3.10.44 \(\int \frac {x}{(1+x^4)^{3/2}} \, dx\) [944]

Optimal. Leaf size=16 \[ \frac {x^2}{2 \sqrt {1+x^4}} \]

[Out]

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

________________________________________________________________________________________

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 = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {270} \begin {gather*} \frac {x^2}{2 \sqrt {x^4+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(1 + x^4)^(3/2),x]

[Out]

x^2/(2*Sqrt[1 + x^4])

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 {x}{\left (1+x^4\right )^{3/2}} \, dx &=\frac {x^2}{2 \sqrt {1+x^4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.08, size = 16, normalized size = 1.00 \begin {gather*} \frac {x^2}{2 \sqrt {1+x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(1 + x^4)^(3/2),x]

[Out]

x^2/(2*Sqrt[1 + x^4])

________________________________________________________________________________________

Maple [A]
time = 0.14, size = 13, normalized size = 0.81

method	result	size
gosper	\(\frac {x^{2}}{2 \sqrt {x^{4}+1}}\)	\(13\)
default	\(\frac {x^{2}}{2 \sqrt {x^{4}+1}}\)	\(13\)
trager	\(\frac {x^{2}}{2 \sqrt {x^{4}+1}}\)	\(13\)
meijerg	\(\frac {x^{2}}{2 \sqrt {x^{4}+1}}\)	\(13\)
risch	\(\frac {x^{2}}{2 \sqrt {x^{4}+1}}\)	\(13\)
elliptic	\(\frac {x^{2}}{2 \sqrt {x^{4}+1}}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 12, normalized size = 0.75 \begin {gather*} \frac {x^{2}}{2 \, \sqrt {x^{4} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2/sqrt(x^4 + 1)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (12) = 24\).
time = 0.36, size = 25, normalized size = 1.56 \begin {gather*} \frac {x^{4} + \sqrt {x^{4} + 1} x^{2} + 1}{2 \, {\left (x^{4} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.27, size = 12, normalized size = 0.75 \begin {gather*} \frac {x^{2}}{2 \sqrt {x^{4} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/(2*sqrt(x**4 + 1))

________________________________________________________________________________________

Giac [A]
time = 1.57, size = 12, normalized size = 0.75 \begin {gather*} \frac {x^{2}}{2 \, \sqrt {x^{4} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2/sqrt(x^4 + 1)

________________________________________________________________________________________

Mupad [B]
time = 1.05, size = 12, normalized size = 0.75 \begin {gather*} \frac {x^2}{2\,\sqrt {x^4+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^4 + 1)^(3/2),x)

[Out]

x^2/(2*(x^4 + 1)^(1/2))

________________________________________________________________________________________