Optimal. Leaf size=225 \[ \frac{(a+4) \left ((x-1)^2+2\right ) (x-1)}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}+\frac{(x-1)^2+1}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac{\left (a+\sqrt{a+4}+4\right ) \tan ^{-1}\left (\frac{x-1}{\sqrt{1-\sqrt{a+4}}}\right )}{8 (a+3) (a+4) \sqrt{1-\sqrt{a+4}}}-\frac{\left (a-\sqrt{a+4}+4\right ) \tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )}{8 (a+3) (a+4) \sqrt{\sqrt{a+4}+1}}+\frac{\tanh ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a+4}}\right )}{2 (a+4)^{3/2}} \]
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Rubi [A] time = 0.212609, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {1680, 1673, 1178, 1166, 204, 12, 1107, 614, 618, 206} \[ \frac{(a+4) \left ((x-1)^2+2\right ) (x-1)}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}+\frac{(x-1)^2+1}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac{\left (a+\sqrt{a+4}+4\right ) \tan ^{-1}\left (\frac{x-1}{\sqrt{1-\sqrt{a+4}}}\right )}{8 (a+3) (a+4) \sqrt{1-\sqrt{a+4}}}-\frac{\left (a-\sqrt{a+4}+4\right ) \tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )}{8 (a+3) (a+4) \sqrt{\sqrt{a+4}+1}}+\frac{\tanh ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a+4}}\right )}{2 (a+4)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1680
Rule 1673
Rule 1178
Rule 1166
Rule 204
Rule 12
Rule 1107
Rule 614
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx &=\operatorname{Subst}\left (\int \frac{(1+x)^2}{\left (3+a-2 x^2-x^4\right )^2} \, dx,x,-1+x\right )\\ &=\operatorname{Subst}\left (\int \frac{2 x}{\left (3+a-2 x^2-x^4\right )^2} \, dx,x,-1+x\right )+\operatorname{Subst}\left (\int \frac{1+x^2}{\left (3+a-2 x^2-x^4\right )^2} \, dx,x,-1+x\right )\\ &=\frac{(4+a) \left (2+(-1+x)^2\right ) (-1+x)}{4 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+2 \operatorname{Subst}\left (\int \frac{x}{\left (3+a-2 x^2-x^4\right )^2} \, dx,x,-1+x\right )-\frac{\operatorname{Subst}\left (\int \frac{-4 (4+a)-2 (4+a) x^2}{3+a-2 x^2-x^4} \, dx,x,-1+x\right )}{8 \left (12+7 a+a^2\right )}\\ &=\frac{(4+a) \left (2+(-1+x)^2\right ) (-1+x)}{4 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+\frac{\left (4+a-\sqrt{4+a}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-\sqrt{4+a}-x^2} \, dx,x,-1+x\right )}{8 \left (12+7 a+a^2\right )}+\frac{\left (4+a+\sqrt{4+a}\right ) \operatorname{Subst}\left (\int \frac{1}{-1+\sqrt{4+a}-x^2} \, dx,x,-1+x\right )}{8 \left (12+7 a+a^2\right )}+\operatorname{Subst}\left (\int \frac{1}{\left (3+a-2 x-x^2\right )^2} \, dx,x,(-1+x)^2\right )\\ &=\frac{1+(-1+x)^2}{2 (4+a) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+\frac{(4+a) \left (2+(-1+x)^2\right ) (-1+x)}{4 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+\frac{\left (4+a+\sqrt{4+a}\right ) \tan ^{-1}\left (\frac{1-x}{\sqrt{1-\sqrt{4+a}}}\right )}{8 \left (12+7 a+a^2\right ) \sqrt{1-\sqrt{4+a}}}+\frac{\left (4+a-\sqrt{4+a}\right ) \tan ^{-1}\left (\frac{1-x}{\sqrt{1+\sqrt{4+a}}}\right )}{8 \left (12+7 a+a^2\right ) \sqrt{1+\sqrt{4+a}}}+\frac{\operatorname{Subst}\left (\int \frac{1}{3+a-2 x-x^2} \, dx,x,(-1+x)^2\right )}{2 (4+a)}\\ &=\frac{1+(-1+x)^2}{2 (4+a) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+\frac{(4+a) \left (2+(-1+x)^2\right ) (-1+x)}{4 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+\frac{\left (4+a+\sqrt{4+a}\right ) \tan ^{-1}\left (\frac{1-x}{\sqrt{1-\sqrt{4+a}}}\right )}{8 \left (12+7 a+a^2\right ) \sqrt{1-\sqrt{4+a}}}+\frac{\left (4+a-\sqrt{4+a}\right ) \tan ^{-1}\left (\frac{1-x}{\sqrt{1+\sqrt{4+a}}}\right )}{8 \left (12+7 a+a^2\right ) \sqrt{1+\sqrt{4+a}}}-\frac{\operatorname{Subst}\left (\int \frac{1}{4 (4+a)-x^2} \, dx,x,-2 \left (1+(-1+x)^2\right )\right )}{4+a}\\ &=\frac{1+(-1+x)^2}{2 (4+a) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+\frac{(4+a) \left (2+(-1+x)^2\right ) (-1+x)}{4 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+\frac{\left (4+a+\sqrt{4+a}\right ) \tan ^{-1}\left (\frac{1-x}{\sqrt{1-\sqrt{4+a}}}\right )}{8 \left (12+7 a+a^2\right ) \sqrt{1-\sqrt{4+a}}}+\frac{\left (4+a-\sqrt{4+a}\right ) \tan ^{-1}\left (\frac{1-x}{\sqrt{1+\sqrt{4+a}}}\right )}{8 \left (12+7 a+a^2\right ) \sqrt{1+\sqrt{4+a}}}+\frac{\tanh ^{-1}\left (\frac{1+(-1+x)^2}{\sqrt{4+a}}\right )}{2 (4+a)^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0612668, size = 182, normalized size = 0.81 \[ \frac{a \left (x^3-x^2+x+1\right )+2 x \left (2 x^2-3 x+4\right )}{4 (a+3) (a+4) \left (a-x \left (x^3-4 x^2+8 x-8\right )\right )}-\frac{\text{RootSum}\left [-\text{$\#$1}^4+4 \text{$\#$1}^3-8 \text{$\#$1}^2+8 \text{$\#$1}+a\& ,\frac{\text{$\#$1}^2 a \log (x-\text{$\#$1})+4 \text{$\#$1}^2 \log (x-\text{$\#$1})+2 \text{$\#$1} a \log (x-\text{$\#$1})-a \log (x-\text{$\#$1})+4 \text{$\#$1} \log (x-\text{$\#$1})}{\text{$\#$1}^3-3 \text{$\#$1}^2+4 \text{$\#$1}-2}\& \right ]}{16 \left (a^2+7 a+12\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 160, normalized size = 0.7 \begin{align*}{\frac{1}{{x}^{4}-4\,{x}^{3}+8\,{x}^{2}-a-8\,x} \left ( -{\frac{{x}^{3}}{12+4\,a}}+{\frac{ \left ( 6+a \right ){x}^{2}}{ \left ( 12+4\,a \right ) \left ( 4+a \right ) }}-{\frac{ \left ( 8+a \right ) x}{ \left ( 12+4\,a \right ) \left ( 4+a \right ) }}-{\frac{a}{ \left ( 12+4\,a \right ) \left ( 4+a \right ) }} \right ) }+{\frac{1}{ \left ( 48+16\,a \right ) \left ( 4+a \right ) }\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}-4\,{{\it \_Z}}^{3}+8\,{{\it \_Z}}^{2}-8\,{\it \_Z}-a \right ) }{\frac{ \left ( \left ( -a-4 \right ){{\it \_R}}^{2}+2\, \left ( -a-2 \right ){\it \_R}+a \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{3}-3\,{{\it \_R}}^{2}+4\,{\it \_R}-2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 18.0023, size = 559, normalized size = 2.48 \begin{align*} - \frac{a + x^{3} \left (a + 4\right ) + x^{2} \left (- a - 6\right ) + x \left (a + 8\right )}{- 4 a^{3} - 28 a^{2} - 48 a + x^{4} \left (4 a^{2} + 28 a + 48\right ) + x^{3} \left (- 16 a^{2} - 112 a - 192\right ) + x^{2} \left (32 a^{2} + 224 a + 384\right ) + x \left (- 32 a^{2} - 224 a - 384\right )} + \operatorname{RootSum}{\left (t^{4} \left (65536 a^{9} + 2162688 a^{8} + 31653888 a^{7} + 269680640 a^{6} + 1473773568 a^{5} + 5357174784 a^{4} + 12952010752 a^{3} + 20082327552 a^{2} + 18119393280 a + 7247757312\right ) + t^{2} \left (- 9728 a^{6} - 209408 a^{5} - 1878016 a^{4} - 8986624 a^{3} - 24215552 a^{2} - 34865152 a - 20971520\right ) + t \left (256 a^{5} + 5888 a^{4} + 53248 a^{3} + 237568 a^{2} + 524288 a + 458752\right ) - a^{4} + 144 a^{3} + 1024 a^{2} + 1792 a, \left ( t \mapsto t \log{\left (x + \frac{4096 t^{3} a^{12} - 61440 t^{3} a^{11} - 5480448 t^{3} a^{10} - 111403008 t^{3} a^{9} - 1227173888 t^{3} a^{8} - 8682876928 t^{3} a^{7} - 42187440128 t^{3} a^{6} - 144630284288 t^{3} a^{5} - 350972280832 t^{3} a^{4} - 591750234112 t^{3} a^{3} - 660716126208 t^{3} a^{2} - 439848271872 t^{3} a - 132271570944 t^{3} - 28672 t^{2} a^{10} - 993280 t^{2} a^{9} - 15400960 t^{2} a^{8} - 140742656 t^{2} a^{7} - 839462912 t^{2} a^{6} - 3414427648 t^{2} a^{5} - 9590087680 t^{2} a^{4} - 18363547648 t^{2} a^{3} - 22938255360 t^{2} a^{2} - 16873684992 t^{2} a - 5549064192 t^{2} - 848 t a^{9} - 6096 t a^{8} + 174608 t a^{7} + 3323792 t a^{6} + 26276224 t a^{5} + 119009280 t a^{4} + 332017664 t a^{3} + 566497280 t a^{2} + 544112640 t a + 225837056 t + 11 a^{8} + 958 a^{7} + 17419 a^{6} + 142964 a^{5} + 632632 a^{4} + 1567552 a^{3} + 2049792 a^{2} + 1100800 a}{a^{8} + 870 a^{7} + 18289 a^{6} + 165176 a^{5} + 824560 a^{4} + 2452288 a^{3} + 4340224 a^{2} + 4229120 a + 1748992} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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