Optimal. Leaf size=154 \[ -\frac {1}{7 x^7}+\frac {1}{3 x^3}+\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{4} \tan ^{-1}\left (\sqrt {3}-2 x\right )-\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{4} \tan ^{-1}\left (\sqrt {3}+2 x\right )-\frac {1}{8} \log \left (1-x+x^2\right )+\frac {1}{8} \log \left (1+x+x^2\right )+\frac {\log \left (1-\sqrt {3} x+x^2\right )}{8 \sqrt {3}}-\frac {\log \left (1+\sqrt {3} x+x^2\right )}{8 \sqrt {3}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {1382, 1518,
12, 1387, 1141, 1175, 632, 210, 1178, 642} \begin {gather*} \frac {\text {ArcTan}\left (\frac {1-2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{4} \text {ArcTan}\left (\sqrt {3}-2 x\right )-\frac {\text {ArcTan}\left (\frac {2 x+1}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{4} \text {ArcTan}\left (2 x+\sqrt {3}\right )-\frac {1}{7 x^7}+\frac {1}{3 x^3}-\frac {1}{8} \log \left (x^2-x+1\right )+\frac {1}{8} \log \left (x^2+x+1\right )+\frac {\log \left (x^2-\sqrt {3} x+1\right )}{8 \sqrt {3}}-\frac {\log \left (x^2+\sqrt {3} x+1\right )}{8 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 210
Rule 632
Rule 642
Rule 1141
Rule 1175
Rule 1178
Rule 1382
Rule 1387
Rule 1518
Rubi steps
\begin {align*} \int \frac {1}{x^8 \left (1+x^4+x^8\right )} \, dx &=-\frac {1}{7 x^7}+\frac {1}{7} \int \frac {-7-7 x^4}{x^4 \left (1+x^4+x^8\right )} \, dx\\ &=-\frac {1}{7 x^7}+\frac {1}{3 x^3}-\frac {1}{21} \int -\frac {21 x^4}{1+x^4+x^8} \, dx\\ &=-\frac {1}{7 x^7}+\frac {1}{3 x^3}+\int \frac {x^4}{1+x^4+x^8} \, dx\\ &=-\frac {1}{7 x^7}+\frac {1}{3 x^3}+\frac {1}{2} \int \frac {x^2}{1-x^2+x^4} \, dx-\frac {1}{2} \int \frac {x^2}{1+x^2+x^4} \, dx\\ &=-\frac {1}{7 x^7}+\frac {1}{3 x^3}-\frac {1}{4} \int \frac {1-x^2}{1-x^2+x^4} \, dx+\frac {1}{4} \int \frac {1+x^2}{1-x^2+x^4} \, dx+\frac {1}{4} \int \frac {1-x^2}{1+x^2+x^4} \, dx-\frac {1}{4} \int \frac {1+x^2}{1+x^2+x^4} \, dx\\ &=-\frac {1}{7 x^7}+\frac {1}{3 x^3}-\frac {1}{8} \int \frac {1+2 x}{-1-x-x^2} \, dx-\frac {1}{8} \int \frac {1-2 x}{-1+x-x^2} \, dx-\frac {1}{8} \int \frac {1}{1-x+x^2} \, dx-\frac {1}{8} \int \frac {1}{1+x+x^2} \, dx+\frac {1}{8} \int \frac {1}{1-\sqrt {3} x+x^2} \, dx+\frac {1}{8} \int \frac {1}{1+\sqrt {3} x+x^2} \, dx+\frac {\int \frac {\sqrt {3}+2 x}{-1-\sqrt {3} x-x^2} \, dx}{8 \sqrt {3}}+\frac {\int \frac {\sqrt {3}-2 x}{-1+\sqrt {3} x-x^2} \, dx}{8 \sqrt {3}}\\ &=-\frac {1}{7 x^7}+\frac {1}{3 x^3}-\frac {1}{8} \log \left (1-x+x^2\right )+\frac {1}{8} \log \left (1+x+x^2\right )+\frac {\log \left (1-\sqrt {3} x+x^2\right )}{8 \sqrt {3}}-\frac {\log \left (1+\sqrt {3} x+x^2\right )}{8 \sqrt {3}}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+2 x\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+2 x\right )\\ &=-\frac {1}{7 x^7}+\frac {1}{3 x^3}+\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{4} \tan ^{-1}\left (\sqrt {3}-2 x\right )-\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{4} \tan ^{-1}\left (\sqrt {3}+2 x\right )-\frac {1}{8} \log \left (1-x+x^2\right )+\frac {1}{8} \log \left (1+x+x^2\right )+\frac {\log \left (1-\sqrt {3} x+x^2\right )}{8 \sqrt {3}}-\frac {\log \left (1+\sqrt {3} x+x^2\right )}{8 \sqrt {3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.21, size = 171, normalized size = 1.11 \begin {gather*} -\frac {1}{7 x^7}+\frac {1}{3 x^3}+\frac {\left (i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) x\right )}{2 \sqrt {-6+6 i \sqrt {3}}}+\frac {\left (-i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {1}{2} \left (1+i \sqrt {3}\right ) x\right )}{2 \sqrt {-6-6 i \sqrt {3}}}-\frac {\tan ^{-1}\left (\frac {-1+2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{8} \log \left (1-x+x^2\right )+\frac {1}{8} \log \left (1+x+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.05, size = 131, normalized size = 0.85
method | result | size |
risch | \(\frac {\frac {x^{4}}{3}-\frac {1}{7}}{x^{7}}-\frac {\ln \left (x^{2}-x +1\right )}{8}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x -\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{12}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (9 \textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (6 \textit {\_R}^{3}+\textit {\_R} +x \right )\right )}{4}+\frac {\ln \left (x^{2}+x +1\right )}{8}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x +\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{12}\) | \(90\) |
default | \(\frac {\ln \left (x^{2}+x +1\right )}{8}-\frac {\arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{12}-\frac {1}{7 x^{7}}+\frac {1}{3 x^{3}}-\frac {\sqrt {3}\, \left (-\frac {\ln \left (1+x^{2}-x \sqrt {3}\right )}{2}-\sqrt {3}\, \arctan \left (2 x -\sqrt {3}\right )\right )}{12}-\frac {\sqrt {3}\, \left (\frac {\ln \left (1+x^{2}+x \sqrt {3}\right )}{2}-\sqrt {3}\, \arctan \left (2 x +\sqrt {3}\right )\right )}{12}-\frac {\ln \left (x^{2}-x +1\right )}{8}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{12}\) | \(131\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 250 vs.
