Optimal. Leaf size=76 \[ -\frac{1}{6} \tanh ^{-1}\left (\frac{x}{2}\right ) (d+4 f+16 h)+\frac{1}{3} \tanh ^{-1}(x) (d+f+h)-\frac{1}{6} \log \left (1-x^2\right ) (e+g+i)+\frac{1}{6} \log \left (4-x^2\right ) (e+4 g+16 i)+h x+\frac{i x^2}{2} \]
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Rubi [A] time = 0.191523, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {1673, 1676, 1166, 207, 1663, 1657, 632, 31} \[ -\frac{1}{6} \tanh ^{-1}\left (\frac{x}{2}\right ) (d+4 f+16 h)+\frac{1}{3} \tanh ^{-1}(x) (d+f+h)-\frac{1}{6} \log \left (1-x^2\right ) (e+g+i)+\frac{1}{6} \log \left (4-x^2\right ) (e+4 g+16 i)+h x+\frac{i x^2}{2} \]
Antiderivative was successfully verified.
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Rule 1673
Rule 1676
Rule 1166
Rule 207
Rule 1663
Rule 1657
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{d+e x+f x^2+g x^3+h x^4+14 x^5}{4-5 x^2+x^4} \, dx &=\int \frac{x \left (e+g x^2+14 x^4\right )}{4-5 x^2+x^4} \, dx+\int \frac{d+f x^2+h x^4}{4-5 x^2+x^4} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{e+g x+14 x^2}{4-5 x+x^2} \, dx,x,x^2\right )+\int \left (h+\frac{d-4 h+(f+5 h) x^2}{4-5 x^2+x^4}\right ) \, dx\\ &=h x+\frac{1}{2} \operatorname{Subst}\left (\int \left (14-\frac{56-e-(70+g) x}{4-5 x+x^2}\right ) \, dx,x,x^2\right )+\int \frac{d-4 h+(f+5 h) x^2}{4-5 x^2+x^4} \, dx\\ &=h x+7 x^2-\frac{1}{2} \operatorname{Subst}\left (\int \frac{56-e-(70+g) x}{4-5 x+x^2} \, dx,x,x^2\right )-\frac{1}{3} (d+f+h) \int \frac{1}{-1+x^2} \, dx+\frac{1}{3} (d+4 f+16 h) \int \frac{1}{-4+x^2} \, dx\\ &=h x+7 x^2-\frac{1}{6} (d+4 f+16 h) \tanh ^{-1}\left (\frac{x}{2}\right )+\frac{1}{3} (d+f+h) \tanh ^{-1}(x)-\frac{1}{6} (-224-e-4 g) \operatorname{Subst}\left (\int \frac{1}{-4+x} \, dx,x,x^2\right )-\frac{1}{6} (14+e+g) \operatorname{Subst}\left (\int \frac{1}{-1+x} \, dx,x,x^2\right )\\ &=h x+7 x^2-\frac{1}{6} (d+4 f+16 h) \tanh ^{-1}\left (\frac{x}{2}\right )+\frac{1}{3} (d+f+h) \tanh ^{-1}(x)-\frac{1}{6} (14+e+g) \log \left (1-x^2\right )+\frac{1}{6} (224+e+4 g) \log \left (4-x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0651369, size = 98, normalized size = 1.29 \[ \frac{1}{12} \left (-2 \log (1-x) (d+e+f+g+h+i)+\log (2-x) (d+2 e+4 (f+2 g+4 h+8 i))+2 \log (x+1) (d-e+f-g+h-i)-\log (x+2) (d-2 (e-2 f+4 g-8 h+16 i))+12 h x+6 i x^2\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 179, normalized size = 2.4 \begin{align*} -{\frac{\ln \left ( 2+x \right ) d}{12}}+{\frac{\ln \left ( 2+x \right ) e}{6}}+{\frac{\ln \left ( 1+x \right ) d}{6}}-{\frac{\ln \left ( 1+x \right ) e}{6}}+{\frac{\ln \left ( x-2 \right ) d}{12}}+{\frac{\ln \left ( x-2 \right ) e}{6}}-{\frac{\ln \left ( x-1 \right ) d}{6}}-{\frac{\ln \left ( x-1 \right ) e}{6}}+{\frac{8\,\ln \left ( x-2 \right ) i}{3}}-{\frac{\ln \left ( x-1 \right ) i}{6}}+{\frac{8\,\ln \left ( 2+x \right ) i}{3}}-{\frac{\ln \left ( 1+x \right ) i}{6}}+{\frac{2\,\ln \left ( 2+x \right ) g}{3}}-{\frac{\ln \left ( 1+x \right ) g}{6}}+{\frac{2\,\ln \left ( x-2 \right ) g}{3}}-{\frac{\ln \left ( x-1 \right ) g}{6}}-{\frac{4\,\ln \left ( 2+x \right ) h}{3}}+{\frac{\ln \left ( 1+x \right ) h}{6}}+{\frac{4\,\ln \left ( x-2 \right ) h}{3}}-{\frac{\ln \left ( x-1 \right ) h}{6}}+{\frac{\ln \left ( x-2 \right ) f}{3}}-{\frac{\ln \left ( x-1 \right ) f}{6}}-{\frac{\ln \left ( 2+x \right ) f}{3}}+{\frac{\ln \left ( 1+x \right ) f}{6}}+{\frac{i{x}^{2}}{2}}+hx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.953738, size = 119, normalized size = 1.57 \begin{align*} \frac{1}{2} \, i x^{2} + h x - \frac{1}{12} \,{\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, 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}{6} \,{\left (d + e + f + g + h + i\right )} \log \left (x - 1\right ) + \frac{1}{12} \,{\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left (x - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 55.1404, size = 279, normalized size = 3.67 \begin{align*} \frac{1}{2} \, i x^{2} + h x - \frac{1}{12} \,{\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, 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}{6} \,{\left (d + e + f + g + h + i\right )} \log \left (x - 1\right ) + \frac{1}{12} \,{\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left (x - 2\right ) \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 = 1.10385, size = 130, normalized size = 1.71 \begin{align*} \frac{1}{2} \, i x^{2} + h x - \frac{1}{12} \,{\left (d + 4 \, f - 8 \, g + 16 \, h - 32 \, i - 2 \, 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}{6} \,{\left (d + f + g + h + i + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac{1}{12} \,{\left (d + 4 \, f + 8 \, g + 16 \, h + 32 \, i + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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