Optimal. Leaf size=122 \[ \frac{d-2 e+4 f-8 g+16 h-32 i}{12 (x+2)}-\frac{1}{18} \log (1-x) (d+e+f+g+h+i)+\frac{1}{48} \log (2-x) (d+2 e+4 f+8 g+16 h+32 i)+\frac{1}{6} \log (x+1) (d-e+f-g+h-i)-\frac{1}{144} \log (x+2) (19 d-26 e+28 f-8 g-80 h+352 i)+i x \]
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Rubi [A] time = 0.314953, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 51, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.039, Rules used = {1586, 6742} \[ \frac{d-2 e+4 f-8 g+16 h-32 i}{12 (x+2)}-\frac{1}{18} \log (1-x) (d+e+f+g+h+i)+\frac{1}{48} \log (2-x) (d+2 e+4 f+8 g+16 h+32 i)+\frac{1}{6} \log (x+1) (d-e+f-g+h-i)-\frac{1}{144} \log (x+2) (19 d-26 e+28 f-8 g-80 h+352 i)+i x \]
Antiderivative was successfully verified.
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Rule 1586
Rule 6742
Rubi steps
\begin{align*} \int \frac{\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3+h x^4+90 x^5\right )}{\left (4-5 x^2+x^4\right )^2} \, dx &=\int \frac{d+e x+f x^2+g x^3+h x^4+90 x^5}{(2+x)^2 \left (2-x-2 x^2+x^3\right )} \, dx\\ &=\int \left (90+\frac{2880+d+2 e+4 f+8 g+16 h}{48 (-2+x)}+\frac{-90-d-e-f-g-h}{18 (-1+x)}+\frac{-90+d-e+f-g+h}{6 (1+x)}+\frac{2880-d+2 e-4 f+8 g-16 h}{12 (2+x)^2}+\frac{-31680-19 d+26 e-28 f+8 g+80 h}{144 (2+x)}\right ) \, dx\\ &=90 x-\frac{2880-d+2 e-4 f+8 g-16 h}{12 (2+x)}-\frac{1}{18} (90+d+e+f+g+h) \log (1-x)+\frac{1}{48} (2880+d+2 e+4 f+8 g+16 h) \log (2-x)-\frac{1}{6} (90-d+e-f+g-h) \log (1+x)-\frac{1}{144} (31680+19 d-26 e+28 f-8 g-80 h) \log (2+x)\\ \end{align*}
Mathematica [A] time = 0.0703025, size = 118, normalized size = 0.97 \[ \frac{1}{144} \left (\frac{12 (d-2 (e-2 f+4 g-8 h+16 i))}{x+2}-8 \log (1-x) (d+e+f+g+h+i)+3 \log (2-x) (d+2 e+4 (f+2 g+4 h+8 i))+24 \log (x+1) (d-e+f-g+h-i)+\log (x+2) (-19 d+26 e-28 f+8 g+80 h-352 i)+144 i x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 221, normalized size = 1.8 \begin{align*} -{\frac{19\,\ln \left ( 2+x \right ) d}{144}}+{\frac{13\,\ln \left ( 2+x \right ) e}{72}}+{\frac{\ln \left ( 1+x \right ) d}{6}}-{\frac{\ln \left ( 1+x \right ) e}{6}}+{\frac{\ln \left ( x-2 \right ) d}{48}}+{\frac{\ln \left ( x-2 \right ) e}{24}}-{\frac{\ln \left ( x-1 \right ) d}{18}}-{\frac{\ln \left ( x-1 \right ) e}{18}}-{\frac{8\,i}{6+3\,x}}+{\frac{4\,h}{6+3\,x}}-{\frac{2\,g}{6+3\,x}}+{\frac{d}{24+12\,x}}-{\frac{e}{12+6\,x}}+{\frac{f}{6+3\,x}}+{\frac{2\,\ln \left ( x-2 \right ) i}{3}}-{\frac{\ln \left ( x-1 \right ) i}{18}}-{\frac{22\,\ln \left ( 2+x \right ) i}{9}}-{\frac{\ln \left ( 1+x \right ) i}{6}}+{\frac{\ln \left ( 2+x \right ) g}{18}}-{\frac{\ln \left ( 1+x \right ) g}{6}}+{\frac{\ln \left ( x-2 \right ) g}{6}}-{\frac{\ln \left ( x-1 \right ) g}{18}}+{\frac{5\,\ln \left ( 2+x \right ) h}{9}}+{\frac{\ln \left ( 1+x \right ) h}{6}}+{\frac{\ln \left ( x-2 \right ) h}{3}}-{\frac{\ln \left ( x-1 \right ) h}{18}}+{\frac{\ln \left ( x-2 \right ) f}{12}}-{\frac{\ln \left ( x-1 \right ) f}{18}}-{\frac{7\,\ln \left ( 2+x \right ) f}{36}}+{\frac{\ln \left ( 1+x \right ) f}{6}}+ix \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.952286, size = 146, normalized size = 1.2 \begin{align*} i x - \frac{1}{144} \,{\left (19 \, d - 26 \, e + 28 \, f - 8 \, g - 80 \, h + 352 \, i\right )} \log \left (x + 2\right ) + \frac{1}{6} \,{\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) - \frac{1}{18} \,{\left (d + e + f + g + h + i\right )} \log \left (x - 1\right ) + \frac{1}{48} \,{\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left (x - 2\right ) + \frac{d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, i}{12 \,{\left (x + 2\right )}} \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 [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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Giac [A] time = 1.08429, size = 158, normalized size = 1.3 \begin{align*} i x - \frac{1}{144} \,{\left (19 \, d + 28 \, f - 8 \, g - 80 \, h + 352 \, i - 26 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac{1}{6} \,{\left (d + f - g + h - i - e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{18} \,{\left (d + f + g + h + i + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac{1}{48} \,{\left (d + 4 \, f + 8 \, g + 16 \, h + 32 \, i + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac{d + 4 \, f - 8 \, g + 16 \, h - 32 \, i - 2 \, e}{12 \,{\left (x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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