\(2 (118) = 236\).
time = 0.36, size = 250, normalized size = 1.62 \begin {gather*} -\frac {28 \, \sqrt {6} \sqrt {3} \sqrt {2} x^{7} \arctan \left (-\frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2} x + \frac {1}{18} \, \sqrt {6} \sqrt {3} \sqrt {2} \sqrt {-18 \, \sqrt {6} \sqrt {2} x + 36 \, x^{2} + 36} + \sqrt {3}\right ) + 28 \, \sqrt {6} \sqrt {3} \sqrt {2} x^{7} \arctan \left (-\frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2} x + \frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {\sqrt {6} \sqrt {2} x + 2 \, x^{2} + 2} - \sqrt {3}\right ) + 7 \, \sqrt {6} \sqrt {2} x^{7} \log \left (18 \, \sqrt {6} \sqrt {2} x + 36 \, x^{2} + 36\right ) - 7 \, \sqrt {6} \sqrt {2} x^{7} \log \left (-18 \, \sqrt {6} \sqrt {2} x + 36 \, x^{2} + 36\right ) + 28 \, \sqrt {3} x^{7} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + 28 \, \sqrt {3} x^{7} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 42 \, x^{7} \log \left (x^{2} + x + 1\right ) + 42 \, x^{7} \log \left (x^{2} - x + 1\right ) - 112 \, x^{4} + 48}{336 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.43, size = 209, normalized size = 1.36 \begin {gather*} \left (\frac {1}{8} - \frac {\sqrt {3} i}{24}\right ) \log {\left (x - \frac {1}{2} + \frac {\sqrt {3} i}{6} - 18432 \left (\frac {1}{8} - \frac {\sqrt {3} i}{24}\right )^{5} \right )} + \left (\frac {1}{8} + \frac {\sqrt {3} i}{24}\right ) \log {\left (x - \frac {1}{2} - 18432 \left (\frac {1}{8} + \frac {\sqrt {3} i}{24}\right )^{5} - \frac {\sqrt {3} i}{6} \right )} + \left (- \frac {1}{8} - \frac {\sqrt {3} i}{24}\right ) \log {\left (x + \frac {1}{2} + \frac {\sqrt {3} i}{6} - 18432 \left (- \frac {1}{8} - \frac {\sqrt {3} i}{24}\right )^{5} \right )} + \left (- \frac {1}{8} + \frac {\sqrt {3} i}{24}\right ) \log {\left (x + \frac {1}{2} - 18432 \left (- \frac {1}{8} + \frac {\sqrt {3} i}{24}\right )^{5} - \frac {\sqrt {3} i}{6} \right )} + \operatorname {RootSum} {\left (2304 t^{4} + 48 t^{2} + 1, \left ( t \mapsto t \log {\left (- 18432 t^{5} - 4 t + x \right )} \right )\right )} + \frac {7 x^{4} - 3}{21 x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.95, size = 120, normalized size = 0.78 \begin {gather*} -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{24} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} x + 1\right ) + \frac {1}{24} \, \sqrt {3} \log \left (x^{2} - \sqrt {3} x + 1\right ) + \frac {7 \, x^{4} - 3}{21 \, x^{7}} + \frac {1}{4} \, \arctan \left (2 \, x + \sqrt {3}\right ) + \frac {1}{4} \, \arctan \left (2 \, x - \sqrt {3}\right ) + \frac {1}{8} \, \log \left (x^{2} + x + 1\right ) - \frac {1}{8} \, \log \left (x^{2} - x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.03, size = 110, normalized size = 0.71 \begin {gather*} \frac {\frac {x^4}{3}-\frac {1}{7}}{x^7}-\mathrm {atan}\left (\frac {2\,x}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\mathrm {atan}\left (\frac {x\,2{}\mathrm {i}}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{12}-\frac {1}{4}{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {x\,2{}\mathrm {i}}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{12}+\frac {1}{4}{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {2\,x}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________