Optimal. Leaf size=79 \[ \frac{x \left (11 x^2+9\right )}{8 \left (x^4+3 x^2+2\right )^2}-\frac{x \left (347 x^2+547\right )}{32 \left (x^4+3 x^2+2\right )}-\frac{1}{2 x}+\frac{189}{8} \tan ^{-1}(x)-\frac{1119 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{32 \sqrt{2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.103181, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {1669, 1664, 203} \[ \frac{x \left (11 x^2+9\right )}{8 \left (x^4+3 x^2+2\right )^2}-\frac{x \left (347 x^2+547\right )}{32 \left (x^4+3 x^2+2\right )}-\frac{1}{2 x}+\frac{189}{8} \tan ^{-1}(x)-\frac{1119 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{32 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1669
Rule 1664
Rule 203
Rubi steps
\begin{align*} \int \frac{4+x^2+3 x^4+5 x^6}{x^2 \left (2+3 x^2+x^4\right )^3} \, dx &=\frac{x \left (9+11 x^2\right )}{8 \left (2+3 x^2+x^4\right )^2}-\frac{1}{8} \int \frac{-16+29 x^2-55 x^4}{x^2 \left (2+3 x^2+x^4\right )^2} \, dx\\ &=\frac{x \left (9+11 x^2\right )}{8 \left (2+3 x^2+x^4\right )^2}-\frac{x \left (547+347 x^2\right )}{32 \left (2+3 x^2+x^4\right )}+\frac{1}{32} \int \frac{32+441 x^2-347 x^4}{x^2 \left (2+3 x^2+x^4\right )} \, dx\\ &=\frac{x \left (9+11 x^2\right )}{8 \left (2+3 x^2+x^4\right )^2}-\frac{x \left (547+347 x^2\right )}{32 \left (2+3 x^2+x^4\right )}+\frac{1}{32} \int \left (\frac{16}{x^2}+\frac{756}{1+x^2}-\frac{1119}{2+x^2}\right ) \, dx\\ &=-\frac{1}{2 x}+\frac{x \left (9+11 x^2\right )}{8 \left (2+3 x^2+x^4\right )^2}-\frac{x \left (547+347 x^2\right )}{32 \left (2+3 x^2+x^4\right )}+\frac{189}{8} \int \frac{1}{1+x^2} \, dx-\frac{1119}{32} \int \frac{1}{2+x^2} \, dx\\ &=-\frac{1}{2 x}+\frac{x \left (9+11 x^2\right )}{8 \left (2+3 x^2+x^4\right )^2}-\frac{x \left (547+347 x^2\right )}{32 \left (2+3 x^2+x^4\right )}+\frac{189}{8} \tan ^{-1}(x)-\frac{1119 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{32 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0678454, size = 63, normalized size = 0.8 \[ \frac{1}{64} \left (-\frac{2 \left (363 x^8+1684 x^6+2499 x^4+1250 x^2+64\right )}{x \left (x^4+3 x^2+2\right )^2}+1512 \tan ^{-1}(x)-1119 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.014, size = 58, normalized size = 0.7 \begin{align*} -{\frac{1}{2\, \left ({x}^{2}+2 \right ) ^{2}} \left ({\frac{207\,{x}^{3}}{16}}+{\frac{233\,x}{8}} \right ) }-{\frac{1119\,\sqrt{2}}{64}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) }+{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{35\,{x}^{3}}{8}}-{\frac{37\,x}{8}} \right ) }+{\frac{189\,\arctan \left ( x \right ) }{8}}-{\frac{1}{2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.48621, size = 88, normalized size = 1.11 \begin{align*} -\frac{1119}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{363 \, x^{8} + 1684 \, x^{6} + 2499 \, x^{4} + 1250 \, x^{2} + 64}{32 \,{\left (x^{9} + 6 \, x^{7} + 13 \, x^{5} + 12 \, x^{3} + 4 \, x\right )}} + \frac{189}{8} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.56684, size = 302, normalized size = 3.82 \begin{align*} -\frac{726 \, x^{8} + 3368 \, x^{6} + 4998 \, x^{4} + 1119 \, \sqrt{2}{\left (x^{9} + 6 \, x^{7} + 13 \, x^{5} + 12 \, x^{3} + 4 \, x\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 2500 \, x^{2} - 1512 \,{\left (x^{9} + 6 \, x^{7} + 13 \, x^{5} + 12 \, x^{3} + 4 \, x\right )} \arctan \left (x\right ) + 128}{64 \,{\left (x^{9} + 6 \, x^{7} + 13 \, x^{5} + 12 \, x^{3} + 4 \, x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.260194, size = 70, normalized size = 0.89 \begin{align*} - \frac{363 x^{8} + 1684 x^{6} + 2499 x^{4} + 1250 x^{2} + 64}{32 x^{9} + 192 x^{7} + 416 x^{5} + 384 x^{3} + 128 x} + \frac{189 \operatorname{atan}{\left (x \right )}}{8} - \frac{1119 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{64} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12745, size = 74, normalized size = 0.94 \begin{align*} -\frac{1119}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{347 \, x^{7} + 1588 \, x^{5} + 2291 \, x^{3} + 1058 \, x}{32 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} - \frac{1}{2 \, x} + \frac{189}{8} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]