Optimal. Leaf size=49 \[ \frac {b \log \left (a+b x^2\right )}{a^3}-\frac {2 b \log (x)}{a^3}-\frac {b}{2 a^2 \left (a+b x^2\right )}-\frac {1}{2 a^2 x^2} \]
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Rubi [A] time = 0.05, antiderivative size = 49, 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 {b}{2 a^2 \left (a+b x^2\right )}+\frac {b \log \left (a+b x^2\right )}{a^3}-\frac {2 b \log (x)}{a^3}-\frac {1}{2 a^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx &=b^2 \int \frac {1}{x^3 \left (a b+b^2 x^2\right )^2} \, dx\\ &=\frac {1}{2} b^2 \operatorname {Subst}\left (\int \frac {1}{x^2 \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^2}-\frac {2}{a^3 b x}+\frac {1}{a^2 (a+b x)^2}+\frac {2}{a^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 a^2 x^2}-\frac {b}{2 a^2 \left (a+b x^2\right )}-\frac {2 b \log (x)}{a^3}+\frac {b \log \left (a+b x^2\right )}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 41, normalized size = 0.84 \begin {gather*} -\frac {a \left (\frac {b}{a+b x^2}+\frac {1}{x^2}\right )-2 b \log \left (a+b x^2\right )+4 b \log (x)}{2 a^3} \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^3 \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 = 1.01, size = 73, normalized size = 1.49 \begin {gather*} -\frac {2 \, a b x^{2} + a^{2} - 2 \, {\left (b^{2} x^{4} + a b x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \, {\left (b^{2} x^{4} + a b x^{2}\right )} \log \relax (x)}{2 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 51, normalized size = 1.04 \begin {gather*} -\frac {b \log \left (x^{2}\right )}{a^{3}} + \frac {b \log \left ({\left | b x^{2} + a \right |}\right )}{a^{3}} - \frac {2 \, b x^{2} + a}{2 \, {\left (b x^{4} + a x^{2}\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 = 46, normalized size = 0.94 \begin {gather*} -\frac {b}{2 \left (b \,x^{2}+a \right ) a^{2}}-\frac {2 b \ln \relax (x )}{a^{3}}+\frac {b \ln \left (b \,x^{2}+a \right )}{a^{3}}-\frac {1}{2 a^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 52, normalized size = 1.06 \begin {gather*} -\frac {2 \, b x^{2} + a}{2 \, {\left (a^{2} b x^{4} + a^{3} x^{2}\right )}} + \frac {b \log \left (b x^{2} + a\right )}{a^{3}} - \frac {b \log \left (x^{2}\right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 51, normalized size = 1.04 \begin {gather*} \frac {b\,\ln \left (b\,x^2+a\right )}{a^3}-\frac {\frac {1}{2\,a}+\frac {b\,x^2}{a^2}}{b\,x^4+a\,x^2}-\frac {2\,b\,\ln \relax (x)}{a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.39, size = 51, normalized size = 1.04 \begin {gather*} \frac {- a - 2 b x^{2}}{2 a^{3} x^{2} + 2 a^{2} b x^{4}} - \frac {2 b \log {\relax (x )}}{a^{3}} + \frac {b \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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