Optimal. Leaf size=49 \[ \frac{c \log \left (b+c x^2\right )}{b^3}-\frac{2 c \log (x)}{b^3}-\frac{c}{2 b^2 \left (b+c x^2\right )}-\frac{1}{2 b^2 x^2} \]
[Out]
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Rubi [A] time = 0.0922975, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{c \log \left (b+c x^2\right )}{b^3}-\frac{2 c \log (x)}{b^3}-\frac{c}{2 b^2 \left (b+c x^2\right )}-\frac{1}{2 b^2 x^2} \]
Antiderivative was successfully verified.
[In] Int[x/(b*x^2 + c*x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 14.7485, size = 46, normalized size = 0.94 \[ - \frac{c}{2 b^{2} \left (b + c x^{2}\right )} - \frac{1}{2 b^{2} x^{2}} - \frac{c \log{\left (x^{2} \right )}}{b^{3}} + \frac{c \log{\left (b + c x^{2} \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(c*x**4+b*x**2)**2,x)
[Out]
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Mathematica [A] time = 0.061829, size = 41, normalized size = 0.84 \[ -\frac{b \left (\frac{c}{b+c x^2}+\frac{1}{x^2}\right )-2 c \log \left (b+c x^2\right )+4 c \log (x)}{2 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[x/(b*x^2 + c*x^4)^2,x]
[Out]
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Maple [A] time = 0.019, size = 46, normalized size = 0.9 \[ -{\frac{1}{2\,{b}^{2}{x}^{2}}}-{\frac{c}{2\,{b}^{2} \left ( c{x}^{2}+b \right ) }}-2\,{\frac{c\ln \left ( x \right ) }{{b}^{3}}}+{\frac{c\ln \left ( c{x}^{2}+b \right ) }{{b}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(c*x^4+b*x^2)^2,x)
[Out]
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Maxima [A] time = 0.702168, size = 70, normalized size = 1.43 \[ -\frac{2 \, c x^{2} + b}{2 \,{\left (b^{2} c x^{4} + b^{3} x^{2}\right )}} + \frac{c \log \left (c x^{2} + b\right )}{b^{3}} - \frac{c \log \left (x^{2}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(c*x^4 + b*x^2)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.25666, size = 99, normalized size = 2.02 \[ -\frac{2 \, b c x^{2} + b^{2} - 2 \,{\left (c^{2} x^{4} + b c x^{2}\right )} \log \left (c x^{2} + b\right ) + 4 \,{\left (c^{2} x^{4} + b c x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{3} c x^{4} + b^{4} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(c*x^4 + b*x^2)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.03763, size = 49, normalized size = 1. \[ - \frac{b + 2 c x^{2}}{2 b^{3} x^{2} + 2 b^{2} c x^{4}} - \frac{2 c \log{\left (x \right )}}{b^{3}} + \frac{c \log{\left (\frac{b}{c} + x^{2} \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(c*x**4+b*x**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.271084, size = 68, normalized size = 1.39 \[ \frac{c{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{b^{3}} - \frac{2 \, c{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} - \frac{2 \, c x^{2} + b}{2 \,{\left (c x^{4} + b x^{2}\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(c*x^4 + b*x^2)^2,x, algorithm="giac")
[Out]