Optimal. Leaf size=48 \[ -\frac {1}{4 x^4}-\frac {\tan ^{-1}\left (\frac {1+2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}-\log (x)+\frac {1}{8} \log \left (1+x^4+x^8\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1371, 723, 814,
648, 632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {2 x^4+1}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{4 x^4}+\frac {1}{8} \log \left (x^8+x^4+1\right )-\log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 632
Rule 642
Rule 648
Rule 723
Rule 814
Rule 1371
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (1+x^4+x^8\right )} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{x^2 \left (1+x+x^2\right )} \, dx,x,x^4\right )\\ &=-\frac {1}{4 x^4}+\frac {1}{4} \text {Subst}\left (\int \frac {-1-x}{x \left (1+x+x^2\right )} \, dx,x,x^4\right )\\ &=-\frac {1}{4 x^4}+\frac {1}{4} \text {Subst}\left (\int \left (-\frac {1}{x}+\frac {x}{1+x+x^2}\right ) \, dx,x,x^4\right )\\ &=-\frac {1}{4 x^4}-\log (x)+\frac {1}{4} \text {Subst}\left (\int \frac {x}{1+x+x^2} \, dx,x,x^4\right )\\ &=-\frac {1}{4 x^4}-\log (x)-\frac {1}{8} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^4\right )+\frac {1}{8} \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,x^4\right )\\ &=-\frac {1}{4 x^4}-\log (x)+\frac {1}{8} \log \left (1+x^4+x^8\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^4\right )\\ &=-\frac {1}{4 x^4}-\frac {\tan ^{-1}\left (\frac {1+2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}-\log (x)+\frac {1}{8} \log \left (1+x^4+x^8\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.07, size = 141, normalized size = 2.94 \begin {gather*} \frac {1}{24} \left (-\frac {6}{x^4}+2 \sqrt {3} \tan ^{-1}\left (\frac {-1+2 x}{\sqrt {3}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )-24 \log (x)+\sqrt {3} \left (i+\sqrt {3}\right ) \log \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}+x^2\right )+\sqrt {3} \left (-i+\sqrt {3}\right ) \log \left (\frac {1}{2} i \left (i+\sqrt {3}\right )+x^2\right )+3 \log \left (1-x+x^2\right )+3 \log \left (1+x+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(93\) vs.
\(2(39)=78\).
time = 0.03, size = 94, normalized size = 1.96
method | result | size |
risch | \(-\frac {1}{4 x^{4}}-\ln \left (x \right )+\frac {\ln \left (x^{8}+x^{4}+1\right )}{8}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x^{4}+\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{12}\) | \(38\) |
default | \(\frac {\ln \left (x^{2}+x +1\right )}{8}-\frac {\arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{12}-\frac {1}{4 x^{4}}-\ln \left (x \right )+\frac {\ln \left (x^{4}-x^{2}+1\right )}{8}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{2}-1\right ) \sqrt {3}}{3}\right )}{12}+\frac {\ln \left (x^{2}-x +1\right )}{8}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{12}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 41, normalized size = 0.85 \begin {gather*} -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) - \frac {1}{4 \, x^{4}} + \frac {1}{8} \, \log \left (x^{8} + x^{4} + 1\right ) - \frac {1}{4} \, \log \left (x^{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 49, normalized size = 1.02 \begin {gather*} -\frac {2 \, \sqrt {3} x^{4} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) - 3 \, x^{4} \log \left (x^{8} + x^{4} + 1\right ) + 24 \, x^{4} \log \left (x\right ) + 6}{24 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.07, size = 48, normalized size = 1.00 \begin {gather*} - \log {\left (x \right )} + \frac {\log {\left (x^{8} + x^{4} + 1 \right )}}{8} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{4}}{3} + \frac {\sqrt {3}}{3} \right )}}{12} - \frac {1}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.10, size = 46, normalized size = 0.96 \begin {gather*} -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) + \frac {x^{4} - 1}{4 \, x^{4}} + \frac {1}{8} \, \log \left (x^{8} + x^{4} + 1\right ) - \frac {1}{4} \, \log \left (x^{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.06, size = 41, normalized size = 0.85 \begin {gather*} \frac {\ln \left (x^8+x^4+1\right )}{8}-\ln \left (x\right )-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x^4}{3}+\frac {\sqrt {3}}{3}\right )}{12}-\frac {1}{4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________