Optimal. Leaf size=73 \[ -\frac {a+b x^2}{4 b \log ^2\left (c \left (a+b x^2\right )\right )}-\frac {a+b x^2}{4 b \log \left (c \left (a+b x^2\right )\right )}+\frac {\text {li}\left (c \left (a+b x^2\right )\right )}{4 b c} \]
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Rubi [A]
time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2504, 2436,
2334, 2335} \begin {gather*} \frac {\text {li}\left (c \left (b x^2+a\right )\right )}{4 b c}-\frac {a+b x^2}{4 b \log ^2\left (c \left (a+b x^2\right )\right )}-\frac {a+b x^2}{4 b \log \left (c \left (a+b x^2\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2334
Rule 2335
Rule 2436
Rule 2504
Rubi steps
\begin {align*} \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\log ^3(c (a+b x))} \, dx,x,x^2\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{\log ^3(c x)} \, dx,x,a+b x^2\right )}{2 b}\\ &=-\frac {a+b x^2}{4 b \log ^2\left (c \left (a+b x^2\right )\right )}+\frac {\text {Subst}\left (\int \frac {1}{\log ^2(c x)} \, dx,x,a+b x^2\right )}{4 b}\\ &=-\frac {a+b x^2}{4 b \log ^2\left (c \left (a+b x^2\right )\right )}-\frac {a+b x^2}{4 b \log \left (c \left (a+b x^2\right )\right )}+\frac {\text {Subst}\left (\int \frac {1}{\log (c x)} \, dx,x,a+b x^2\right )}{4 b}\\ &=-\frac {a+b x^2}{4 b \log ^2\left (c \left (a+b x^2\right )\right )}-\frac {a+b x^2}{4 b \log \left (c \left (a+b x^2\right )\right )}+\frac {\text {li}\left (c \left (a+b x^2\right )\right )}{4 b c}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 56, normalized size = 0.77 \begin {gather*} \frac {\text {Ei}\left (\log \left (c \left (a+b x^2\right )\right )\right )-\frac {c \left (a+b x^2\right ) \left (1+\log \left (c \left (a+b x^2\right )\right )\right )}{\log ^2\left (c \left (a+b x^2\right )\right )}}{4 b c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 70, normalized size = 0.96
method | result | size |
derivativedivides | \(\frac {-\frac {c \left (b \,x^{2}+a \right )}{2 \ln \left (c \left (b \,x^{2}+a \right )\right )^{2}}-\frac {c \left (b \,x^{2}+a \right )}{2 \ln \left (c \left (b \,x^{2}+a \right )\right )}-\frac {\expIntegral \left (1, -\ln \left (c \left (b \,x^{2}+a \right )\right )\right )}{2}}{2 b c}\) | \(70\) |
default | \(\frac {-\frac {c \left (b \,x^{2}+a \right )}{2 \ln \left (c \left (b \,x^{2}+a \right )\right )^{2}}-\frac {c \left (b \,x^{2}+a \right )}{2 \ln \left (c \left (b \,x^{2}+a \right )\right )}-\frac {\expIntegral \left (1, -\ln \left (c \left (b \,x^{2}+a \right )\right )\right )}{2}}{2 b c}\) | \(70\) |
risch | \(-\frac {\ln \left (c \left (b \,x^{2}+a \right )\right ) b \,x^{2}+b \,x^{2}+\ln \left (c \left (b \,x^{2}+a \right )\right ) a +a}{4 b \ln \left (c \left (b \,x^{2}+a \right )\right )^{2}}-\frac {\expIntegral \left (1, -\ln \left (c \left (b \,x^{2}+a \right )\right )\right )}{4 b c}\) | \(75\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 79, normalized size = 1.08 \begin {gather*} -\frac {b c x^{2} - \log \left (b c x^{2} + a c\right )^{2} \operatorname {log\_integral}\left (b c x^{2} + a c\right ) + a c + {\left (b c x^{2} + a c\right )} \log \left (b c x^{2} + a c\right )}{4 \, b c \log \left (b c x^{2} + a c\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.95, size = 70, normalized size = 0.96 \begin {gather*} \frac {\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}}{2} + \frac {- a - b x^{2} + \left (- a - b x^{2}\right ) \log {\left (c \left (a + b x^{2}\right ) \right )}}{4 b \log {\left (c \left (a + b x^{2}\right ) \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.05, size = 71, normalized size = 0.97 \begin {gather*} -\frac {\frac {b c x^{2} + a c}{\log \left (b c x^{2} + a c\right )} + \frac {b c x^{2} + a c}{\log \left (b c x^{2} + a c\right )^{2}} - {\rm Ei}\left (\log \left (b c x^{2} + a c\right )\right )}{4 \, b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.45, size = 74, normalized size = 1.01 \begin {gather*} \frac {\mathrm {logint}\left (c\,\left (b\,x^2+a\right )\right )}{4\,b\,c}-\frac {\frac {a\,c}{4}+\ln \left (c\,\left (b\,x^2+a\right )\right )\,\left (\frac {b\,c\,x^2}{4}+\frac {a\,c}{4}\right )+\frac {b\,c\,x^2}{4}}{b\,c\,{\ln \left (c\,\left (b\,x^2+a\right )\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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