3.10.24 \(\int \frac {1}{x^3 \sqrt {1+x^4}} \, dx\) [924]

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

[Out]

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

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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \begin {gather*} -\frac {\sqrt {x^4+1}}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {1}{x^3 \sqrt {1+x^4}} \, dx &=-\frac {\sqrt {1+x^4}}{2 x^2}\\ \end {align*}

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

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[1 + x^4]),x]

[Out]

-1/2*Sqrt[1 + x^4]/x^2

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(x^4 + 1)/x^2

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Fricas [A]
time = 0.36, size = 16, normalized size = 1.00 \begin {gather*} -\frac {x^{2} + \sqrt {x^{4} + 1}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(x^2 + sqrt(x^4 + 1))/x^2

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Sympy [A]
time = 0.30, size = 12, normalized size = 0.75 \begin {gather*} - \frac {\sqrt {1 + \frac {1}{x^{4}}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(1 + x**(-4))/2

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Giac [A]
time = 2.13, size = 19, normalized size = 1.19 \begin {gather*} \frac {1}{{\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/((x^2 - sqrt(x^4 + 1))^2 - 1)

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Mupad [B]
time = 1.11, size = 12, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {x^4+1}}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(x^4 + 1)^(1/2)),x)

[Out]

-(x^4 + 1)^(1/2)/(2*x^2)

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