3.14.93 \(\int \frac {1}{x^4 \sqrt {2+x^6}} \, dx\) [1393]

Optimal. Leaf size=16 \[ -\frac {\sqrt {2+x^6}}{6 x^3} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \begin {gather*} -\frac {\sqrt {x^6+2}}{6 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[2 + x^6]),x]

[Out]

-1/6*Sqrt[2 + x^6]/x^3

Rule 270

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] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{x^4 \sqrt {2+x^6}} \, dx &=-\frac {\sqrt {2+x^6}}{6 x^3}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 16, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2+x^6}}{6 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[2 + x^6]),x]

[Out]

-1/6*Sqrt[2 + x^6]/x^3

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Maple [A]
time = 0.16, size = 13, normalized size = 0.81

method	result	size
gosper	\(-\frac {\sqrt {x^{6}+2}}{6 x^{3}}\)	\(13\)
trager	\(-\frac {\sqrt {x^{6}+2}}{6 x^{3}}\)	\(13\)
risch	\(-\frac {\sqrt {x^{6}+2}}{6 x^{3}}\)	\(13\)
meijerg	\(-\frac {\sqrt {2}\, \sqrt {1+\frac {x^{6}}{2}}}{6 x^{3}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(x^6+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 12, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {x^{6} + 2}}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(x^6+2)^(1/2),x, algorithm="maxima")

[Out]

-1/6*sqrt(x^6 + 2)/x^3

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Fricas [A]
time = 0.35, size = 16, normalized size = 1.00 \begin {gather*} -\frac {x^{3} + \sqrt {x^{6} + 2}}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(x^6+2)^(1/2),x, algorithm="fricas")

[Out]

-1/6*(x^3 + sqrt(x^6 + 2))/x^3

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Sympy [A]
time = 0.32, size = 12, normalized size = 0.75 \begin {gather*} - \frac {\sqrt {1 + \frac {2}{x^{6}}}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(x**6+2)**(1/2),x)

[Out]

-sqrt(1 + 2/x**6)/6

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Giac [A]
time = 1.45, size = 20, normalized size = 1.25 \begin {gather*} -\frac {\sqrt {\frac {2}{x^{6}} + 1}}{6 \, \mathrm {sgn}\left (x\right )} + \frac {1}{6} \, \mathrm {sgn}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(x^6+2)^(1/2),x, algorithm="giac")

[Out]

-1/6*sqrt(2/x^6 + 1)/sgn(x) + 1/6*sgn(x)

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Mupad [B]
time = 1.12, size = 12, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {x^6+2}}{6\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(x^6 + 2)^(1/2)),x)

[Out]

-(x^6 + 2)^(1/2)/(6*x^3)

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