\(\int \frac {1}{x^3 \sqrt {1-x^4}} \, dx\) [881]

Optimal result

Integrand size = 15, antiderivative size = 18 \[ \int \frac {1}{x^3 \sqrt {1-x^4}} \, dx=-\frac {\sqrt {1-x^4}}{2 x^2} \]

[Out]

-1/2*(-x^4+1)^(1/2)/x^2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \[ \int \frac {1}{x^3 \sqrt {1-x^4}} \, dx=-\frac {\sqrt {1-x^4}}{2 x^2} \]

[In]

Int[1/(x^3*Sqrt[1 - x^4]),x]

[Out]

-1/2*Sqrt[1 - x^4]/x^2

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^4}}{2 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \sqrt {1-x^4}} \, dx=-\frac {\sqrt {1-x^4}}{2 x^2} \]

[In]

Integrate[1/(x^3*Sqrt[1 - x^4]),x]

[Out]

-1/2*Sqrt[1 - x^4]/x^2

Maple [A] (verified)

Time = 4.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83

method	result	size
trager	\(-\frac {\sqrt {-x^{4}+1}}{2 x^{2}}\)	\(15\)
meijerg	\(-\frac {\sqrt {-x^{4}+1}}{2 x^{2}}\)	\(15\)
pseudoelliptic	\(-\frac {\sqrt {-x^{4}+1}}{2 x^{2}}\)	\(15\)
risch	\(\frac {x^{4}-1}{2 x^{2} \sqrt {-x^{4}+1}}\)	\(20\)
default	\(\frac {\left (x^{2}-1\right ) \left (x^{2}+1\right )}{2 x^{2} \sqrt {-x^{4}+1}}\)	\(25\)
elliptic	\(\frac {\left (x^{2}-1\right ) \left (x^{2}+1\right )}{2 x^{2} \sqrt {-x^{4}+1}}\)	\(25\)
gosper	\(\frac {\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}{2 x^{2} \sqrt {-x^{4}+1}}\)	\(26\)

[In]

int(1/x^3/(-x^4+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*(-x^4+1)^(1/2)/x^2

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^3 \sqrt {1-x^4}} \, dx=-\frac {\sqrt {-x^{4} + 1}}{2 \, x^{2}} \]

[In]

integrate(1/x^3/(-x^4+1)^(1/2),x, algorithm="fricas")

[Out]

-1/2*sqrt(-x^4 + 1)/x^2

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.36 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.89 \[ \int \frac {1}{x^3 \sqrt {1-x^4}} \, dx=\begin {cases} - \frac {i \sqrt {x^{4} - 1}}{2 x^{2}} & \text {for}\: \left |{x^{4}}\right | > 1 \\- \frac {\sqrt {1 - x^{4}}}{2 x^{2}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/x**3/(-x**4+1)**(1/2),x)

[Out]

Piecewise((-I*sqrt(x**4 - 1)/(2*x**2), Abs(x**4) > 1), (-sqrt(1 - x**4)/(2*x**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^3 \sqrt {1-x^4}} \, dx=-\frac {\sqrt {-x^{4} + 1}}{2 \, x^{2}} \]

[In]

integrate(1/x^3/(-x^4+1)^(1/2),x, algorithm="maxima")

[Out]

-1/2*sqrt(-x^4 + 1)/x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (14) = 28\).

Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.94 \[ \int \frac {1}{x^3 \sqrt {1-x^4}} \, dx=\frac {x^{2}}{4 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}} - \frac {\sqrt {-x^{4} + 1} - 1}{4 \, x^{2}} \]

[In]

integrate(1/x^3/(-x^4+1)^(1/2),x, algorithm="giac")

[Out]

1/4*x^2/(sqrt(-x^4 + 1) - 1) - 1/4*(sqrt(-x^4 + 1) - 1)/x^2

Mupad [B] (verification not implemented)

Time = 5.47 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^3 \sqrt {1-x^4}} \, dx=-\frac {\sqrt {1-x^4}}{2\,x^2} \]

[In]

int(1/(x^3*(1 - x^4)^(1/2)),x)

[Out]

-(1 - x^4)^(1/2)/(2*x^2)