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 = 0.364424, 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 \[ -\frac{1}{6} \tanh ^{-1}\left (\frac{x}{2}\right ) (d+4 f+16 h)+\frac{1}{3} \tanh ^{-1}(x) (d+f+h)-\frac{1}{6} \log \left (1-x^2\right ) (e+g+i)+\frac{1}{6} \log \left (4-x^2\right ) (e+4 g+16 i)+h x+\frac{i x^2}{2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(4 - 5*x^2 + x^4),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \left (\frac{d}{6} + \frac{2 f}{3} + \frac{8 h}{3}\right ) \operatorname{atanh}{\left (\frac{x}{2} \right )} + \left (\frac{d}{3} + \frac{f}{3} + \frac{h}{3}\right ) \operatorname{atanh}{\left (x \right )} - \left (\frac{e}{6} + \frac{g}{6} + \frac{i}{6}\right ) \log{\left (- x^{2} + 1 \right )} + \left (\frac{e}{6} + \frac{2 g}{3} + \frac{8 i}{3}\right ) \log{\left (- x^{2} + 4 \right )} + \int h\, dx + \frac{\int ^{x^{2}} i\, dx}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((i*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(x**4-5*x**2+4),x)
[Out]
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Mathematica [A] time = 0.134652, 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.
[In] Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(4 - 5*x^2 + x^4),x]
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Maple [B] time = 0.014, size = 179, normalized size = 2.4 \[ hx+{\frac{\ln \left ( 1+x \right ) d}{6}}-{\frac{\ln \left ( 1+x \right ) e}{6}}-{\frac{\ln \left ( -1+x \right ) d}{6}}-{\frac{\ln \left ( -1+x \right ) e}{6}}+{\frac{8\,\ln \left ( x-2 \right ) i}{3}}-{\frac{\ln \left ( 1+x \right ) i}{6}}-{\frac{\ln \left ( -1+x \right ) i}{6}}+{\frac{8\,\ln \left ( 2+x \right ) i}{3}}+{\frac{4\,\ln \left ( x-2 \right ) h}{3}}+{\frac{\ln \left ( 1+x \right ) h}{6}}-{\frac{4\,\ln \left ( 2+x \right ) h}{3}}-{\frac{\ln \left ( -1+x \right ) h}{6}}-{\frac{\ln \left ( 1+x \right ) g}{6}}+{\frac{2\,\ln \left ( x-2 \right ) g}{3}}-{\frac{\ln \left ( -1+x \right ) g}{6}}+{\frac{2\,\ln \left ( 2+x \right ) g}{3}}+{\frac{\ln \left ( x-2 \right ) d}{12}}+{\frac{\ln \left ( x-2 \right ) e}{6}}+{\frac{\ln \left ( 2+x \right ) e}{6}}+{\frac{\ln \left ( x-2 \right ) f}{3}}-{\frac{\ln \left ( 2+x \right ) d}{12}}+{\frac{\ln \left ( 1+x \right ) f}{6}}-{\frac{\ln \left ( -1+x \right ) f}{6}}-{\frac{\ln \left ( 2+x \right ) f}{3}}+{\frac{i{x}^{2}}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4),x)
[Out]
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Maxima [A] time = 0.70752, size = 119, normalized size = 1.57 \[ \frac{1}{2} \, i x^{2} + h x - \frac{1}{12} \,{\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, i\right )} \log \left (x + 2\right ) + \frac{1}{6} \,{\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) - \frac{1}{6} \,{\left (d + e + f + g + h + i\right )} \log \left (x - 1\right ) + \frac{1}{12} \,{\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left (x - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((i*x^5 + h*x^4 + g*x^3 + f*x^2 + e*x + d)/(x^4 - 5*x^2 + 4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 6.91263, size = 119, normalized size = 1.57 \[ \frac{1}{2} \, i x^{2} + h x - \frac{1}{12} \,{\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, i\right )} \log \left (x + 2\right ) + \frac{1}{6} \,{\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) - \frac{1}{6} \,{\left (d + e + f + g + h + i\right )} \log \left (x - 1\right ) + \frac{1}{12} \,{\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left (x - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((i*x^5 + h*x^4 + g*x^3 + f*x^2 + e*x + d)/(x^4 - 5*x^2 + 4),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((i*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(x**4-5*x**2+4),x)
[Out]
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GIAC/XCAS [A] time = 0.301763, size = 130, normalized size = 1.71 \[ \frac{1}{2} \, i x^{2} + h x - \frac{1}{12} \,{\left (d + 4 \, f - 8 \, g + 16 \, h - 32 \, i - 2 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{1}{6} \,{\left (d + f - g + h - i - e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{6} \,{\left (d + f + g + h + i + e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{12} \,{\left (d + 4 \, f + 8 \, g + 16 \, h + 32 \, i + 2 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((i*x^5 + h*x^4 + g*x^3 + f*x^2 + e*x + d)/(x^4 - 5*x^2 + 4),x, algorithm="giac")
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