Integrand size = 16, antiderivative size = 51 \[ \int \frac {x^3}{a+b \log \left (c x^n\right )} \, dx=\frac {e^{-\frac {4 a}{b n}} x^4 \left (c x^n\right )^{-4/n} \operatorname {ExpIntegralEi}\left (\frac {4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n} \]
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Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2347, 2209} \[ \int \frac {x^3}{a+b \log \left (c x^n\right )} \, dx=\frac {x^4 e^{-\frac {4 a}{b n}} \left (c x^n\right )^{-4/n} \operatorname {ExpIntegralEi}\left (\frac {4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n} \]
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Rule 2209
Rule 2347
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^4 \left (c x^n\right )^{-4/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {4 x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {e^{-\frac {4 a}{b n}} x^4 \left (c x^n\right )^{-4/n} \text {Ei}\left (\frac {4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{a+b \log \left (c x^n\right )} \, dx=\frac {e^{-\frac {4 a}{b n}} x^4 \left (c x^n\right )^{-4/n} \operatorname {ExpIntegralEi}\left (\frac {4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.38 (sec) , antiderivative size = 242, normalized size of antiderivative = 4.75
method | result | size |
risch | \(-\frac {x^{4} c^{-\frac {4}{n}} \left (x^{n}\right )^{-\frac {4}{n}} {\mathrm e}^{-\frac {2 \left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 a \right )}{b n}} \operatorname {Ei}_{1}\left (-4 \ln \left (x \right )-\frac {2 \left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+2 a \right )}{b n}\right )}{b n}\) | \(242\) |
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Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.82 \[ \int \frac {x^3}{a+b \log \left (c x^n\right )} \, dx=\frac {e^{\left (-\frac {4 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \operatorname {log\_integral}\left (x^{4} e^{\left (\frac {4 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )}{b n} \]
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\[ \int \frac {x^3}{a+b \log \left (c x^n\right )} \, dx=\int \frac {x^{3}}{a + b \log {\left (c x^{n} \right )}}\, dx \]
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\[ \int \frac {x^3}{a+b \log \left (c x^n\right )} \, dx=\int { \frac {x^{3}}{b \log \left (c x^{n}\right ) + a} \,d x } \]
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Time = 0.35 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.94 \[ \int \frac {x^3}{a+b \log \left (c x^n\right )} \, dx=\frac {{\rm Ei}\left (\frac {4 \, \log \left (c\right )}{n} + \frac {4 \, a}{b n} + 4 \, \log \left (x\right )\right ) e^{\left (-\frac {4 \, a}{b n}\right )}}{b c^{\frac {4}{n}} n} \]
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Timed out. \[ \int \frac {x^3}{a+b \log \left (c x^n\right )} \, dx=\int \frac {x^3}{a+b\,\ln \left (c\,x^n\right )} \,d x \]
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