\(\int \frac {1}{x^2 (1+x^4)^{3/4}} \, dx\) [38]

Optimal result

Integrand size = 13, antiderivative size = 14 \[ \int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx=-\frac {\sqrt [4]{1+x^4}}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx=-\frac {\sqrt [4]{x^4+1}}{x} \]

[In]

Int[1/(x^2*(1 + x^4)^(3/4)),x]

[Out]

-((1 + x^4)^(1/4)/x)

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*} \text {integral}& = -\frac {\sqrt [4]{1+x^4}}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx=-\frac {\sqrt [4]{1+x^4}}{x} \]

[In]

Integrate[1/(x^2*(1 + x^4)^(3/4)),x]

[Out]

-((1 + x^4)^(1/4)/x)

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 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	\(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}\)	\(13\)
pseudoelliptic	\(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}\)	\(13\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx=-\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} \]

[In]

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

[Out]

-(x^4 + 1)^(1/4)/x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (10) = 20\).

Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx=\frac {\sqrt [4]{1 + \frac {1}{x^{4}}} \Gamma \left (- \frac {1}{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx=-\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} \]

[In]

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

[Out]

-(x^4 + 1)^(1/4)/x

Giac [F]

\[ \int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx=\int { \frac {1}{{\left (x^{4} + 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \]

[In]

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

[Out]

integrate(1/((x^4 + 1)^(3/4)*x^2), x)

Mupad [B] (verification not implemented)

Time = 15.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx=-\frac {{\left (x^4+1\right )}^{1/4}}{x} \]

[In]

int(1/(x^2*(x^4 + 1)^(3/4)),x)

[Out]

-(x^4 + 1)^(1/4)/x