Optimal. Leaf size=37 \[ \frac {\sqrt {x^4+1}}{x}+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.33, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {6725, 277, 305, 220, 1196, 1209, 1211, 1699, 203} \begin {gather*} \frac {\sqrt {x^4+1}}{x}+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 220
Rule 277
Rule 305
Rule 1196
Rule 1209
Rule 1211
Rule 1699
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{x^2 \left (1+x^2\right )} \, dx &=\int \left (-\frac {\sqrt {1+x^4}}{x^2}+\frac {2 \sqrt {1+x^4}}{1+x^2}\right ) \, dx\\ &=2 \int \frac {\sqrt {1+x^4}}{1+x^2} \, dx-\int \frac {\sqrt {1+x^4}}{x^2} \, dx\\ &=\frac {\sqrt {1+x^4}}{x}-2 \int \frac {x^2}{\sqrt {1+x^4}} \, dx-2 \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+4 \int \frac {1}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx\\ &=\frac {\sqrt {1+x^4}}{x}+\frac {2 x \sqrt {1+x^4}}{1+x^2}-\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {1+x^4}}+2 \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+2 \int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx\\ &=\frac {\sqrt {1+x^4}}{x}+2 \operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )\\ &=\frac {\sqrt {1+x^4}}{x}+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.12, size = 72, normalized size = 1.95 \begin {gather*} \frac {1}{\sqrt {x^4+1} x}+\frac {x^3}{\sqrt {x^4+1}}+2 \sqrt [4]{-1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-4 \sqrt [4]{-1} \Pi \left (-i;\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.26, size = 37, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^4}}{x}+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.55, size = 30, normalized size = 0.81 \begin {gather*} \frac {\sqrt {2} x \arctan \left (\frac {\sqrt {2} x}{\sqrt {x^{4} + 1}}\right ) + \sqrt {x^{4} + 1}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + 1} {\left (x^{2} - 1\right )}}{{\left (x^{2} + 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.33, size = 39, normalized size = 1.05
method | result | size |
elliptic | \(\frac {\left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x}-2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )\right ) \sqrt {2}}{2}\) | \(39\) |
trager | \(\frac {\sqrt {x^{4}+1}}{x}-\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) x +\sqrt {x^{4}+1}}{x^{2}+1}\right )\) | \(46\) |
risch | \(\frac {\sqrt {x^{4}+1}}{x}-\frac {2 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}-\frac {4 \left (-1\right )^{\frac {3}{4}} \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i, -\sqrt {-i}\, \left (-1\right )^{\frac {3}{4}}\right )}{\sqrt {x^{4}+1}}\) | \(123\) |
default | \(\frac {\sqrt {x^{4}+1}}{x}-\frac {2 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (\EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-\EllipticE \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}-\frac {2 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}+\frac {2 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}-\frac {2 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticE \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}-\frac {4 \left (-1\right )^{\frac {3}{4}} \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i, -\sqrt {-i}\, \left (-1\right )^{\frac {3}{4}}\right )}{\sqrt {x^{4}+1}}\) | \(326\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + 1} {\left (x^{2} - 1\right )}}{{\left (x^{2} + 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\left (x^2-1\right )\,\sqrt {x^4+1}}{x^2\,\left (x^2+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right ) \sqrt {x^{4} + 1}}{x^{2} \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________