\(\int \frac {3+x^4}{x^4 \sqrt {1+x^4}} \, dx\) [54]

Optimal result

Integrand size = 18, antiderivative size = 14 \[ \int \frac {3+x^4}{x^4 \sqrt {1+x^4}} \, dx=-\frac {\sqrt {1+x^4}}{x^3} \]

[Out]

-(x^4+1)^(1/2)/x^3

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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {460} \[ \int \frac {3+x^4}{x^4 \sqrt {1+x^4}} \, dx=-\frac {\sqrt {x^4+1}}{x^3} \]

[In]

Int[(3 + x^4)/(x^4*Sqrt[1 + x^4]),x]

[Out]

-(Sqrt[1 + x^4]/x^3)

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^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {3+x^4}{x^4 \sqrt {1+x^4}} \, dx=-\frac {\sqrt {1+x^4}}{x^3} \]

[In]

Integrate[(3 + x^4)/(x^4*Sqrt[1 + x^4]),x]

[Out]

-(Sqrt[1 + x^4]/x^3)

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
gosper	\(-\frac {\sqrt {x^{4}+1}}{x^{3}}\)	\(13\)
default	\(-\frac {\sqrt {x^{4}+1}}{x^{3}}\)	\(13\)
trager	\(-\frac {\sqrt {x^{4}+1}}{x^{3}}\)	\(13\)
risch	\(-\frac {\sqrt {x^{4}+1}}{x^{3}}\)	\(13\)
elliptic	\(-\frac {\sqrt {x^{4}+1}}{x^{3}}\)	\(13\)
meijerg	\(x \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {5}{4}\right ], -x^{4}\right )-\frac {\operatorname {hypergeom}\left (\left [-\frac {3}{4}, \frac {1}{2}\right ], \left [\frac {1}{4}\right ], -x^{4}\right )}{x^{3}}\)	\(31\)

[In]

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

[Out]

-(x^4+1)^(1/2)/x^3

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {3+x^4}{x^4 \sqrt {1+x^4}} \, dx=-\frac {\sqrt {x^{4} + 1}}{x^{3}} \]

[In]

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

[Out]

-sqrt(x^4 + 1)/x^3

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.76 (sec) , antiderivative size = 63, normalized size of antiderivative = 4.50 \[ \int \frac {3+x^4}{x^4 \sqrt {1+x^4}} \, dx=\frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {3 \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} \]

[In]

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

[Out]

x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), x**4*exp_polar(I*pi))/(4*gamma(5/4)) + 3*gamma(-3/4)*hyper((-3/4, 1/2),
(1/4,), x**4*exp_polar(I*pi))/(4*x**3*gamma(1/4))

Maxima [A] (verification not implemented)

none

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

[In]

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

[Out]

-sqrt(x^4 + 1)/x^3

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {3+x^4}{x^4 \sqrt {1+x^4}} \, dx=-\frac {\sqrt {\frac {1}{x^{4}} + 1}}{x} \]

[In]

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

[Out]

-sqrt(1/x^4 + 1)/x

Mupad [B] (verification not implemented)

Time = 4.75 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {3+x^4}{x^4 \sqrt {1+x^4}} \, dx=-\frac {\sqrt {x^4+1}}{x^3} \]

[In]

int((x^4 + 3)/(x^4*(x^4 + 1)^(1/2)),x)

[Out]

-(x^4 + 1)^(1/2)/x^3