Optimal. Leaf size=189 \[ -\frac {1}{7 x^7}-\frac {1}{x^3}-\frac {\sqrt [4]{\frac {1}{2} \left (39603-17711 \sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (39603+17711 \sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (39603-17711 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (39603+17711 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1368, 1504, 1422, 212, 206, 203} \[ -\frac {1}{x^3}-\frac {1}{7 x^7}-\frac {\sqrt [4]{\frac {1}{2} \left (39603-17711 \sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (39603+17711 \sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (39603-17711 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (39603+17711 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 206
Rule 212
Rule 1368
Rule 1422
Rule 1504
Rubi steps
\begin {align*} \int \frac {1}{x^8 \left (1-3 x^4+x^8\right )} \, dx &=-\frac {1}{7 x^7}+\frac {1}{7} \int \frac {21-7 x^4}{x^4 \left (1-3 x^4+x^8\right )} \, dx\\ &=-\frac {1}{7 x^7}-\frac {1}{x^3}-\frac {1}{21} \int \frac {-168+63 x^4}{1-3 x^4+x^8} \, dx\\ &=-\frac {1}{7 x^7}-\frac {1}{x^3}-\frac {1}{10} \left (15-7 \sqrt {5}\right ) \int \frac {1}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx-\frac {1}{10} \left (15+7 \sqrt {5}\right ) \int \frac {1}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx\\ &=-\frac {1}{7 x^7}-\frac {1}{x^3}-\frac {\left (-15+7 \sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx}{10 \sqrt {3+\sqrt {5}}}-\frac {\left (-15+7 \sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx}{10 \sqrt {3+\sqrt {5}}}+\frac {1}{2} \sqrt {\frac {1}{5} \left (123+55 \sqrt {5}\right )} \int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx+\frac {1}{2} \sqrt {\frac {1}{5} \left (123+55 \sqrt {5}\right )} \int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx\\ &=-\frac {1}{7 x^7}-\frac {1}{x^3}-\frac {\sqrt [4]{\frac {1}{2} \left (39603-17711 \sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (39603+17711 \sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (39603-17711 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (39603+17711 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.27, size = 189, normalized size = 1.00 \[ -\frac {1}{7 x^7}-\frac {1}{x^3}+\frac {\left (11+5 \sqrt {5}\right ) \tan ^{-1}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )}{2 \sqrt {10 \left (\sqrt {5}-1\right )}}+\frac {\left (11-5 \sqrt {5}\right ) \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {\left (-11-5 \sqrt {5}\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )}{2 \sqrt {10 \left (\sqrt {5}-1\right )}}-\frac {\left (5 \sqrt {5}-11\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.04, size = 332, normalized size = 1.76 \[ \frac {28 \, \sqrt {10} x^{7} \sqrt {89 \, \sqrt {5} + 199} \arctan \left (\frac {1}{40} \, {\left (\sqrt {10} \sqrt {2 \, x^{2} + \sqrt {5} - 1} {\left (11 \, \sqrt {5} \sqrt {2} - 25 \, \sqrt {2}\right )} - 2 \, \sqrt {10} {\left (11 \, \sqrt {5} x - 25 \, x\right )}\right )} \sqrt {89 \, \sqrt {5} + 199}\right ) + 28 \, \sqrt {10} x^{7} \sqrt {89 \, \sqrt {5} - 199} \arctan \left (\frac {1}{40} \, {\left (\sqrt {10} \sqrt {2 \, x^{2} + \sqrt {5} + 1} {\left (11 \, \sqrt {5} \sqrt {2} + 25 \, \sqrt {2}\right )} - 2 \, \sqrt {10} {\left (11 \, \sqrt {5} x + 25 \, x\right )}\right )} \sqrt {89 \, \sqrt {5} - 199}\right ) - 7 \, \sqrt {10} x^{7} \sqrt {89 \, \sqrt {5} - 199} \log \left (\sqrt {10} \sqrt {89 \, \sqrt {5} - 199} {\left (9 \, \sqrt {5} + 20\right )} + 10 \, x\right ) + 7 \, \sqrt {10} x^{7} \sqrt {89 \, \sqrt {5} - 199} \log \left (-\sqrt {10} \sqrt {89 \, \sqrt {5} - 199} {\left (9 \, \sqrt {5} + 20\right )} + 10 \, x\right ) + 7 \, \sqrt {10} x^{7} \sqrt {89 \, \sqrt {5} + 199} \log \left (\sqrt {10} \sqrt {89 \, \sqrt {5} + 199} {\left (9 \, \sqrt {5} - 20\right )} + 10 \, x\right ) - 7 \, \sqrt {10} x^{7} \sqrt {89 \, \sqrt {5} + 199} \log \left (-\sqrt {10} \sqrt {89 \, \sqrt {5} + 199} {\left (9 \, \sqrt {5} - 20\right )} + 10 \, x\right ) - 280 \, x^{4} - 40}{280 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.56, size = 159, normalized size = 0.84 \[ -\frac {1}{20} \, \sqrt {890 \, \sqrt {5} - 1990} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{20} \, \sqrt {890 \, \sqrt {5} + 1990} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{40} \, \sqrt {890 \, \sqrt {5} - 1990} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {890 \, \sqrt {5} - 1990} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {890 \, \sqrt {5} + 1990} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{40} \, \sqrt {890 \, \sqrt {5} + 1990} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {7 \, x^{4} + 1}{7 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 216, normalized size = 1.14 \[ \frac {11 \sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{10 \sqrt {-2+2 \sqrt {5}}}+\frac {5 \arctanh \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{2 \sqrt {-2+2 \sqrt {5}}}+\frac {11 \sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{10 \sqrt {2+2 \sqrt {5}}}-\frac {5 \arctanh \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{2 \sqrt {2+2 \sqrt {5}}}+\frac {11 \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{10 \sqrt {-2+2 \sqrt {5}}}+\frac {5 \arctan \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{2 \sqrt {-2+2 \sqrt {5}}}+\frac {11 \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{10 \sqrt {2+2 \sqrt {5}}}-\frac {5 \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{2 \sqrt {2+2 \sqrt {5}}}-\frac {1}{x^{3}}-\frac {1}{7 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {7 \, x^{4} + 1}{7 \, x^{7}} - \frac {1}{2} \, \int \frac {5 \, x^{2} + 8}{x^{4} + x^{2} - 1}\,{d x} + \frac {1}{2} \, \int \frac {5 \, x^{2} - 8}{x^{4} - x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.21, size = 291, normalized size = 1.54 \[ -\frac {x^4+\frac {1}{7}}{x^7}+\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {-89\,\sqrt {5}-199}\,6677047{}\mathrm {i}}{2\,\left (74049691\,\sqrt {5}+165580139\right )}+\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {-89\,\sqrt {5}-199}\,14930373{}\mathrm {i}}{10\,\left (74049691\,\sqrt {5}+165580139\right )}\right )\,\sqrt {-89\,\sqrt {5}-199}\,1{}\mathrm {i}}{20}+\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {199-89\,\sqrt {5}}\,6677047{}\mathrm {i}}{2\,\left (74049691\,\sqrt {5}-165580139\right )}-\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {199-89\,\sqrt {5}}\,14930373{}\mathrm {i}}{10\,\left (74049691\,\sqrt {5}-165580139\right )}\right )\,\sqrt {199-89\,\sqrt {5}}\,1{}\mathrm {i}}{20}-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {89\,\sqrt {5}-199}\,6677047{}\mathrm {i}}{2\,\left (74049691\,\sqrt {5}-165580139\right )}-\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {89\,\sqrt {5}-199}\,14930373{}\mathrm {i}}{10\,\left (74049691\,\sqrt {5}-165580139\right )}\right )\,\sqrt {89\,\sqrt {5}-199}\,1{}\mathrm {i}}{20}-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {89\,\sqrt {5}+199}\,6677047{}\mathrm {i}}{2\,\left (74049691\,\sqrt {5}+165580139\right )}+\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {89\,\sqrt {5}+199}\,14930373{}\mathrm {i}}{10\,\left (74049691\,\sqrt {5}+165580139\right )}\right )\,\sqrt {89\,\sqrt {5}+199}\,1{}\mathrm {i}}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.30, size = 70, normalized size = 0.37 \[ \operatorname {RootSum} {\left (6400 t^{4} - 15920 t^{2} - 1, \left (t \mapsto t \log {\left (\frac {460800 t^{5}}{17711} - \frac {2842588 t}{17711} + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 15920 t^{2} - 1, \left (t \mapsto t \log {\left (\frac {460800 t^{5}}{17711} - \frac {2842588 t}{17711} + x \right )} \right )\right )} + \frac {- 7 x^{4} - 1}{7 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________