Optimal. Leaf size=165 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {-1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (-1+\sqrt {3}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (1+\sqrt {3}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {-1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (-1+\sqrt {3}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (1+\sqrt {3}\right )}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1433, 1107,
213, 209} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt {\sqrt {3}-1}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (\sqrt {3}-1\right )}}+\frac {\text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (1+\sqrt {3}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {\sqrt {3}-1}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (\sqrt {3}-1\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (1+\sqrt {3}\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 209
Rule 213
Rule 1107
Rule 1433
Rubi steps
\begin {align*} \int \frac {1-x^4}{1-4 x^4+x^8} \, dx &=-\left (\frac {1}{2} \int \frac {1}{-1-\sqrt {2} x^2+x^4} \, dx\right )-\frac {1}{2} \int \frac {1}{-1+\sqrt {2} x^2+x^4} \, dx\\ &=-\frac {\int \frac {1}{-\sqrt {\frac {3}{2}}-\frac {1}{\sqrt {2}}+x^2} \, dx}{2 \sqrt {6}}+\frac {\int \frac {1}{\sqrt {\frac {3}{2}}-\frac {1}{\sqrt {2}}+x^2} \, dx}{2 \sqrt {6}}-\frac {\int \frac {1}{-\sqrt {\frac {3}{2}}+\frac {1}{\sqrt {2}}+x^2} \, dx}{2 \sqrt {6}}+\frac {\int \frac {1}{\sqrt {\frac {3}{2}}+\frac {1}{\sqrt {2}}+x^2} \, dx}{2 \sqrt {6}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {-1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (-1+\sqrt {3}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (1+\sqrt {3}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {-1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (-1+\sqrt {3}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (1+\sqrt {3}\right )}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.01, size = 55, normalized size = 0.33 \begin {gather*} -\frac {1}{8} \text {RootSum}\left [1-4 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{-2 \text {$\#$1}^3+\text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.02, size = 42, normalized size = 0.25
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{4}+1\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}-2 \textit {\_R}^{3}}\right )}{8}\) | \(42\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{4}+1\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}-2 \textit {\_R}^{3}}\right )}{8}\) | \(42\) |
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. 302 vs.
\(2 (109) = 218\).
time = 0.36, size = 302, normalized size = 1.83 \begin {gather*} -\frac {1}{6} \, \sqrt {6} {\left (-\sqrt {3} + 2\right )}^{\frac {1}{4}} \arctan \left (\frac {1}{6} \, \sqrt {6} \sqrt {x^{2} + {\left (\sqrt {3} + 2\right )} \sqrt {-\sqrt {3} + 2}} {\left (\sqrt {3} + 3\right )} {\left (-\sqrt {3} + 2\right )}^{\frac {3}{4}} - \frac {1}{6} \, \sqrt {6} {\left (\sqrt {3} x + 3 \, x\right )} {\left (-\sqrt {3} + 2\right )}^{\frac {3}{4}}\right ) + \frac {1}{6} \, \sqrt {6} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}} \arctan \left (\frac {1}{6} \, {\left (\sqrt {6} \sqrt {x^{2} - \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )}} \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 3\right )} - \sqrt {6} {\left (\sqrt {3} x - 3 \, x\right )} \sqrt {\sqrt {3} + 2}\right )} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}}\right ) - \frac {1}{24} \, \sqrt {6} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (\sqrt {6} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}} {\left (\sqrt {3} - 3\right )} + 6 \, x\right ) + \frac {1}{24} \, \sqrt {6} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (-\sqrt {6} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}} {\left (\sqrt {3} - 3\right )} + 6 \, x\right ) + \frac {1}{24} \, \sqrt {6} {\left (-\sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (\sqrt {6} {\left (\sqrt {3} + 3\right )} {\left (-\sqrt {3} + 2\right )}^{\frac {1}{4}} + 6 \, x\right ) - \frac {1}{24} \, \sqrt {6} {\left (-\sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (-\sqrt {6} {\left (\sqrt {3} + 3\right )} {\left (-\sqrt {3} + 2\right )}^{\frac {1}{4}} + 6 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.08, size = 26, normalized size = 0.16 \begin {gather*} - \operatorname {RootSum} {\left (84934656 t^{8} - 36864 t^{4} + 1, \left ( t \mapsto t \log {\left (36864 t^{5} - 20 t + x \right )} \right )\right )} \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.18, size = 399, normalized size = 2.42 \begin {gather*} \frac {\sqrt {6}\,\mathrm {atan}\left (\frac {64\,\sqrt {6}\,x\,{\left (\sqrt {3}+2\right )}^{1/4}}{80\,\sqrt {\sqrt {3}+2}+48\,\sqrt {3}\,\sqrt {\sqrt {3}+2}}+\frac {112\,\sqrt {3}\,\sqrt {6}\,x\,{\left (\sqrt {3}+2\right )}^{1/4}}{3\,\left (80\,\sqrt {\sqrt {3}+2}+48\,\sqrt {3}\,\sqrt {\sqrt {3}+2}\right )}\right )\,{\left (\sqrt {3}+2\right )}^{1/4}}{12}+\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x\,{\left (2-\sqrt {3}\right )}^{1/4}\,64{}\mathrm {i}}{48\,\sqrt {3}\,\sqrt {2-\sqrt {3}}-80\,\sqrt {2-\sqrt {3}}}-\frac {\sqrt {3}\,\sqrt {6}\,x\,{\left (2-\sqrt {3}\right )}^{1/4}\,112{}\mathrm {i}}{3\,\left (48\,\sqrt {3}\,\sqrt {2-\sqrt {3}}-80\,\sqrt {2-\sqrt {3}}\right )}\right )\,{\left (2-\sqrt {3}\right )}^{1/4}\,1{}\mathrm {i}}{12}-\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {64\,\sqrt {6}\,x\,{\left (2-\sqrt {3}\right )}^{1/4}}{48\,\sqrt {3}\,\sqrt {2-\sqrt {3}}-80\,\sqrt {2-\sqrt {3}}}-\frac {112\,\sqrt {3}\,\sqrt {6}\,x\,{\left (2-\sqrt {3}\right )}^{1/4}}{3\,\left (48\,\sqrt {3}\,\sqrt {2-\sqrt {3}}-80\,\sqrt {2-\sqrt {3}}\right )}\right )\,{\left (2-\sqrt {3}\right )}^{1/4}}{12}-\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x\,{\left (\sqrt {3}+2\right )}^{1/4}\,64{}\mathrm {i}}{80\,\sqrt {\sqrt {3}+2}+48\,\sqrt {3}\,\sqrt {\sqrt {3}+2}}+\frac {\sqrt {3}\,\sqrt {6}\,x\,{\left (\sqrt {3}+2\right )}^{1/4}\,112{}\mathrm {i}}{3\,\left (80\,\sqrt {\sqrt {3}+2}+48\,\sqrt {3}\,\sqrt {\sqrt {3}+2}\right )}\right )\,{\left (\sqrt {3}+2\right )}^{1/4}\,1{}\mathrm {i}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________