\(\int \frac {-1+x^4}{x^3 \sqrt {1+x^4}} \, dx\) [432]

Optimal result

Integrand size = 18, antiderivative size = 35 \[ \int \frac {-1+x^4}{x^3 \sqrt {1+x^4}} \, dx=\frac {\sqrt {1+x^4}}{2 x^2}+\frac {1}{2} \log \left (x^2+\sqrt {1+x^4}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.71, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {462, 281, 221} \[ \int \frac {-1+x^4}{x^3 \sqrt {1+x^4}} \, dx=\frac {\text {arcsinh}\left (x^2\right )}{2}+\frac {\sqrt {x^4+1}}{2 x^2} \]

[In]

Int[(-1 + x^4)/(x^3*Sqrt[1 + x^4]),x]

[Out]

Sqrt[1 + x^4]/(2*x^2) + ArcSinh[x^2]/2

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 462

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] + Dist[d/e^n, Int[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a,
b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && (IntegerQ[n] || GtQ[e, 0]) && (
(GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1]))

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+x^4}}{2 x^2}+\int \frac {x}{\sqrt {1+x^4}} \, dx \\ & = \frac {\sqrt {1+x^4}}{2 x^2}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right ) \\ & = \frac {\sqrt {1+x^4}}{2 x^2}+\frac {\text {arcsinh}\left (x^2\right )}{2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {-1+x^4}{x^3 \sqrt {1+x^4}} \, dx=\frac {\sqrt {1+x^4}}{2 x^2}-\frac {1}{2} \log \left (-x^2+\sqrt {1+x^4}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.57

method	result	size
default	\(\frac {\operatorname {arcsinh}\left (x^{2}\right )}{2}+\frac {\sqrt {x^{4}+1}}{2 x^{2}}\)	\(20\)
meijerg	\(\frac {\operatorname {arcsinh}\left (x^{2}\right )}{2}+\frac {\sqrt {x^{4}+1}}{2 x^{2}}\)	\(20\)
risch	\(\frac {\operatorname {arcsinh}\left (x^{2}\right )}{2}+\frac {\sqrt {x^{4}+1}}{2 x^{2}}\)	\(20\)
elliptic	\(\frac {\operatorname {arcsinh}\left (x^{2}\right )}{2}+\frac {\sqrt {x^{4}+1}}{2 x^{2}}\)	\(20\)
pseudoelliptic	\(\frac {\operatorname {arcsinh}\left (x^{2}\right ) x^{2}+\sqrt {x^{4}+1}}{2 x^{2}}\)	\(22\)
trager	\(\frac {\sqrt {x^{4}+1}}{2 x^{2}}-\frac {\ln \left (x^{2}-\sqrt {x^{4}+1}\right )}{2}\)	\(30\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.09 \[ \int \frac {-1+x^4}{x^3 \sqrt {1+x^4}} \, dx=-\frac {x^{2} \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) - x^{2} - \sqrt {x^{4} + 1}}{2 \, x^{2}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.65 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.54 \[ \int \frac {-1+x^4}{x^3 \sqrt {1+x^4}} \, dx=\frac {\operatorname {asinh}{\left (x^{2} \right )}}{2} + \frac {\sqrt {x^{4} + 1}}{2 x^{2}} \]

[In]

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

[Out]

asinh(x**2)/2 + sqrt(x**4 + 1)/(2*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.29 \[ \int \frac {-1+x^4}{x^3 \sqrt {1+x^4}} \, dx=\frac {\sqrt {x^{4} + 1}}{2 \, x^{2}} + \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} + 1}}{x^{2}} + 1\right ) - \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} + 1}}{x^{2}} - 1\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.09 \[ \int \frac {-1+x^4}{x^3 \sqrt {1+x^4}} \, dx=-\frac {1}{{\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} - 1} - \frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.46 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.54 \[ \int \frac {-1+x^4}{x^3 \sqrt {1+x^4}} \, dx=\frac {\mathrm {asinh}\left (x^2\right )}{2}+\frac {\sqrt {x^4+1}}{2\,x^2} \]

[In]

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

[Out]

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