Optimal. Leaf size=87 \[ -\frac {2}{27 x^6}+\frac {13}{108 x^4}-\frac {13}{54 x^2}+\frac {25 \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}-\frac {1237 \tan ^{-1}\left (\frac {1+x^2}{\sqrt {2}}\right )}{1944 \sqrt {2}}+\frac {61 \log (x)}{243}-\frac {61}{972} \log \left (3+2 x^2+x^4\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {1677, 1660,
1642, 648, 632, 210, 642} \begin {gather*} -\frac {1237 \text {ArcTan}\left (\frac {x^2+1}{\sqrt {2}}\right )}{1944 \sqrt {2}}-\frac {2}{27 x^6}+\frac {13}{108 x^4}-\frac {13}{54 x^2}+\frac {25 \left (1-7 x^2\right )}{648 \left (x^4+2 x^2+3\right )}-\frac {61}{972} \log \left (x^4+2 x^2+3\right )+\frac {61 \log (x)}{243} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 632
Rule 642
Rule 648
Rule 1642
Rule 1660
Rule 1677
Rubi steps
\begin {align*} \int \frac {4+x^2+3 x^4+5 x^6}{x^7 \left (3+2 x^2+x^4\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {4+x+3 x^2+5 x^3}{x^4 \left (3+2 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {25 \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac {1}{16} \text {Subst}\left (\int \frac {\frac {32}{3}-\frac {40 x}{9}+\frac {200 x^2}{27}+\frac {800 x^3}{81}-\frac {350 x^4}{81}}{x^4 \left (3+2 x+x^2\right )} \, dx,x,x^2\right )\\ &=\frac {25 \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac {1}{16} \text {Subst}\left (\int \left (\frac {32}{9 x^4}-\frac {104}{27 x^3}+\frac {104}{27 x^2}+\frac {488}{243 x}-\frac {2 (1481+244 x)}{243 \left (3+2 x+x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {2}{27 x^6}+\frac {13}{108 x^4}-\frac {13}{54 x^2}+\frac {25 \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac {61 \log (x)}{243}-\frac {\text {Subst}\left (\int \frac {1481+244 x}{3+2 x+x^2} \, dx,x,x^2\right )}{1944}\\ &=-\frac {2}{27 x^6}+\frac {13}{108 x^4}-\frac {13}{54 x^2}+\frac {25 \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac {61 \log (x)}{243}-\frac {61}{972} \text {Subst}\left (\int \frac {2+2 x}{3+2 x+x^2} \, dx,x,x^2\right )-\frac {1237 \text {Subst}\left (\int \frac {1}{3+2 x+x^2} \, dx,x,x^2\right )}{1944}\\ &=-\frac {2}{27 x^6}+\frac {13}{108 x^4}-\frac {13}{54 x^2}+\frac {25 \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac {61 \log (x)}{243}-\frac {61}{972} \log \left (3+2 x^2+x^4\right )+\frac {1237}{972} \text {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,2 \left (1+x^2\right )\right )\\ &=-\frac {2}{27 x^6}+\frac {13}{108 x^4}-\frac {13}{54 x^2}+\frac {25 \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}-\frac {1237 \tan ^{-1}\left (\frac {1+x^2}{\sqrt {2}}\right )}{1944 \sqrt {2}}+\frac {61 \log (x)}{243}-\frac {61}{972} \log \left (3+2 x^2+x^4\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.05, size = 110, normalized size = 1.26 \begin {gather*} \frac {-\frac {576}{x^6}+\frac {936}{x^4}-\frac {1872}{x^2}-\frac {300 \left (-1+7 x^2\right )}{3+2 x^2+x^4}+1952 \log (x)+\sqrt {2} \left (1237 i-244 \sqrt {2}\right ) \log \left (1-i \sqrt {2}+x^2\right )-\sqrt {2} \left (1237 i+244 \sqrt {2}\right ) \log \left (1+i \sqrt {2}+x^2\right )}{7776} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.03, size = 73, normalized size = 0.84
method | result | size |
default | \(-\frac {\frac {525 x^{2}}{4}-\frac {75}{4}}{486 \left (x^{4}+2 x^{2}+3\right )}-\frac {61 \ln \left (x^{4}+2 x^{2}+3\right )}{972}-\frac {1237 \sqrt {2}\, \arctan \left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4}\right )}{3888}-\frac {2}{27 x^{6}}+\frac {13}{108 x^{4}}-\frac {13}{54 x^{2}}+\frac {61 \ln \left (x \right )}{243}\) | \(73\) |
risch | \(\frac {-\frac {331}{648} x^{8}-\frac {209}{648} x^{6}-\frac {5}{9} x^{4}+\frac {23}{108} x^{2}-\frac {2}{9}}{x^{6} \left (x^{4}+2 x^{2}+3\right )}+\frac {61 \ln \left (x \right )}{243}-\frac {61 \ln \left (1530169 x^{4}+3060338 x^{2}+4590507\right )}{972}-\frac {1237 \sqrt {2}\, \arctan \left (\frac {\left (1237 x^{2}+1237\right ) \sqrt {2}}{2474}\right )}{3888}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 76, normalized size = 0.87 \begin {gather*} -\frac {1237}{3888} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) - \frac {331 \, x^{8} + 209 \, x^{6} + 360 \, x^{4} - 138 \, x^{2} + 144}{648 \, {\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )}} - \frac {61}{972} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) + \frac {61}{486} \, \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 115, normalized size = 1.32 \begin {gather*} -\frac {1986 \, x^{8} + 1254 \, x^{6} + 2160 \, x^{4} + 1237 \, \sqrt {2} {\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) - 828 \, x^{2} + 244 \, {\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) - 976 \, {\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )} \log \left (x\right ) + 864}{3888 \, {\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.10, size = 85, normalized size = 0.98 \begin {gather*} \frac {61 \log {\left (x \right )}}{243} - \frac {61 \log {\left (x^{4} + 2 x^{2} + 3 \right )}}{972} - \frac {1237 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x^{2}}{2} + \frac {\sqrt {2}}{2} \right )}}{3888} + \frac {- 331 x^{8} - 209 x^{6} - 360 x^{4} + 138 x^{2} - 144}{648 x^{10} + 1296 x^{8} + 1944 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 4.09, size = 84, normalized size = 0.97 \begin {gather*} -\frac {1237}{3888} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) + \frac {122 \, x^{4} - 281 \, x^{2} + 441}{1944 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac {671 \, x^{6} + 702 \, x^{4} - 351 \, x^{2} + 216}{2916 \, x^{6}} - \frac {61}{972} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) + \frac {61}{486} \, \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.07, size = 78, normalized size = 0.90 \begin {gather*} \frac {61\,\ln \left (x\right )}{243}-\frac {61\,\ln \left (x^4+2\,x^2+3\right )}{972}-\frac {\frac {331\,x^8}{648}+\frac {209\,x^6}{648}+\frac {5\,x^4}{9}-\frac {23\,x^2}{108}+\frac {2}{9}}{x^{10}+2\,x^8+3\,x^6}-\frac {1237\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x^2}{2}+\frac {\sqrt {2}}{2}\right )}{3888} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________