Optimal. Leaf size=47 \[ \frac{\log \left (b^2+x^2\right )}{2 \left (a^2-b^2\right )}-\frac{\log \left (a^2+x^2\right )}{2 \left (a^2-b^2\right )} \]
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Rubi [A] time = 0.0197179, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {444, 36, 31} \[ \frac{\log \left (b^2+x^2\right )}{2 \left (a^2-b^2\right )}-\frac{\log \left (a^2+x^2\right )}{2 \left (a^2-b^2\right )} \]
Antiderivative was successfully verified.
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Rule 444
Rule 36
Rule 31
Rubi steps
\begin{align*} \int \frac{x}{\left (a^2+x^2\right ) \left (b^2+x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\left (a^2+x\right ) \left (b^2+x\right )} \, dx,x,x^2\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{a^2+x} \, dx,x,x^2\right )}{2 \left (a^2-b^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{b^2+x} \, dx,x,x^2\right )}{2 \left (a^2-b^2\right )}\\ &=-\frac{\log \left (a^2+x^2\right )}{2 \left (a^2-b^2\right )}+\frac{\log \left (b^2+x^2\right )}{2 \left (a^2-b^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0105171, size = 34, normalized size = 0.72 \[ \frac{\log \left (b^2+x^2\right )-\log \left (a^2+x^2\right )}{2 \left (a^2-b^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 44, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ({a}^{2}+{x}^{2} \right ) }{2\,{a}^{2}-2\,{b}^{2}}}+{\frac{\ln \left ({b}^{2}+{x}^{2} \right ) }{2\,{a}^{2}-2\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.948073, size = 58, normalized size = 1.23 \begin{align*} -\frac{\log \left (a^{2} + x^{2}\right )}{2 \,{\left (a^{2} - b^{2}\right )}} + \frac{\log \left (b^{2} + x^{2}\right )}{2 \,{\left (a^{2} - b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01894, size = 70, normalized size = 1.49 \begin{align*} -\frac{\log \left (a^{2} + x^{2}\right ) - \log \left (b^{2} + x^{2}\right )}{2 \,{\left (a^{2} - b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.540466, size = 121, normalized size = 2.57 \begin{align*} \frac{\log{\left (- \frac{a^{4}}{2 \left (a - b\right ) \left (a + b\right )} + \frac{a^{2} b^{2}}{\left (a - b\right ) \left (a + b\right )} + \frac{a^{2}}{2} - \frac{b^{4}}{2 \left (a - b\right ) \left (a + b\right )} + \frac{b^{2}}{2} + x^{2} \right )}}{2 \left (a - b\right ) \left (a + b\right )} - \frac{\log{\left (\frac{a^{4}}{2 \left (a - b\right ) \left (a + b\right )} - \frac{a^{2} b^{2}}{\left (a - b\right ) \left (a + b\right )} + \frac{a^{2}}{2} + \frac{b^{4}}{2 \left (a - b\right ) \left (a + b\right )} + \frac{b^{2}}{2} + x^{2} \right )}}{2 \left (a - b\right ) \left (a + b\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14083, size = 88, normalized size = 1.87 \begin{align*} \frac{\log \left (\frac{{\left | a^{2} + b^{2} + 2 \, x^{2} -{\left | a^{2} - b^{2} \right |} \right |}}{a^{2} + b^{2} + 2 \, x^{2} +{\left | a^{2} - b^{2} \right |}}\right )}{2 \,{\left | a^{2} - b^{2} \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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