Optimal. Leaf size=99 \[ -\frac {\tan ^{-1}\left (\frac {x-1}{\sqrt {1-\sqrt {a+4}}}\right )}{2 \sqrt {1-\sqrt {a+4}}}-\frac {\tan ^{-1}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )}{2 \sqrt {\sqrt {a+4}+1}}+\frac {\tanh ^{-1}\left (\frac {(x-1)^2+1}{\sqrt {a+4}}\right )}{\sqrt {a+4}} \]
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Rubi [A] time = 0.09, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1680, 1673, 1166, 204, 12, 1107, 618, 206} \[ -\frac {\tan ^{-1}\left (\frac {x-1}{\sqrt {1-\sqrt {a+4}}}\right )}{2 \sqrt {1-\sqrt {a+4}}}-\frac {\tan ^{-1}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )}{2 \sqrt {\sqrt {a+4}+1}}+\frac {\tanh ^{-1}\left (\frac {(x-1)^2+1}{\sqrt {a+4}}\right )}{\sqrt {a+4}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 206
Rule 618
Rule 1107
Rule 1166
Rule 1673
Rule 1680
Rubi steps
\begin {align*} \int \frac {x^2}{a+8 x-8 x^2+4 x^3-x^4} \, dx &=\operatorname {Subst}\left (\int \frac {(1+x)^2}{3+a-2 x^2-x^4} \, dx,x,-1+x\right )\\ &=\operatorname {Subst}\left (\int \frac {2 x}{3+a-2 x^2-x^4} \, dx,x,-1+x\right )+\operatorname {Subst}\left (\int \frac {1+x^2}{3+a-2 x^2-x^4} \, dx,x,-1+x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-1-\sqrt {4+a}-x^2} \, dx,x,-1+x\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-1+\sqrt {4+a}-x^2} \, dx,x,-1+x\right )+2 \operatorname {Subst}\left (\int \frac {x}{3+a-2 x^2-x^4} \, dx,x,-1+x\right )\\ &=\frac {\tan ^{-1}\left (\frac {1-x}{\sqrt {1-\sqrt {4+a}}}\right )}{2 \sqrt {1-\sqrt {4+a}}}+\frac {\tan ^{-1}\left (\frac {1-x}{\sqrt {1+\sqrt {4+a}}}\right )}{2 \sqrt {1+\sqrt {4+a}}}+\operatorname {Subst}\left (\int \frac {1}{3+a-2 x-x^2} \, dx,x,(-1+x)^2\right )\\ &=\frac {\tan ^{-1}\left (\frac {1-x}{\sqrt {1-\sqrt {4+a}}}\right )}{2 \sqrt {1-\sqrt {4+a}}}+\frac {\tan ^{-1}\left (\frac {1-x}{\sqrt {1+\sqrt {4+a}}}\right )}{2 \sqrt {1+\sqrt {4+a}}}-2 \operatorname {Subst}\left (\int \frac {1}{4 (4+a)-x^2} \, dx,x,-2 \left (1+(-1+x)^2\right )\right )\\ &=\frac {\tan ^{-1}\left (\frac {1-x}{\sqrt {1-\sqrt {4+a}}}\right )}{2 \sqrt {1-\sqrt {4+a}}}+\frac {\tan ^{-1}\left (\frac {1-x}{\sqrt {1+\sqrt {4+a}}}\right )}{2 \sqrt {1+\sqrt {4+a}}}+\frac {\tanh ^{-1}\left (\frac {1+(-1+x)^2}{\sqrt {4+a}}\right )}{\sqrt {4+a}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 61, normalized size = 0.62 \[ -\frac {1}{4} \text {RootSum}\left [-\text {$\#$1}^4+4 \text {$\#$1}^3-8 \text {$\#$1}^2+8 \text {$\#$1}+a\& ,\frac {\text {$\#$1}^2 \log (x-\text {$\#$1})}{\text {$\#$1}^3-3 \text {$\#$1}^2+4 \text {$\#$1}-2}\& \right ] \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x^{2}}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.00, size = 52, normalized size = 0.53 \[ -\frac {\RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )^{2} \ln \left (-\RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )+x \right )}{4 \left (\RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )^{3}-3 \RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )^{2}+4 \RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )-2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {x^{2}}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.78, size = 878, normalized size = 8.87 \[ \sum _{k=1}^4\ln \left (64\,\mathrm {root}\left (2816\,a^2\,z^4+256\,a^3\,z^4+10240\,a\,z^4+12288\,z^4-160\,a^2\,z^2-1152\,a\,z^2-2048\,z^2+32\,a^2\,z+256\,a\,z+512\,z-a^2,z,k\right )-a-8\,x+\mathrm {root}\left (2816\,a^2\,z^4+256\,a^3\,z^4+10240\,a\,z^4+12288\,z^4-160\,a^2\,z^2-1152\,a\,z^2-2048\,z^2+32\,a^2\,z+256\,a\,z+512\,z-a^2,z,k\right )\,a\,20-{\mathrm {root}\left (2816\,a^2\,z^4+256\,a^3\,z^4+10240\,a\,z^4+12288\,z^4-160\,a^2\,z^2-1152\,a\,z^2-2048\,z^2+32\,a^2\,z+256\,a\,z+512\,z-a^2,z,k\right )}^2\,a\,48+{\mathrm {root}\left (2816\,a^2\,z^4+256\,a^3\,z^4+10240\,a\,z^4+12288\,z^4-160\,a^2\,z^2-1152\,a\,z^2-2048\,z^2+32\,a^2\,z+256\,a\,z+512\,z-a^2,z,k\right )}^3\,a\,64+{\mathrm {root}\left (2816\,a^2\,z^4+256\,a^3\,z^4+10240\,a\,z^4+12288\,z^4-160\,a^2\,z^2-1152\,a\,z^2-2048\,z^2+32\,a^2\,z+256\,a\,z+512\,z-a^2,z,k\right )}^2\,x\,128-{\mathrm {root}\left (2816\,a^2\,z^4+256\,a^3\,z^4+10240\,a\,z^4+12288\,z^4-160\,a^2\,z^2-1152\,a\,z^2-2048\,z^2+32\,a^2\,z+256\,a\,z+512\,z-a^2,z,k\right )}^3\,x\,256-192\,{\mathrm {root}\left (2816\,a^2\,z^4+256\,a^3\,z^4+10240\,a\,z^4+12288\,z^4-160\,a^2\,z^2-1152\,a\,z^2-2048\,z^2+32\,a^2\,z+256\,a\,z+512\,z-a^2,z,k\right )}^2+256\,{\mathrm {root}\left (2816\,a^2\,z^4+256\,a^3\,z^4+10240\,a\,z^4+12288\,z^4-160\,a^2\,z^2-1152\,a\,z^2-2048\,z^2+32\,a^2\,z+256\,a\,z+512\,z-a^2,z,k\right )}^3-\mathrm {root}\left (2816\,a^2\,z^4+256\,a^3\,z^4+10240\,a\,z^4+12288\,z^4-160\,a^2\,z^2-1152\,a\,z^2-2048\,z^2+32\,a^2\,z+256\,a\,z+512\,z-a^2,z,k\right )\,a\,x\,4+{\mathrm {root}\left (2816\,a^2\,z^4+256\,a^3\,z^4+10240\,a\,z^4+12288\,z^4-160\,a^2\,z^2-1152\,a\,z^2-2048\,z^2+32\,a^2\,z+256\,a\,z+512\,z-a^2,z,k\right )}^2\,a\,x\,32-{\mathrm {root}\left (2816\,a^2\,z^4+256\,a^3\,z^4+10240\,a\,z^4+12288\,z^4-160\,a^2\,z^2-1152\,a\,z^2-2048\,z^2+32\,a^2\,z+256\,a\,z+512\,z-a^2,z,k\right )}^3\,a\,x\,64\right )\,\mathrm {root}\left (2816\,a^2\,z^4+256\,a^3\,z^4+10240\,a\,z^4+12288\,z^4-160\,a^2\,z^2-1152\,a\,z^2-2048\,z^2+32\,a^2\,z+256\,a\,z+512\,z-a^2,z,k\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.61, size = 172, normalized size = 1.74 \[ - \operatorname {RootSum} {\left (t^{4} \left (256 a^{3} + 2816 a^{2} + 10240 a + 12288\right ) + t^{2} \left (- 160 a^{2} - 1152 a - 2048\right ) + t \left (- 32 a^{2} - 256 a - 512\right ) - a^{2}, \left (t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{4} - 448 t^{3} a^{3} - 256 t^{3} a^{2} + 3584 t^{3} a + 6144 t^{3} - 224 t^{2} a^{3} - 2208 t^{2} a^{2} - 7168 t^{2} a - 7680 t^{2} + 56 t a^{3} + 400 t a^{2} + 864 t a + 512 t + 5 a^{3} + 34 a^{2} + 56 a}{a^{3} + 60 a^{2} + 320 a + 448} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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