sqrt {x^{4}+1}}{3 x^{3}}-\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{3 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\)	\(74\)
risch	\(-\frac {\sqrt {x^{4}+1}}{3 x^{3}}-\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{3 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\)	\(74\)
elliptic	\(-\frac {\sqrt {x^{4}+1}}{3 x^{3}}-\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{3 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(x^4+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*(x^4+1)^(1/2)/x^3-1/3/(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x^4 + 1)*x^4), x)

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Fricas [C] Result contains complex when optimal does not.
time = 0.07, size = 31, normalized size = 0.52 \begin {gather*} \frac {i \, \sqrt {i} x^{3} F(\arcsin \left (\sqrt {i} x\right )\,|\,-1) - \sqrt {x^{4} + 1}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(I*sqrt(I)*x^3*elliptic_f(arcsin(sqrt(I)*x), -1) - sqrt(x^4 + 1))/x^3

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Sympy [C] Result contains complex when optimal does not.
time = 0.36, size = 32, normalized size = 0.53 \begin {gather*} \frac {\Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

gamma(-3/4)*hyper((-3/4, 1/2), (1/4,), x**4*exp_polar(I*pi))/(4*x**3*gamma(1/4))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x^4 + 1)*x^4), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x^4\,\sqrt {x^4+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(x^4 + 1)^(1/2)),x)

[Out]

int(1/(x^4*(x^4 + 1)^(1/2)), x)

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