\(\int \frac {x^3}{\sqrt {1-x^4}} \, dx\) [876]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, 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 = {267} \[ \int \frac {x^3}{\sqrt {1-x^4}} \, dx=-\frac {1}{2} \sqrt {1-x^4} \]

[In]

Int[x^3/Sqrt[1 - x^4],x]

[Out]

-1/2*Sqrt[1 - x^4]

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2} \sqrt {1-x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{\sqrt {1-x^4}} \, dx=-\frac {1}{2} \sqrt {1-x^4} \]

[In]

Integrate[x^3/Sqrt[1 - x^4],x]

[Out]

-1/2*Sqrt[1 - x^4]

Maple [A] (verified)

Time = 4.16 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(-\frac {\sqrt {-x^{4}+1}}{2}\)	\(12\)
default	\(-\frac {\sqrt {-x^{4}+1}}{2}\)	\(12\)
trager	\(-\frac {\sqrt {-x^{4}+1}}{2}\)	\(12\)
pseudoelliptic	\(-\frac {\sqrt {-x^{4}+1}}{2}\)	\(12\)
risch	\(\frac {x^{4}-1}{2 \sqrt {-x^{4}+1}}\)	\(17\)
elliptic	\(\frac {\left (x^{2}-1\right ) \left (x^{2}+1\right )}{2 \sqrt {-x^{4}+1}}\)	\(22\)
gosper	\(\frac {\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}{2 \sqrt {-x^{4}+1}}\)	\(23\)
meijerg	\(-\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-x^{4}+1}}{4 \sqrt {\pi }}\)	\(26\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {x^3}{\sqrt {1-x^4}} \, dx=-\frac {1}{2} \, \sqrt {-x^{4} + 1} \]

[In]

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

[Out]

-1/2*sqrt(-x^4 + 1)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {x^3}{\sqrt {1-x^4}} \, dx=- \frac {\sqrt {1 - x^{4}}}{2} \]

[In]

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

[Out]

-sqrt(1 - x**4)/2

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {x^3}{\sqrt {1-x^4}} \, dx=-\frac {1}{2} \, \sqrt {-x^{4} + 1} \]

[In]

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

[Out]

-1/2*sqrt(-x^4 + 1)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {x^3}{\sqrt {1-x^4}} \, dx=-\frac {1}{2} \, \sqrt {-x^{4} + 1} \]

[In]

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

[Out]

-1/2*sqrt(-x^4 + 1)

Mupad [B] (verification not implemented)

Time = 5.66 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {x^3}{\sqrt {1-x^4}} \, dx=-\frac {\sqrt {1-x^4}}{2} \]

[In]

int(x^3/(1 - x^4)^(1/2),x)

[Out]

-(1 - x^4)^(1/2)/2