Optimal. Leaf size=47 \[ \frac {\text {li}\left (c \left (b x^2+a\right )\right )}{2 b c}-\frac {a+b x^2}{2 b \log \left (c \left (a+b x^2\right )\right )} \]
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Rubi [A] time = 0.04, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2454, 2389, 2297, 2298} \[ \frac {\text {li}\left (c \left (b x^2+a\right )\right )}{2 b c}-\frac {a+b x^2}{2 b \log \left (c \left (a+b x^2\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2297
Rule 2298
Rule 2389
Rule 2454
Rubi steps
\begin {align*} \int \frac {x}{\log ^2\left (c \left (a+b x^2\right )\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\log ^2(c (a+b x))} \, dx,x,x^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{\log ^2(c x)} \, dx,x,a+b x^2\right )}{2 b}\\ &=-\frac {a+b x^2}{2 b \log \left (c \left (a+b x^2\right )\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{\log (c x)} \, dx,x,a+b x^2\right )}{2 b}\\ &=-\frac {a+b x^2}{2 b \log \left (c \left (a+b x^2\right )\right )}+\frac {\text {li}\left (c \left (a+b x^2\right )\right )}{2 b c}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 0.91 \[ \frac {\frac {\text {li}\left (c \left (b x^2+a\right )\right )}{c}-\frac {a+b x^2}{\log \left (c \left (a+b x^2\right )\right )}}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 55, normalized size = 1.17 \[ -\frac {b c x^{2} + a c - \log \left (b c x^{2} + a c\right ) \operatorname {log\_integral}\left (b c x^{2} + a c\right )}{2 \, b c \log \left (b c x^{2} + a c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 45, normalized size = 0.96 \[ -\frac {\frac {b c x^{2} + a c}{\log \left ({\left (b x^{2} + a\right )} c\right )} - {\rm Ei}\left (\log \left ({\left (b x^{2} + a\right )} c\right )\right )}{2 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 59, normalized size = 1.26 \[ -\frac {x^{2}}{2 \ln \left (\left (b \,x^{2}+a \right ) c \right )}-\frac {a}{2 b \ln \left (\left (b \,x^{2}+a \right ) c \right )}-\frac {\Ei \left (1, -\ln \left (\left (b \,x^{2}+a \right ) c \right )\right )}{2 b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b x^{2} + a}{2 \, {\left (b \log \left (b x^{2} + a\right ) + b \log \relax (c)\right )}} + \int \frac {x}{\log \left (b x^{2} + a\right ) + \log \relax (c)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 46, normalized size = 0.98 \[ \frac {\mathrm {logint}\left (c\,\left (b\,x^2+a\right )\right )}{2\,b\,c}-\frac {\frac {b\,x^2}{2}+\frac {a}{2}}{b\,\ln \left (c\,\left (b\,x^2+a\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.16, size = 49, normalized size = 1.04 \[ \begin {cases} \frac {x^{2}}{2 \log {\left (a c \right )}} & \text {for}\: b = 0 \\0 & \text {for}\: c = 0 \\\frac {\operatorname {Ei}{\left (\log {\left (a c + b c x^{2} \right )} \right )}}{2 b c} & \text {otherwise} \end {cases} + \frac {- a - b x^{2}}{2 b \log {\left (c \left (a + b x^{2}\right ) \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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