3.2.3 \(\int \frac {1}{x^4 \sqrt {1+x^6}} \, dx\)

Optimal. Leaf size=16 \[ -\frac {\sqrt {x^6+1}}{3 x^3} \]

________________________________________________________________________________________

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 = {264} \begin {gather*} -\frac {\sqrt {x^6+1}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[1 + x^6]),x]

[Out]

-1/3*Sqrt[1 + x^6]/x^3

Rule 264

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {x^6+1}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[1 + x^6]),x]

[Out]

-1/3*Sqrt[1 + x^6]/x^3

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.12, size = 16, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1+x^6}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^4*Sqrt[1 + x^6]),x]

[Out]

-1/3*Sqrt[1 + x^6]/x^3

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 16, normalized size = 1.00 \begin {gather*} -\frac {x^{3} + \sqrt {x^{6} + 1}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(x^3 + sqrt(x^6 + 1))/x^3

________________________________________________________________________________________

giac [A]  time = 0.37, size = 18, normalized size = 1.12 \begin {gather*} -\frac {\sqrt {\frac {1}{x^{6}} + 1}}{3 \, \mathrm {sgn}\relax (x)} + \frac {1}{3} \, \mathrm {sgn}\relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(1/x^6 + 1)/sgn(x) + 1/3*sgn(x)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 13, normalized size = 0.81

method	result	size
trager	\(-\frac {\sqrt {x^{6}+1}}{3 x^{3}}\)	\(13\)
meijerg	\(-\frac {\sqrt {x^{6}+1}}{3 x^{3}}\)	\(13\)
risch	\(-\frac {\sqrt {x^{6}+1}}{3 x^{3}}\)	\(13\)
gosper	\(-\frac {\left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right )}{3 x^{3} \sqrt {x^{6}+1}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.38, size = 12, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {x^{6} + 1}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(x^6 + 1)/x^3

________________________________________________________________________________________

mupad [B]  time = 0.22, size = 12, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {x^6+1}}{3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(x^6 + 1)^(1/2)),x)

[Out]

-(x^6 + 1)^(1/2)/(3*x^3)

________________________________________________________________________________________

sympy [A]  time = 0.55, size = 12, normalized size = 0.75 \begin {gather*} - \frac {\sqrt {1 + \frac {1}{x^{6}}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(1 + x**(-6))/3

________________________________________________________________________________________