Optimal. Leaf size=229 \[ -\frac{25 x \left (x^2+5\right )}{72 \left (x^4+2 x^2+3\right )}-\frac{1}{96} \sqrt{\frac{1}{6} \left (965+699 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{1}{96} \sqrt{\frac{1}{6} \left (965+699 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{4}{9 x}+\frac{1}{48} \sqrt{\frac{1}{6} \left (699 \sqrt{3}-965\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{1}{48} \sqrt{\frac{1}{6} \left (699 \sqrt{3}-965\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.310033, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {1669, 1664, 1169, 634, 618, 204, 628} \[ -\frac{25 x \left (x^2+5\right )}{72 \left (x^4+2 x^2+3\right )}-\frac{1}{96} \sqrt{\frac{1}{6} \left (965+699 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{1}{96} \sqrt{\frac{1}{6} \left (965+699 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{4}{9 x}+\frac{1}{48} \sqrt{\frac{1}{6} \left (699 \sqrt{3}-965\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{1}{48} \sqrt{\frac{1}{6} \left (699 \sqrt{3}-965\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1669
Rule 1664
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{4+x^2+3 x^4+5 x^6}{x^2 \left (3+2 x^2+x^4\right )^2} \, dx &=-\frac{25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}+\frac{1}{48} \int \frac{64+\frac{170 x^2}{3}-\frac{50 x^4}{3}}{x^2 \left (3+2 x^2+x^4\right )} \, dx\\ &=-\frac{25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}+\frac{1}{48} \int \left (\frac{64}{3 x^2}-\frac{2 \left (-7+19 x^2\right )}{3+2 x^2+x^4}\right ) \, dx\\ &=-\frac{4}{9 x}-\frac{25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}-\frac{1}{24} \int \frac{-7+19 x^2}{3+2 x^2+x^4} \, dx\\ &=-\frac{4}{9 x}-\frac{25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}-\frac{\int \frac{-7 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (-7-19 \sqrt{3}\right ) x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{48 \sqrt{6 \left (-1+\sqrt{3}\right )}}-\frac{\int \frac{-7 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (-7-19 \sqrt{3}\right ) x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{48 \sqrt{6 \left (-1+\sqrt{3}\right )}}\\ &=-\frac{4}{9 x}-\frac{25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}-\frac{1}{48} \sqrt{\frac{1}{6} \left (566-133 \sqrt{3}\right )} \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx-\frac{1}{48} \sqrt{\frac{1}{6} \left (566-133 \sqrt{3}\right )} \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx-\frac{1}{96} \sqrt{\frac{1}{6} \left (965+699 \sqrt{3}\right )} \int \frac{-\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx+\frac{1}{96} \sqrt{\frac{1}{6} \left (965+699 \sqrt{3}\right )} \int \frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx\\ &=-\frac{4}{9 x}-\frac{25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}-\frac{1}{96} \sqrt{\frac{1}{6} \left (965+699 \sqrt{3}\right )} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{1}{96} \sqrt{\frac{1}{6} \left (965+699 \sqrt{3}\right )} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{1}{24} \sqrt{\frac{1}{6} \left (566-133 \sqrt{3}\right )} \operatorname{Subst}\left (\int \frac{1}{-2 \left (1+\sqrt{3}\right )-x^2} \, dx,x,-\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x\right )+\frac{1}{24} \sqrt{\frac{1}{6} \left (566-133 \sqrt{3}\right )} \operatorname{Subst}\left (\int \frac{1}{-2 \left (1+\sqrt{3}\right )-x^2} \, dx,x,\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x\right )\\ &=-\frac{4}{9 x}-\frac{25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}+\frac{1}{48} \sqrt{\frac{1}{6} \left (-965+699 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{1}{48} \sqrt{\frac{1}{6} \left (-965+699 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{1}{96} \sqrt{\frac{1}{6} \left (965+699 \sqrt{3}\right )} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{1}{96} \sqrt{\frac{1}{6} \left (965+699 \sqrt{3}\right )} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )\\ \end{align*}
Mathematica [C] time = 0.184795, size = 126, normalized size = 0.55 \[ -\frac{25 x \left (x^2+5\right )}{72 \left (x^4+2 x^2+3\right )}-\frac{4}{9 x}-\frac{\left (19 \sqrt{2}+26 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{48 \sqrt{2-2 i \sqrt{2}}}-\frac{\left (19 \sqrt{2}-26 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{48 \sqrt{2+2 i \sqrt{2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.023, size = 414, normalized size = 1.8 \begin{align*} -{\frac{1}{9\,{x}^{4}+18\,{x}^{2}+27} \left ({\frac{25\,{x}^{3}}{8}}+{\frac{125\,x}{8}} \right ) }-{\frac{\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{18}}-{\frac{13\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{192}}-{\frac{ \left ( -2+2\,\sqrt{3} \right ) \sqrt{3}}{9\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-26+26\,\sqrt{3}}{96\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{7\,\sqrt{3}}{72\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{18}}+{\frac{13\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{192}}-{\frac{ \left ( -2+2\,\sqrt{3} \right ) \sqrt{3}}{9\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-26+26\,\sqrt{3}}{96\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{7\,\sqrt{3}}{72\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{4}{9\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{19 \, x^{4} + 63 \, x^{2} + 32}{24 \,{\left (x^{5} + 2 \, x^{3} + 3 \, x\right )}} - \frac{1}{24} \, \int \frac{19 \, x^{2} - 7}{x^{4} + 2 \, x^{2} + 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.70574, size = 1871, normalized size = 8.17 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.550947, size = 53, normalized size = 0.23 \begin{align*} - \frac{19 x^{4} + 63 x^{2} + 32}{24 x^{5} + 48 x^{3} + 72 x} + \operatorname{RootSum}{\left (28311552 t^{4} - 1976320 t^{2} + 54289, \left ( t \mapsto t \log{\left (- \frac{28311552 t^{3}}{120461} + \frac{1103968 t}{120461} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{5 \, x^{6} + 3 \, x^{4} + x^{2} + 4}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]