\(\int \frac {1}{x^4 \sqrt {1+x^6}} \, dx\) [103]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \frac {1}{x^4 \sqrt {1+x^6}} \, dx=-\frac {\sqrt {1+x^6}}{3 x^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \int \frac {1}{x^4 \sqrt {1+x^6}} \, dx=-\frac {\sqrt {x^6+1}}{3 x^3} \]

[In]

Int[1/(x^4*Sqrt[1 + x^6]),x]

[Out]

-1/3*Sqrt[1 + 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*} \text {integral}& = -\frac {\sqrt {1+x^6}}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 \sqrt {1+x^6}} \, dx=-\frac {\sqrt {1+x^6}}{3 x^3} \]

[In]

Integrate[1/(x^4*Sqrt[1 + x^6]),x]

[Out]

-1/3*Sqrt[1 + x^6]/x^3

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 13, normalized size of antiderivative = 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	\(-\frac {\sqrt {x^{6}+1}}{3 x^{3}}\)	\(13\)
pseudoelliptic	\(-\frac {\sqrt {x^{6}+1}}{3 x^{3}}\)	\(13\)
gosper	\(-\frac {\left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right )}{3 x^{3} \sqrt {x^{6}+1}}\)	\(28\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 \sqrt {1+x^6}} \, dx=-\frac {x^{3} + \sqrt {x^{6} + 1}}{3 \, x^{3}} \]

[In]

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

[Out]

-1/3*(x^3 + sqrt(x^6 + 1))/x^3

Sympy [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^4 \sqrt {1+x^6}} \, dx=- \frac {\sqrt {1 + \frac {1}{x^{6}}}}{3} \]

[In]

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

[Out]

-sqrt(1 + x**(-6))/3

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^4 \sqrt {1+x^6}} \, dx=-\frac {\sqrt {x^{6} + 1}}{3 \, x^{3}} \]

[In]

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

[Out]

-1/3*sqrt(x^6 + 1)/x^3

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^4 \sqrt {1+x^6}} \, dx=-\frac {\sqrt {\frac {1}{x^{6}} + 1}}{3 \, \mathrm {sgn}\left (x\right )} + \frac {1}{3} \, \mathrm {sgn}\left (x\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 4.86 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^4 \sqrt {1+x^6}} \, dx=-\frac {\sqrt {x^6+1}}{3\,x^3} \]

[In]

int(1/(x^4*(x^6 + 1)^(1/2)),x)

[Out]

-(x^6 + 1)^(1/2)/(3*x^3)