Optimal. Leaf size=93 \[ -\frac{(2 a e+3 c d) \log (a-c x)}{4 a^5}+\frac{(3 c d-2 a e) \log (a+c x)}{4 a^5}-\frac{3 d}{2 a^4 x}+\frac{e \log (x)}{a^4}+\frac{d+e x}{2 a^2 x \left (a^2-c^2 x^2\right )} \]
[Out]
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Rubi [A] time = 0.220097, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{(2 a e+3 c d) \log (a-c x)}{4 a^5}+\frac{(3 c d-2 a e) \log (a+c x)}{4 a^5}-\frac{3 d}{2 a^4 x}+\frac{e \log (x)}{a^4}+\frac{d+e x}{2 a^2 x \left (a^2-c^2 x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(x^2*(a^2 - c^2*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 37.3784, size = 83, normalized size = 0.89 \[ \frac{d + e x}{2 a^{2} x \left (a^{2} - c^{2} x^{2}\right )} - \frac{3 d}{2 a^{4} x} + \frac{e \log{\left (x \right )}}{a^{4}} - \frac{\left (a e - \frac{3 c d}{2}\right ) \log{\left (a + c x \right )}}{2 a^{5}} - \frac{\left (a e + \frac{3 c d}{2}\right ) \log{\left (a - c x \right )}}{2 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/x**2/(-c**2*x**2+a**2)**2,x)
[Out]
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Mathematica [A] time = 0.193328, size = 77, normalized size = 0.83 \[ \frac{-a e \log \left (a^2-c^2 x^2\right )+\frac{a^3 e+a c^2 d x}{a^2-c^2 x^2}+3 c d \tanh ^{-1}\left (\frac{c x}{a}\right )-\frac{2 a d}{x}+2 a e \log (x)}{2 a^5} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(x^2*(a^2 - c^2*x^2)^2),x]
[Out]
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Maple [A] time = 0.023, size = 130, normalized size = 1.4 \[ -{\frac{d}{{a}^{4}x}}+{\frac{e\ln \left ( x \right ) }{{a}^{4}}}-{\frac{\ln \left ( cx+a \right ) e}{2\,{a}^{4}}}+{\frac{3\,\ln \left ( cx+a \right ) cd}{4\,{a}^{5}}}+{\frac{e}{4\,{a}^{3} \left ( cx+a \right ) }}-{\frac{cd}{4\,{a}^{4} \left ( cx+a \right ) }}-{\frac{\ln \left ( cx-a \right ) e}{2\,{a}^{4}}}-{\frac{3\,\ln \left ( cx-a \right ) cd}{4\,{a}^{5}}}-{\frac{e}{4\,{a}^{3} \left ( cx-a \right ) }}-{\frac{cd}{4\,{a}^{4} \left ( cx-a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/x^2/(-c^2*x^2+a^2)^2,x)
[Out]
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Maxima [A] time = 0.697689, size = 126, normalized size = 1.35 \[ -\frac{3 \, c^{2} d x^{2} + a^{2} e x - 2 \, a^{2} d}{2 \,{\left (a^{4} c^{2} x^{3} - a^{6} x\right )}} + \frac{e \log \left (x\right )}{a^{4}} + \frac{{\left (3 \, c d - 2 \, a e\right )} \log \left (c x + a\right )}{4 \, a^{5}} - \frac{{\left (3 \, c d + 2 \, a e\right )} \log \left (c x - a\right )}{4 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/((c^2*x^2 - a^2)^2*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.294118, size = 209, normalized size = 2.25 \[ -\frac{6 \, a c^{2} d x^{2} + 2 \, a^{3} e x - 4 \, a^{3} d -{\left ({\left (3 \, c^{3} d - 2 \, a c^{2} e\right )} x^{3} -{\left (3 \, a^{2} c d - 2 \, a^{3} e\right )} x\right )} \log \left (c x + a\right ) +{\left ({\left (3 \, c^{3} d + 2 \, a c^{2} e\right )} x^{3} -{\left (3 \, a^{2} c d + 2 \, a^{3} e\right )} x\right )} \log \left (c x - a\right ) - 4 \,{\left (a c^{2} e x^{3} - a^{3} e x\right )} \log \left (x\right )}{4 \,{\left (a^{5} c^{2} x^{3} - a^{7} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/((c^2*x^2 - a^2)^2*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.08363, size = 291, normalized size = 3.13 \[ - \frac{- 2 a^{2} d + a^{2} e x + 3 c^{2} d x^{2}}{- 2 a^{6} x + 2 a^{4} c^{2} x^{3}} + \frac{e \log{\left (x \right )}}{a^{4}} - \frac{\left (2 a e - 3 c d\right ) \log{\left (x + \frac{16 a^{4} e^{3} - 4 a^{3} e^{2} \left (2 a e - 3 c d\right ) + 12 a^{2} c^{2} d^{2} e - 2 a^{2} e \left (2 a e - 3 c d\right )^{2} + 3 a c^{2} d^{2} \left (2 a e - 3 c d\right )}{36 a^{2} c^{2} d e^{2} - 9 c^{4} d^{3}} \right )}}{4 a^{5}} - \frac{\left (2 a e + 3 c d\right ) \log{\left (x + \frac{16 a^{4} e^{3} - 4 a^{3} e^{2} \left (2 a e + 3 c d\right ) + 12 a^{2} c^{2} d^{2} e - 2 a^{2} e \left (2 a e + 3 c d\right )^{2} + 3 a c^{2} d^{2} \left (2 a e + 3 c d\right )}{36 a^{2} c^{2} d e^{2} - 9 c^{4} d^{3}} \right )}}{4 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/x**2/(-c**2*x**2+a**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.286393, size = 151, normalized size = 1.62 \[ \frac{e{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} - \frac{3 \, c^{2} d x^{2} + a^{2} x e - 2 \, a^{2} d}{2 \,{\left (c^{2} x^{3} - a^{2} x\right )} a^{4}} + \frac{{\left (3 \, c^{2} d - 2 \, a c e\right )}{\rm ln}\left ({\left | c x + a \right |}\right )}{4 \, a^{5} c} - \frac{{\left (3 \, c^{2} d + 2 \, a c e\right )}{\rm ln}\left ({\left | c x - a \right |}\right )}{4 \, a^{5} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/((c^2*x^2 - a^2)^2*x^2),x, algorithm="giac")
[Out]