Optimal. Leaf size=38 \[ -\frac {\log \left (a+b x^2\right )}{2 a^2}+\frac {\log (x)}{a^2}+\frac {1}{2 a \left (a+b x^2\right )} \]
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Rubi [A] time = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 266, 44} \begin {gather*} -\frac {\log \left (a+b x^2\right )}{2 a^2}+\frac {\log (x)}{a^2}+\frac {1}{2 a \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx &=b^2 \int \frac {1}{x \left (a b+b^2 x^2\right )^2} \, dx\\ &=\frac {1}{2} b^2 \operatorname {Subst}\left (\int \frac {1}{x \left (a b+b^2 x\right )^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} b^2 \operatorname {Subst}\left (\int \left (\frac {1}{a^2 b^2 x}-\frac {1}{a b (a+b x)^2}-\frac {1}{a^2 b (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2 a \left (a+b x^2\right )}+\frac {\log (x)}{a^2}-\frac {\log \left (a+b x^2\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.87 \begin {gather*} \frac {\frac {a}{a+b x^2}-\log \left (a+b x^2\right )+2 \log (x)}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.91, size = 47, normalized size = 1.24 \begin {gather*} -\frac {{\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (b x^{2} + a\right )} \log \relax (x) - a}{2 \, {\left (a^{2} b x^{2} + a^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 47, normalized size = 1.24 \begin {gather*} \frac {\log \left (x^{2}\right )}{2 \, a^{2}} - \frac {\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2}} + \frac {b x^{2} + 2 \, a}{2 \, {\left (b x^{2} + a\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 35, normalized size = 0.92 \begin {gather*} \frac {1}{2 \left (b \,x^{2}+a \right ) a}+\frac {\ln \relax (x )}{a^{2}}-\frac {\ln \left (b \,x^{2}+a \right )}{2 a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 37, normalized size = 0.97 \begin {gather*} \frac {1}{2 \, {\left (a b x^{2} + a^{2}\right )}} - \frac {\log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac {\log \left (x^{2}\right )}{2 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.39, size = 34, normalized size = 0.89 \begin {gather*} \frac {\ln \relax (x)}{a^2}+\frac {1}{2\,a\,\left (b\,x^2+a\right )}-\frac {\ln \left (b\,x^2+a\right )}{2\,a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 34, normalized size = 0.89 \begin {gather*} \frac {1}{2 a^{2} + 2 a b x^{2}} + \frac {\log {\relax (x )}}{a^{2}} - \frac {\log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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