Integrand size = 16, antiderivative size = 100 \[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {2 e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^3 n^3 x^2}-\frac {1}{2 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}+\frac {1}{b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )} \]
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Time = 0.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2343, 2347, 2209} \[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {2 e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^3 n^3 x^2}+\frac {1}{b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}-\frac {1}{2 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2} \]
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Rule 2209
Rule 2343
Rule 2347
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}-\frac {\int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^2} \, dx}{b n} \\ & = -\frac {1}{2 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}+\frac {1}{b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}+\frac {2 \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )} \, dx}{b^2 n^2} \\ & = -\frac {1}{2 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}+\frac {1}{b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}+\frac {\left (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 x^2} \\ & = \frac {2 e^{\frac {2 a}{b n}} \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 x^2}-\frac {1}{2 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}+\frac {1}{b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {4 e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\frac {b n \left (2 a-b n+2 b \log \left (c x^n\right )\right )}{\left (a+b \log \left (c x^n\right )\right )^2}}{2 b^3 n^3 x^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.62 (sec) , antiderivative size = 454, normalized size of antiderivative = 4.54
method | result | size |
risch | \(\frac {-2 b n +4 a +4 b \ln \left (c \right )+4 \ln \left (x^{n}\right ) b -2 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+2 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+2 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-2 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{{\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} x^{2}}-\frac {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} x^{2}}\) | \(454\) |
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Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (97) = 194\).
Time = 0.33 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.21 \[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {2 \, b^{2} n^{2} \log \left (x\right ) - b^{2} n^{2} + 2 \, b^{2} n \log \left (c\right ) + 2 \, a b n + 4 \, {\left (b^{2} n^{2} x^{2} \log \left (x\right )^{2} + b^{2} x^{2} \log \left (c\right )^{2} + 2 \, a b x^{2} \log \left (c\right ) + a^{2} x^{2} + 2 \, {\left (b^{2} n x^{2} \log \left (c\right ) + a b n x^{2}\right )} \log \left (x\right )\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \operatorname {log\_integral}\left (\frac {e^{\left (-\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{x^{2}}\right )}{2 \, {\left (b^{5} n^{5} x^{2} \log \left (x\right )^{2} + b^{5} n^{3} x^{2} \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} x^{2} \log \left (c\right ) + a^{2} b^{3} n^{3} x^{2} + 2 \, {\left (b^{5} n^{4} x^{2} \log \left (c\right ) + a b^{4} n^{4} x^{2}\right )} \log \left (x\right )\right )}} \]
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\[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {1}{x^{3} \left (a + b \log {\left (c x^{n} \right )}\right )^{3}}\, dx \]
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\[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int { \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{3}} \,d x } \]
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\[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int { \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {1}{x^3\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \]
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