\(\int \frac {-1+x^4}{x^2 \sqrt {1+x^4}} \, dx\) [31]

Optimal result

Integrand size = 18, antiderivative size = 13 \[ \int \frac {-1+x^4}{x^2 \sqrt {1+x^4}} \, dx=\frac {\sqrt {1+x^4}}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {460} \[ \int \frac {-1+x^4}{x^2 \sqrt {1+x^4}} \, dx=\frac {\sqrt {x^4+1}}{x} \]

[In]

Int[(-1 + x^4)/(x^2*Sqrt[1 + x^4]),x]

[Out]

Sqrt[1 + x^4]/x

Rule 460

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[
a*d*(m + 1) - b*c*(m + n*(p + 1) + 1), 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+x^4}}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^4}{x^2 \sqrt {1+x^4}} \, dx=\frac {\sqrt {1+x^4}}{x} \]

[In]

Integrate[(-1 + x^4)/(x^2*Sqrt[1 + x^4]),x]

[Out]

Sqrt[1 + x^4]/x

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92

method	result	size
gosper	\(\frac {\sqrt {x^{4}+1}}{x}\)	\(12\)
default	\(\frac {\sqrt {x^{4}+1}}{x}\)	\(12\)
trager	\(\frac {\sqrt {x^{4}+1}}{x}\)	\(12\)
risch	\(\frac {\sqrt {x^{4}+1}}{x}\)	\(12\)
elliptic	\(\frac {\sqrt {x^{4}+1}}{x}\)	\(12\)
pseudoelliptic	\(\frac {\sqrt {x^{4}+1}}{x}\)	\(12\)
meijerg	\(\frac {x^{3} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -x^{4}\right )}{3}+\frac {\operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{4}\right ], -x^{4}\right )}{x}\)	\(33\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {-1+x^4}{x^2 \sqrt {1+x^4}} \, dx=\frac {\sqrt {x^{4} + 1}}{x} \]

[In]

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

[Out]

sqrt(x^4 + 1)/x

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.77 (sec) , antiderivative size = 61, normalized size of antiderivative = 4.69 \[ \int \frac {-1+x^4}{x^2 \sqrt {1+x^4}} \, dx=\frac {x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} - \frac {\Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} \]

[In]

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

[Out]

x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), x**4*exp_polar(I*pi))/(4*gamma(7/4)) - gamma(-1/4)*hyper((-1/4, 1/2)
, (3/4,), x**4*exp_polar(I*pi))/(4*x*gamma(3/4))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {-1+x^4}{x^2 \sqrt {1+x^4}} \, dx=\frac {\sqrt {x^{4} + 1}}{x} \]

[In]

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

[Out]

sqrt(x^4 + 1)/x

Giac [F]

\[ \int \frac {-1+x^4}{x^2 \sqrt {1+x^4}} \, dx=\int { \frac {x^{4} - 1}{\sqrt {x^{4} + 1} x^{2}} \,d x } \]

[In]

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

[Out]

integrate((x^4 - 1)/(sqrt(x^4 + 1)*x^2), x)

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {-1+x^4}{x^2 \sqrt {1+x^4}} \, dx=\frac {\sqrt {x^4+1}}{x} \]

[In]

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

[Out]

(x^4 + 1)^(1/2)/x