3.10.79 \(\int \frac {1}{x^4 \sqrt {-1+x^4}} \, dx\) [979]

Optimal. Leaf size=74 \[ \frac {\sqrt {-1+x^4}}{3 x^3}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{3 \sqrt {2} \sqrt {-1+x^4}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 74, 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, 228} \begin {gather*} \frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\text {ArcSin}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{3 \sqrt {2} \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]

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

Rule 228

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[(-a)*b, 2]}, Simp[Sqrt[-a + q*x^2]*(Sqrt[(a + q*x^2
)/q]/(Sqrt[2]*Sqrt[-a]*Sqrt[a + b*x^4]))*EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2], x] /; IntegerQ[q]]
/; FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]

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 {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{3 \sqrt {2} \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 = 10.02, size = 40, normalized size = 0.54 \begin {gather*} -\frac {\sqrt {1-x^4} \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {1}{4};x^4\right )}{3 x^3 \sqrt {-1+x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[-1 + x^4]),x]

[Out]

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

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Maple [C] Result contains complex when optimal does not.
time = 0.16, size = 47, normalized size = 0.64

method	result	size
meijerg	\(-\frac {\sqrt {-\mathrm {signum}\left (x^{4}-1\right )}\, \hypergeom \left (\left [-\frac {3}{4}, \frac {1}{2}\right ], \left [\frac {1}{4}\right ], x^{4}\right )}{3 \sqrt {\mathrm {signum}\left (x^{4}-1\right )}\, x^{3}}\)	\(33\)
default	\(\frac {\sqrt {x^{4}-1}}{3 x^{3}}-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{3 \sqrt {x^{4}-1}}\)	\(47\)
risch	\(\frac {\sqrt {x^{4}-1}}{3 x^{3}}-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{3 \sqrt {x^{4}-1}}\)	\(47\)
elliptic	\(\frac {\sqrt {x^{4}-1}}{3 x^{3}}-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{3 \sqrt {x^{4}-1}}\)	\(47\)

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*I*(x^2+1)^(1/2)*(-x^2+1)^(1/2)/(x^4-1)^(1/2)*EllipticF(I*x,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 [A]
time = 0.07, size = 12, normalized size = 0.16 \begin {gather*} \frac {\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*sqrt(x^4 - 1)/x^3

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Sympy [C] Result contains complex when optimal does not.
time = 0.38, size = 31, normalized size = 0.42 \begin {gather*} - \frac {i \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}} \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]

-I*gamma(-3/4)*hyper((-3/4, 1/2), (1/4,), x**4)/(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 = -0.01 \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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