Integrand size = 16, antiderivative size = 71 \[ \int \frac {x^3}{\log ^2\left (c \left (a+b x^2\right )\right )} \, dx=\frac {\operatorname {ExpIntegralEi}\left (2 \log \left (c \left (a+b x^2\right )\right )\right )}{b^2 c^2}-\frac {x^2 \left (a+b x^2\right )}{2 b \log \left (c \left (a+b x^2\right )\right )}-\frac {a \operatorname {LogIntegral}\left (c \left (a+b x^2\right )\right )}{2 b^2 c} \]
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Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2504, 2447, 2446, 2436, 2335, 2437, 2346, 2209} \[ \int \frac {x^3}{\log ^2\left (c \left (a+b x^2\right )\right )} \, dx=\frac {\operatorname {ExpIntegralEi}\left (2 \log \left (c \left (b x^2+a\right )\right )\right )}{b^2 c^2}-\frac {a \operatorname {LogIntegral}\left (c \left (b x^2+a\right )\right )}{2 b^2 c}-\frac {x^2 \left (a+b x^2\right )}{2 b \log \left (c \left (a+b x^2\right )\right )} \]
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Rule 2209
Rule 2335
Rule 2346
Rule 2436
Rule 2437
Rule 2446
Rule 2447
Rule 2504
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x}{\log ^2(c (a+b x))} \, dx,x,x^2\right ) \\ & = -\frac {x^2 \left (a+b x^2\right )}{2 b \log \left (c \left (a+b x^2\right )\right )}+\frac {a \text {Subst}\left (\int \frac {1}{\log (c (a+b x))} \, dx,x,x^2\right )}{2 b}+\text {Subst}\left (\int \frac {x}{\log (c (a+b x))} \, dx,x,x^2\right ) \\ & = -\frac {x^2 \left (a+b x^2\right )}{2 b \log \left (c \left (a+b x^2\right )\right )}+\frac {a \text {Subst}\left (\int \frac {1}{\log (c x)} \, dx,x,a+b x^2\right )}{2 b^2}+\text {Subst}\left (\int \left (-\frac {a}{b \log (c (a+b x))}+\frac {a+b x}{b \log (c (a+b x))}\right ) \, dx,x,x^2\right ) \\ & = -\frac {x^2 \left (a+b x^2\right )}{2 b \log \left (c \left (a+b x^2\right )\right )}+\frac {a \text {li}\left (c \left (a+b x^2\right )\right )}{2 b^2 c}+\frac {\text {Subst}\left (\int \frac {a+b x}{\log (c (a+b x))} \, dx,x,x^2\right )}{b}-\frac {a \text {Subst}\left (\int \frac {1}{\log (c (a+b x))} \, dx,x,x^2\right )}{b} \\ & = -\frac {x^2 \left (a+b x^2\right )}{2 b \log \left (c \left (a+b x^2\right )\right )}+\frac {a \text {li}\left (c \left (a+b x^2\right )\right )}{2 b^2 c}+\frac {\text {Subst}\left (\int \frac {x}{\log (c x)} \, dx,x,a+b x^2\right )}{b^2}-\frac {a \text {Subst}\left (\int \frac {1}{\log (c x)} \, dx,x,a+b x^2\right )}{b^2} \\ & = -\frac {x^2 \left (a+b x^2\right )}{2 b \log \left (c \left (a+b x^2\right )\right )}-\frac {a \text {li}\left (c \left (a+b x^2\right )\right )}{2 b^2 c}+\frac {\text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log \left (c \left (a+b x^2\right )\right )\right )}{b^2 c^2} \\ & = \frac {\text {Ei}\left (2 \log \left (c \left (a+b x^2\right )\right )\right )}{b^2 c^2}-\frac {x^2 \left (a+b x^2\right )}{2 b \log \left (c \left (a+b x^2\right )\right )}-\frac {a \text {li}\left (c \left (a+b x^2\right )\right )}{2 b^2 c} \\ \end{align*}
\[ \int \frac {x^3}{\log ^2\left (c \left (a+b x^2\right )\right )} \, dx=\int \frac {x^3}{\log ^2\left (c \left (a+b x^2\right )\right )} \, dx \]
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Time = 2.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.04
method | result | size |
risch | \(-\frac {x^{2} \left (b \,x^{2}+a \right )}{2 b \ln \left (c \left (b \,x^{2}+a \right )\right )}+\frac {a \,\operatorname {Ei}_{1}\left (-\ln \left (c \left (b \,x^{2}+a \right )\right )\right )}{2 c \,b^{2}}-\frac {\operatorname {Ei}_{1}\left (-2 \ln \left (c \left (b \,x^{2}+a \right )\right )\right )}{c^{2} b^{2}}\) | \(74\) |
default | \(\frac {-\frac {c^{2} \left (b \,x^{2}+a \right )^{2}}{\ln \left (c \left (b \,x^{2}+a \right )\right )}-2 \,\operatorname {Ei}_{1}\left (-2 \ln \left (c \left (b \,x^{2}+a \right )\right )\right )-c a \left (-\frac {c \left (b \,x^{2}+a \right )}{\ln \left (c \left (b \,x^{2}+a \right )\right )}-\operatorname {Ei}_{1}\left (-\ln \left (c \left (b \,x^{2}+a \right )\right )\right )\right )}{2 c^{2} b^{2}}\) | \(95\) |
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Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.39 \[ \int \frac {x^3}{\log ^2\left (c \left (a+b x^2\right )\right )} \, dx=-\frac {b^{2} c^{2} x^{4} + a b c^{2} x^{2} + {\left (a c \operatorname {log\_integral}\left (b c x^{2} + a c\right ) - 2 \, \operatorname {log\_integral}\left (b^{2} c^{2} x^{4} + 2 \, a b c^{2} x^{2} + a^{2} c^{2}\right )\right )} \log \left (b c x^{2} + a c\right )}{2 \, b^{2} c^{2} \log \left (b c x^{2} + a c\right )} \]
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\[ \int \frac {x^3}{\log ^2\left (c \left (a+b x^2\right )\right )} \, dx=\frac {- a x^{2} - b x^{4}}{2 b \log {\left (c \left (a + b x^{2}\right ) \right )}} + \frac {\int \frac {a x}{\log {\left (a c + b c x^{2} \right )}}\, dx + \int \frac {2 b x^{3}}{\log {\left (a c + b c x^{2} \right )}}\, dx}{b} \]
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\[ \int \frac {x^3}{\log ^2\left (c \left (a+b x^2\right )\right )} \, dx=\int { \frac {x^{3}}{\log \left ({\left (b x^{2} + a\right )} c\right )^{2}} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.41 \[ \int \frac {x^3}{\log ^2\left (c \left (a+b x^2\right )\right )} \, dx=\frac {a {\left (\frac {b c x^{2} + a c}{\log \left (b c x^{2} + a c\right )} - {\rm Ei}\left (\log \left (b c x^{2} + a c\right )\right )\right )}}{2 \, b^{2} c} - \frac {\frac {{\left (b c x^{2} + a c\right )}^{2}}{\log \left (b c x^{2} + a c\right )} - 2 \, {\rm Ei}\left (2 \, \log \left (b c x^{2} + a c\right )\right )}{2 \, b^{2} c^{2}} \]
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Timed out. \[ \int \frac {x^3}{\log ^2\left (c \left (a+b x^2\right )\right )} \, dx=\int \frac {x^3}{{\ln \left (c\,\left (b\,x^2+a\right )\right )}^2} \,d x \]
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