Integrand size = 14, antiderivative size = 101 \[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {2 e^{-\frac {2 a}{b n}} x^2 \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^3 n^3}-\frac {x^2}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}-\frac {x^2}{b^2 n^2 \left (a+b \log \left (c x^n\right )\right )} \]
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Time = 0.05 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2343, 2347, 2209} \[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {2 x^2 e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^3 n^3}-\frac {x^2}{b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}-\frac {x^2}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]
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Rule 2209
Rule 2343
Rule 2347
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}+\frac {\int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx}{b n} \\ & = -\frac {x^2}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}-\frac {x^2}{b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}+\frac {2 \int \frac {x}{a+b \log \left (c x^n\right )} \, dx}{b^2 n^2} \\ & = -\frac {x^2}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}-\frac {x^2}{b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}+\frac {\left (2 x^2 \left (c x^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{b^2 n^3} \\ & = \frac {2 e^{-\frac {2 a}{b n}} x^2 \left (c x^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^3 n^3}-\frac {x^2}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}-\frac {x^2}{b^2 n^2 \left (a+b \log \left (c x^n\right )\right )} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.88 \[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {x^2 \left (4 e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {b n \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{\left (a+b \log \left (c x^n\right )\right )^2}\right )}{2 b^3 n^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.56 (sec) , antiderivative size = 476, normalized size of antiderivative = 4.71
method | result | size |
risch | \(-\frac {2 \left (b n \,x^{2}-i \pi b \,x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i \pi b \,x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \,x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 \ln \left (c \right ) b \,x^{2}+2 b \,x^{2} \ln \left (x^{n}\right )+2 x^{2} a \right )}{{\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 \ln \left (x^{n}\right ) b +2 a \right )}^{2} b^{2} n^{2}}-\frac {2 x^{2} c^{-\frac {2}{n}} \left (x^{n}\right )^{-\frac {2}{n}} {\mathrm e}^{-\frac {-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}{b n}} \operatorname {Ei}_{1}\left (-2 \ln \left (x \right )-\frac {-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}{b n}\right )}{b^{3} n^{3}}\) | \(476\) |
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Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (100) = 200\).
Time = 0.31 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.09 \[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=-\frac {{\left ({\left (2 \, b^{2} n^{2} x^{2} \log \left (x\right ) + 2 \, b^{2} n x^{2} \log \left (c\right ) + {\left (b^{2} n^{2} + 2 \, a b n\right )} x^{2}\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} - 4 \, {\left (b^{2} n^{2} \log \left (x\right )^{2} + b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2} + 2 \, {\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x\right )\right )} \operatorname {log\_integral}\left (x^{2} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{2 \, {\left (b^{5} n^{5} \log \left (x\right )^{2} + b^{5} n^{3} \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} \log \left (c\right ) + a^{2} b^{3} n^{3} + 2 \, {\left (b^{5} n^{4} \log \left (c\right ) + a b^{4} n^{4}\right )} \log \left (x\right )\right )}} \]
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\[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {x}{\left (a + b \log {\left (c x^{n} \right )}\right )^{3}}\, dx \]
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\[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int { \frac {x}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 1029 vs. \(2 (100) = 200\).
Time = 0.46 (sec) , antiderivative size = 1029, normalized size of antiderivative = 10.19 \[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {x}{{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \]
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