Integrand size = 16, antiderivative size = 45 \[ \int \frac {x^3}{\log \left (c \left (a+b x^2\right )\right )} \, dx=\frac {\operatorname {ExpIntegralEi}\left (2 \log \left (c \left (a+b x^2\right )\right )\right )}{2 b^2 c^2}-\frac {a \operatorname {LogIntegral}\left (c \left (a+b x^2\right )\right )}{2 b^2 c} \]
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Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {2504, 2446, 2436, 2335, 2437, 2346, 2209} \[ \int \frac {x^3}{\log \left (c \left (a+b x^2\right )\right )} \, dx=\frac {\operatorname {ExpIntegralEi}\left (2 \log \left (c \left (b x^2+a\right )\right )\right )}{2 b^2 c^2}-\frac {a \operatorname {LogIntegral}\left (c \left (b x^2+a\right )\right )}{2 b^2 c} \]
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Rule 2209
Rule 2335
Rule 2346
Rule 2436
Rule 2437
Rule 2446
Rule 2504
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x}{\log (c (a+b x))} \, dx,x,x^2\right ) \\ & = \frac {1}{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 {\text {Subst}\left (\int \frac {a+b x}{\log (c (a+b x))} \, dx,x,x^2\right )}{2 b}-\frac {a \text {Subst}\left (\int \frac {1}{\log (c (a+b x))} \, dx,x,x^2\right )}{2 b} \\ & = \frac {\text {Subst}\left (\int \frac {x}{\log (c x)} \, dx,x,a+b x^2\right )}{2 b^2}-\frac {a \text {Subst}\left (\int \frac {1}{\log (c x)} \, dx,x,a+b x^2\right )}{2 b^2} \\ & = -\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 )}{2 b^2 c^2} \\ & = \frac {\text {Ei}\left (2 \log \left (c \left (a+b x^2\right )\right )\right )}{2 b^2 c^2}-\frac {a \text {li}\left (c \left (a+b x^2\right )\right )}{2 b^2 c} \\ \end{align*}
\[ \int \frac {x^3}{\log \left (c \left (a+b x^2\right )\right )} \, dx=\int \frac {x^3}{\log \left (c \left (a+b x^2\right )\right )} \, dx \]
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Time = 1.72 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {-\operatorname {Ei}_{1}\left (-2 \ln \left (c \left (b \,x^{2}+a \right )\right )\right )+c a \,\operatorname {Ei}_{1}\left (-\ln \left (c \left (b \,x^{2}+a \right )\right )\right )}{2 c^{2} b^{2}}\) | \(43\) |
risch | \(\frac {a \,\operatorname {Ei}_{1}\left (-\ln \left (c \left (b \,x^{2}+a \right )\right )\right )}{2 c \,b^{2}}-\frac {\operatorname {Ei}_{1}\left (-2 \ln \left (c \left (b \,x^{2}+a \right )\right )\right )}{2 c^{2} b^{2}}\) | \(47\) |
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Time = 0.34 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.20 \[ \int \frac {x^3}{\log \left (c \left (a+b x^2\right )\right )} \, dx=-\frac {a c \operatorname {log\_integral}\left (b c x^{2} + a c\right ) - \operatorname {log\_integral}\left (b^{2} c^{2} x^{4} + 2 \, a b c^{2} x^{2} + a^{2} c^{2}\right )}{2 \, b^{2} c^{2}} \]
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\[ \int \frac {x^3}{\log \left (c \left (a+b x^2\right )\right )} \, dx=\int \frac {x^{3}}{\log {\left (a c + b c x^{2} \right )}}\, dx \]
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\[ \int \frac {x^3}{\log \left (c \left (a+b x^2\right )\right )} \, dx=\int { \frac {x^{3}}{\log \left ({\left (b x^{2} + a\right )} c\right )} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int \frac {x^3}{\log \left (c \left (a+b x^2\right )\right )} \, dx=-\frac {a {\rm Ei}\left (\log \left (b c x^{2} + a c\right )\right )}{2 \, b^{2} c} + \frac {{\rm Ei}\left (2 \, \log \left (b c x^{2} + a c\right )\right )}{2 \, b^{2} c^{2}} \]
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Timed out. \[ \int \frac {x^3}{\log \left (c \left (a+b x^2\right )\right )} \, dx=\int \frac {x^3}{\ln \left (c\,\left (b\,x^2+a\right )\right )} \,d x \]
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