Optimal. Leaf size=87 \[ \frac {\sqrt {x^{16}-2 x^8+4} \left (x^8-2\right )}{8 x^8}-\frac {1}{8} \log \left (-x^8+\sqrt {x^{16}-2 x^8+4}+2\right )+\frac {1}{4} \tanh ^{-1}\left (\frac {2 x^8}{3}-\frac {2}{3} \sqrt {x^{16}-2 x^8+4}+\frac {1}{3}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 77, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {1474, 812, 843, 619, 215, 724, 206} \begin {gather*} -\frac {1}{8} \sinh ^{-1}\left (\frac {1-x^8}{\sqrt {3}}\right )-\frac {\sqrt {x^{16}-2 x^8+4} \left (2-x^8\right )}{8 x^8}-\frac {1}{8} \tanh ^{-1}\left (\frac {4-x^8}{2 \sqrt {x^{16}-2 x^8+4}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 215
Rule 619
Rule 724
Rule 812
Rule 843
Rule 1474
Rubi steps
\begin {align*} \int \frac {\left (2+x^8\right ) \sqrt {4-2 x^8+x^{16}}}{x^9} \, dx &=\frac {1}{8} \operatorname {Subst}\left (\int \frac {(2+x) \sqrt {4-2 x+x^2}}{x^2} \, dx,x,x^8\right )\\ &=-\frac {\left (2-x^8\right ) \sqrt {4-2 x^8+x^{16}}}{8 x^8}-\frac {1}{16} \operatorname {Subst}\left (\int \frac {-4-2 x}{x \sqrt {4-2 x+x^2}} \, dx,x,x^8\right )\\ &=-\frac {\left (2-x^8\right ) \sqrt {4-2 x^8+x^{16}}}{8 x^8}+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {4-2 x+x^2}} \, dx,x,x^8\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {4-2 x+x^2}} \, dx,x,x^8\right )\\ &=-\frac {\left (2-x^8\right ) \sqrt {4-2 x^8+x^{16}}}{8 x^8}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{16-x^2} \, dx,x,\frac {2 \left (4-x^8\right )}{\sqrt {4-2 x^8+x^{16}}}\right )+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{12}}} \, dx,x,2 \left (-1+x^8\right )\right )}{16 \sqrt {3}}\\ &=-\frac {\left (2-x^8\right ) \sqrt {4-2 x^8+x^{16}}}{8 x^8}-\frac {1}{8} \sinh ^{-1}\left (\frac {1-x^8}{\sqrt {3}}\right )-\frac {1}{8} \tanh ^{-1}\left (\frac {4-x^8}{2 \sqrt {4-2 x^8+x^{16}}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 68, normalized size = 0.78 \begin {gather*} \frac {1}{8} \left (\sinh ^{-1}\left (\frac {x^8-1}{\sqrt {3}}\right )+\frac {\sqrt {x^{16}-2 x^8+4} \left (x^8-2\right )}{x^8}-\tanh ^{-1}\left (\frac {4-x^8}{2 \sqrt {x^{16}-2 x^8+4}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.16, size = 87, normalized size = 1.00 \begin {gather*} \frac {\left (-2+x^8\right ) \sqrt {4-2 x^8+x^{16}}}{8 x^8}+\frac {1}{4} \tanh ^{-1}\left (\frac {1}{3}+\frac {2 x^8}{3}-\frac {2}{3} \sqrt {4-2 x^8+x^{16}}\right )-\frac {1}{8} \log \left (2-x^8+\sqrt {4-2 x^8+x^{16}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 94, normalized size = 1.08 \begin {gather*} -\frac {2 \, x^{8} \log \left (2 \, x^{16} - 5 \, x^{8} - \sqrt {x^{16} - 2 \, x^{8} + 4} {\left (2 \, x^{8} - 3\right )} + 6\right ) - 2 \, x^{8} \log \left (-x^{8} + \sqrt {x^{16} - 2 \, x^{8} + 4} - 2\right ) + 5 \, x^{8} - 2 \, \sqrt {x^{16} - 2 \, x^{8} + 4} {\left (x^{8} - 2\right )}}{16 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 126, normalized size = 1.45 \begin {gather*} \frac {1}{8} \, \sqrt {x^{16} - 2 \, x^{8} + 4} - \frac {x^{8} - \sqrt {x^{16} - 2 \, x^{8} + 4} - 4}{2 \, {\left ({\left (x^{8} - \sqrt {x^{16} - 2 \, x^{8} + 4}\right )}^{2} - 4\right )}} + \frac {1}{8} \, \log \left (x^{8} - \sqrt {x^{16} - 2 \, x^{8} + 4} + 2\right ) - \frac {1}{8} \, \log \left (-x^{8} + \sqrt {x^{16} - 2 \, x^{8} + 4} + 2\right ) - \frac {1}{8} \, \log \left (-x^{8} + \sqrt {x^{16} - 2 \, x^{8} + 4} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.07, size = 50, normalized size = 0.57
method | result | size |
trager | \(\frac {\left (x^{8}-2\right ) \sqrt {x^{16}-2 x^{8}+4}}{8 x^{8}}-\frac {\ln \left (\frac {2-x^{8}+\sqrt {x^{16}-2 x^{8}+4}}{x^{4}}\right )}{4}\) | \(50\) |
risch | \(\frac {x^{24}-4 x^{16}+8 x^{8}-8}{8 x^{8} \sqrt {x^{16}-2 x^{8}+4}}+\frac {\ln \left (\frac {x^{8}+\sqrt {x^{16}-2 x^{8}+4}-2}{x^{4}}\right )}{4}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.45, size = 62, normalized size = 0.71 \begin {gather*} \frac {1}{8} \, \sqrt {x^{16} - 2 \, x^{8} + 4} - \frac {\sqrt {x^{16} - 2 \, x^{8} + 4}}{4 \, x^{8}} + \frac {1}{8} \, \operatorname {arsinh}\left (\frac {1}{3} \, \sqrt {3} {\left (x^{8} - 1\right )}\right ) - \frac {1}{8} \, \operatorname {arsinh}\left (-\frac {1}{3} \, \sqrt {3} + \frac {4 \, \sqrt {3}}{3 \, x^{8}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.97, size = 80, normalized size = 0.92 \begin {gather*} \frac {\ln \left (\sqrt {x^{16}-2\,x^8+4}+x^8-1\right )}{8}-\frac {\ln \left (\frac {2\,\sqrt {x^{16}-2\,x^8+4}-x^8+4}{x^8}\right )}{8}-\frac {\sqrt {x^{16}-2\,x^8+4}}{4\,x^8}+\frac {\sqrt {x^{16}-2\,x^8+4}}{8} \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^{8} + 2\right ) \sqrt {x^{16} - 2 x^{8} + 4}}{x^{9}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________