Optimal. Leaf size=70 \[ \frac{1}{8 a^3 b (a-b x)}-\frac{1}{4 a^3 b (a+b x)}-\frac{1}{8 a^2 b (a+b x)^2}+\frac{3 \tanh ^{-1}\left (\frac{b x}{a}\right )}{8 a^4 b} \]
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Rubi [A] time = 0.0460265, 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, Rules used = {627, 44, 208} \[ \frac{1}{8 a^3 b (a-b x)}-\frac{1}{4 a^3 b (a+b x)}-\frac{1}{8 a^2 b (a+b x)^2}+\frac{3 \tanh ^{-1}\left (\frac{b x}{a}\right )}{8 a^4 b} \]
Antiderivative was successfully verified.
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Rule 627
Rule 44
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(a+b x) \left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac{1}{(a-b x)^2 (a+b x)^3} \, dx\\ &=\int \left (\frac{1}{8 a^3 (a-b x)^2}+\frac{1}{4 a^2 (a+b x)^3}+\frac{1}{4 a^3 (a+b x)^2}+\frac{3}{8 a^3 \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=\frac{1}{8 a^3 b (a-b x)}-\frac{1}{8 a^2 b (a+b x)^2}-\frac{1}{4 a^3 b (a+b x)}+\frac{3 \int \frac{1}{a^2-b^2 x^2} \, dx}{8 a^3}\\ &=\frac{1}{8 a^3 b (a-b x)}-\frac{1}{8 a^2 b (a+b x)^2}-\frac{1}{4 a^3 b (a+b x)}+\frac{3 \tanh ^{-1}\left (\frac{b x}{a}\right )}{8 a^4 b}\\ \end{align*}
Mathematica [A] time = 0.0203476, size = 87, normalized size = 1.24 \[ -\frac{1}{8 a^3 b (b x-a)}-\frac{1}{4 a^3 b (a+b x)}-\frac{1}{8 a^2 b (a+b x)^2}-\frac{3 \log (a-b x)}{16 a^4 b}+\frac{3 \log (a+b x)}{16 a^4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 79, normalized size = 1.1 \begin{align*}{\frac{3\,\ln \left ( bx+a \right ) }{16\,{a}^{4}b}}-{\frac{1}{4\,{a}^{3}b \left ( bx+a \right ) }}-{\frac{1}{8\,b{a}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{3\,\ln \left ( bx-a \right ) }{16\,{a}^{4}b}}-{\frac{1}{8\,{a}^{3}b \left ( bx-a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04205, size = 122, normalized size = 1.74 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.7458, size = 269, normalized size = 3.84 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.578813, size = 85, normalized size = 1.21 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24312, size = 107, normalized size = 1.53 \begin{align*} \frac{3 \, \log \left ({\left | b x + a \right |}\right )}{16 \, a^{4} b} - \frac{3 \, \log \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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