3.1.53 \(\int \frac {1}{x^2 (1+x^4)^{3/4}} \, dx\)

Optimal. Leaf size=14 \[ -\frac {\sqrt [4]{x^4+1}}{x} \]

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Rubi [A]  time = 0.00, antiderivative size = 14, 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 = {264} \begin {gather*} -\frac {\sqrt [4]{x^4+1}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(1 + x^4)^(3/4)),x]

[Out]

-((1 + x^4)^(1/4)/x)

Rule 264

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

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [4]{x^4+1}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(1 + x^4)^(3/4)),x]

[Out]

-((1 + x^4)^(1/4)/x)

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IntegrateAlgebraic [A]  time = 0.14, size = 14, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [4]{1+x^4}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^2*(1 + x^4)^(3/4)),x]

[Out]

-((1 + x^4)^(1/4)/x)

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fricas [A]  time = 0.46, size = 12, normalized size = 0.86 \begin {gather*} -\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^4 + 1)^(1/4)/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{4} + 1\right )}^{\frac {3}{4}} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((x^4 + 1)^(3/4)*x^2), x)

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maple [A]  time = 0.06, size = 13, normalized size = 0.93

method	result	size
gosper	\(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}\)	\(13\)
trager	\(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}\)	\(13\)
meijerg	\(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}\)	\(13\)
risch	\(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.43, size = 12, normalized size = 0.86 \begin {gather*} -\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^4 + 1)^(1/4)/x

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mupad [B]  time = 0.15, size = 12, normalized size = 0.86 \begin {gather*} -\frac {{\left (x^4+1\right )}^{1/4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(x^4 + 1)^(3/4)),x)

[Out]

-(x^4 + 1)^(1/4)/x

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sympy [B]  time = 0.55, size = 22, normalized size = 1.57 \begin {gather*} \frac {\sqrt [4]{1 + \frac {1}{x^{4}}} \Gamma \left (- \frac {1}{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1 + x**(-4))**(1/4)*gamma(-1/4)/(4*gamma(3/4))

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