Optimal. Leaf size=71 \[ \frac {5}{64 \sqrt {2+x^6}}-\frac {1}{24 x^{12} \sqrt {2+x^6}}+\frac {5}{96 x^6 \sqrt {2+x^6}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {2+x^6}}{\sqrt {2}}\right )}{64 \sqrt {2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {272, 44, 53, 65,
213} \begin {gather*} \frac {5}{96 x^6 \sqrt {x^6+2}}+\frac {5}{64 \sqrt {x^6+2}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {x^6+2}}{\sqrt {2}}\right )}{64 \sqrt {2}}-\frac {1}{24 x^{12} \sqrt {x^6+2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 53
Rule 65
Rule 213
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^{13} \left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {1}{x^3 (2+x)^{3/2}} \, dx,x,x^6\right )\\ &=\frac {1}{6 x^{12} \sqrt {2+x^6}}+\frac {5}{12} \text {Subst}\left (\int \frac {1}{x^3 \sqrt {2+x}} \, dx,x,x^6\right )\\ &=\frac {1}{6 x^{12} \sqrt {2+x^6}}-\frac {5 \sqrt {2+x^6}}{48 x^{12}}-\frac {5}{32} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {2+x}} \, dx,x,x^6\right )\\ &=\frac {1}{6 x^{12} \sqrt {2+x^6}}-\frac {5 \sqrt {2+x^6}}{48 x^{12}}+\frac {5 \sqrt {2+x^6}}{64 x^6}+\frac {5}{128} \text {Subst}\left (\int \frac {1}{x \sqrt {2+x}} \, dx,x,x^6\right )\\ &=\frac {1}{6 x^{12} \sqrt {2+x^6}}-\frac {5 \sqrt {2+x^6}}{48 x^{12}}+\frac {5 \sqrt {2+x^6}}{64 x^6}+\frac {5}{64} \text {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\sqrt {2+x^6}\right )\\ &=\frac {1}{6 x^{12} \sqrt {2+x^6}}-\frac {5 \sqrt {2+x^6}}{48 x^{12}}+\frac {5 \sqrt {2+x^6}}{64 x^6}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {2+x^6}}{\sqrt {2}}\right )}{64 \sqrt {2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.10, size = 54, normalized size = 0.76 \begin {gather*} \frac {-8+10 x^6+15 x^{12}}{192 x^{12} \sqrt {2+x^6}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {2+x^6}}{\sqrt {2}}\right )}{64 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.38, size = 51, normalized size = 0.72
method | result | size |
risch | \(\frac {15 x^{12}+10 x^{6}-8}{192 x^{12} \sqrt {x^{6}+2}}+\frac {5 \sqrt {2}\, \ln \left (\frac {\sqrt {x^{6}+2}-\sqrt {2}}{\sqrt {x^{6}}}\right )}{128}\) | \(51\) |
trager | \(\frac {15 x^{12}+10 x^{6}-8}{192 x^{12} \sqrt {x^{6}+2}}+\frac {5 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}-2\right )+\sqrt {x^{6}+2}}{x^{3}}\right )}{128}\) | \(55\) |
meijerg | \(\frac {\sqrt {2}\, \left (\frac {\sqrt {\pi }\, \left (-\frac {47}{4} x^{12}-12 x^{6}+8\right )}{8 x^{12}}-\frac {\sqrt {\pi }\, \left (-15 x^{12}-10 x^{6}+8\right )}{8 x^{12} \sqrt {1+\frac {x^{6}}{2}}}-\frac {15 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {x^{6}}{2}}}{2}\right )}{8}+\frac {15 \left (\frac {47}{30}-3 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }}{16}-\frac {\sqrt {\pi }}{x^{12}}+\frac {3 \sqrt {\pi }}{2 x^{6}}\right )}{48 \sqrt {\pi }}\) | \(109\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 81, normalized size = 1.14 \begin {gather*} \frac {5}{256} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {x^{6} + 2}}{\sqrt {2} + \sqrt {x^{6} + 2}}\right ) - \frac {50 \, x^{6} - 15 \, {\left (x^{6} + 2\right )}^{2} + 68}{192 \, {\left ({\left (x^{6} + 2\right )}^{\frac {5}{2}} - 4 \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} + 4 \, \sqrt {x^{6} + 2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 71, normalized size = 1.00 \begin {gather*} \frac {15 \, \sqrt {2} {\left (x^{18} + 2 \, x^{12}\right )} \log \left (\frac {x^{6} - 2 \, \sqrt {2} \sqrt {x^{6} + 2} + 4}{x^{6}}\right ) + 4 \, {\left (15 \, x^{12} + 10 \, x^{6} - 8\right )} \sqrt {x^{6} + 2}}{768 \, {\left (x^{18} + 2 \, x^{12}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 3.70, size = 68, normalized size = 0.96 \begin {gather*} - \frac {5 \sqrt {2} \operatorname {asinh}{\left (\frac {\sqrt {2}}{x^{3}} \right )}}{128} + \frac {5}{64 x^{3} \sqrt {1 + \frac {2}{x^{6}}}} + \frac {5}{96 x^{9} \sqrt {1 + \frac {2}{x^{6}}}} - \frac {1}{24 x^{15} \sqrt {1 + \frac {2}{x^{6}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.41, size = 68, normalized size = 0.96 \begin {gather*} \frac {5}{256} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {x^{6} + 2}}{\sqrt {2} + \sqrt {x^{6} + 2}}\right ) + \frac {1}{24 \, \sqrt {x^{6} + 2}} + \frac {7 \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} - 18 \, \sqrt {x^{6} + 2}}{192 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.53, size = 65, normalized size = 0.92 \begin {gather*} -\frac {\frac {25\,x^6}{96}-\frac {5\,{\left (x^6+2\right )}^2}{64}+\frac {17}{48}}{4\,\sqrt {x^6+2}-4\,{\left (x^6+2\right )}^{3/2}+{\left (x^6+2\right )}^{5/2}}-\frac {5\,\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {x^6+2}}{2}\right )}{128} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________