Optimal. Leaf size=70 \[ \frac{3 \tanh ^{-1}\left (\frac{b x}{a}\right )}{8 a^4 b}+\frac{1}{8 a^3 b (a-b x)}-\frac{1}{4 a^3 b (a+b x)}-\frac{1}{8 a^2 b (a+b x)^2} \]
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Rubi [A] time = 0.115714, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{3 \tanh ^{-1}\left (\frac{b x}{a}\right )}{8 a^4 b}+\frac{1}{8 a^3 b (a-b x)}-\frac{1}{4 a^3 b (a+b x)}-\frac{1}{8 a^2 b (a+b x)^2} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)*(a^2 - b^2*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 24.6344, size = 56, normalized size = 0.8 \[ - \frac{1}{8 a^{2} b \left (a + b x\right )^{2}} - \frac{1}{4 a^{3} b \left (a + b x\right )} + \frac{1}{8 a^{3} b \left (a - b x\right )} + \frac{3 \operatorname{atanh}{\left (\frac{b x}{a} \right )}}{8 a^{4} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)/(-b**2*x**2+a**2)**2,x)
[Out]
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Mathematica [A] time = 0.0393394, size = 87, normalized size = 1.24 \[ -\frac{3 \log (a-b x)}{16 a^4 b}+\frac{3 \log (a+b x)}{16 a^4 b}-\frac{1}{8 a^3 b (b x-a)}-\frac{1}{4 a^3 b (a+b x)}-\frac{1}{8 a^2 b (a+b x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)*(a^2 - b^2*x^2)^2),x]
[Out]
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Maple [A] time = 0.016, size = 79, normalized size = 1.1 \[ -{\frac{3\,\ln \left ( bx-a \right ) }{16\,{a}^{4}b}}-{\frac{1}{8\,{a}^{3}b \left ( bx-a \right ) }}+{\frac{3\,\ln \left ( bx+a \right ) }{16\,{a}^{4}b}}-{\frac{1}{4\,{a}^{3}b \left ( bx+a \right ) }}-{\frac{1}{8\,{a}^{2}b \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)/(-b^2*x^2+a^2)^2,x)
[Out]
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Maxima [A] time = 0.68271, size = 122, normalized size = 1.74 \[ -\frac{3 \, b^{2} x^{2} + 3 \, a b x - 2 \, a^{2}}{8 \,{\left (a^{3} b^{4} x^{3} + a^{4} b^{3} x^{2} - a^{5} b^{2} x - a^{6} b\right )}} + \frac{3 \, \log \left (b x + a\right )}{16 \, a^{4} b} - \frac{3 \, \log \left (b x - a\right )}{16 \, a^{4} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 - a^2)^2*(b*x + a)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.211504, size = 184, normalized size = 2.63 \[ -\frac{6 \, a b^{2} x^{2} + 6 \, a^{2} b x - 4 \, a^{3} - 3 \,{\left (b^{3} x^{3} + a b^{2} x^{2} - a^{2} b x - a^{3}\right )} \log \left (b x + a\right ) + 3 \,{\left (b^{3} x^{3} + a b^{2} x^{2} - a^{2} b x - a^{3}\right )} \log \left (b x - a\right )}{16 \,{\left (a^{4} b^{4} x^{3} + a^{5} b^{3} x^{2} - a^{6} b^{2} x - a^{7} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 - a^2)^2*(b*x + a)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.24148, size = 85, normalized size = 1.21 \[ - \frac{- 2 a^{2} + 3 a b x + 3 b^{2} x^{2}}{- 8 a^{6} b - 8 a^{5} b^{2} x + 8 a^{4} b^{3} x^{2} + 8 a^{3} b^{4} x^{3}} + \frac{- \frac{3 \log{\left (- \frac{a}{b} + x \right )}}{16} + \frac{3 \log{\left (\frac{a}{b} + x \right )}}{16}}{a^{4} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)/(-b**2*x**2+a**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.21675, size = 107, normalized size = 1.53 \[ \frac{3 \,{\rm ln}\left ({\left | b x + a \right |}\right )}{16 \, a^{4} b} - \frac{3 \,{\rm ln}\left ({\left | b x - a \right |}\right )}{16 \, a^{4} b} - \frac{3 \, a b^{2} x^{2} + 3 \, a^{2} b x - 2 \, a^{3}}{8 \,{\left (b x + a\right )}^{2}{\left (b x - a\right )} a^{4} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 - a^2)^2*(b*x + a)),x, algorithm="giac")
[Out]