3.10.39 \(\int \frac {x^3}{(1+x^4)^{3/2}} \, dx\) [939]

Optimal. Leaf size=13 \[ -\frac {1}{2 \sqrt {1+x^4}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 13, 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 = {267} \begin {gather*} -\frac {1}{2 \sqrt {x^4+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(1 + x^4)^(3/2),x]

[Out]

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

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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{2 \sqrt {1+x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(1 + x^4)^(3/2),x]

[Out]

-1/2*1/Sqrt[1 + x^4]

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Maple [A]
time = 0.15, size = 10, normalized size = 0.77

method	result	size
gosper	\(-\frac {1}{2 \sqrt {x^{4}+1}}\)	\(10\)
derivativedivides	\(-\frac {1}{2 \sqrt {x^{4}+1}}\)	\(10\)
default	\(-\frac {1}{2 \sqrt {x^{4}+1}}\)	\(10\)
trager	\(-\frac {1}{2 \sqrt {x^{4}+1}}\)	\(10\)
risch	\(-\frac {1}{2 \sqrt {x^{4}+1}}\)	\(10\)
elliptic	\(-\frac {1}{2 \sqrt {x^{4}+1}}\)	\(10\)
meijerg	\(\frac {\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {x^{4}+1}}}{2 \sqrt {\pi }}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 9, normalized size = 0.69 \begin {gather*} -\frac {1}{2 \, \sqrt {x^{4} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/sqrt(x^4 + 1)

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Fricas [A]
time = 0.34, size = 9, normalized size = 0.69 \begin {gather*} -\frac {1}{2 \, \sqrt {x^{4} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/sqrt(x^4 + 1)

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Sympy [A]
time = 0.14, size = 12, normalized size = 0.92 \begin {gather*} - \frac {1}{2 \sqrt {x^{4} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(2*sqrt(x**4 + 1))

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Giac [A]
time = 1.84, size = 9, normalized size = 0.69 \begin {gather*} -\frac {1}{2 \, \sqrt {x^{4} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/sqrt(x^4 + 1)

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Mupad [B]
time = 1.17, size = 9, normalized size = 0.69 \begin {gather*} -\frac {1}{2\,\sqrt {x^4+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^4 + 1)^(3/2),x)

[Out]

-1/(2*(x^4 + 1)^(1/2))

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