Optimal. Leaf size=35 \[ \frac {\sqrt {x^4+1}}{2 x^2}+\frac {1}{2} \log \left (\sqrt {x^4+1}+x^2\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 0.71, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {451, 275, 215} \begin {gather*} \frac {1}{2} \sinh ^{-1}\left (x^2\right )+\frac {\sqrt {x^4+1}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 215
Rule 275
Rule 451
Rubi steps
\begin {align*} \int \frac {-1+x^4}{x^3 \sqrt {1+x^4}} \, dx &=\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} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )\\ &=\frac {\sqrt {1+x^4}}{2 x^2}+\frac {1}{2} \sinh ^{-1}\left (x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 22, normalized size = 0.63 \begin {gather*} \frac {1}{2} \left (\sinh ^{-1}\left (x^2\right )+\frac {\sqrt {x^4+1}}{x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.08, size = 37, normalized size = 1.06 \begin {gather*} \frac {\sqrt {x^4+1}}{2 x^2}-\frac {1}{2} \log \left (\sqrt {x^4+1}-x^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.39, size = 38, normalized size = 1.09 \begin {gather*} -\frac {x^{2} \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) - x^{2} - \sqrt {x^{4} + 1}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.31, size = 38, normalized size = 1.09 \begin {gather*} -\frac {1}{{\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} - 1} - \frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 20, normalized size = 0.57 \begin {gather*} \frac {\arcsinh \left (x^{2}\right )}{2}+\frac {\sqrt {x^{4}+1}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.53, size = 45, normalized size = 1.29 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.39, size = 19, normalized size = 0.54 \begin {gather*} \frac {\mathrm {asinh}\left (x^2\right )}{2}+\frac {\sqrt {x^4+1}}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.53, size = 19, normalized size = 0.54 \begin {gather*} \frac {\operatorname {asinh}{\left (x^{2} \right )}}{2} + \frac {\sqrt {x^{4} + 1}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________