Integrand size = 16, antiderivative size = 102 \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )}{2 b^3 n^3 x}-\frac {1}{2 b n x \left (a+b \log \left (c x^n\right )\right )^2}+\frac {1}{2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )} \]
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Time = 0.07 (sec) , antiderivative size = 102, 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} \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )}{2 b^3 n^3 x}+\frac {1}{2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}-\frac {1}{2 b n x \left (a+b \log \left (c x^n\right )\right )^2} \]
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Rule 2209
Rule 2343
Rule 2347
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2 b n x \left (a+b \log \left (c x^n\right )\right )^2}-\frac {\int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx}{2 b n} \\ & = -\frac {1}{2 b n x \left (a+b \log \left (c x^n\right )\right )^2}+\frac {1}{2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}+\frac {\int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )} \, dx}{2 b^2 n^2} \\ & = -\frac {1}{2 b n x \left (a+b \log \left (c x^n\right )\right )^2}+\frac {1}{2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}+\frac {\left (c x^n\right )^{\frac {1}{n}} \text {Subst}\left (\int \frac {e^{-\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 b^2 n^3 x} \\ & = \frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {Ei}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )}{2 b^3 n^3 x}-\frac {1}{2 b n x \left (a+b \log \left (c x^n\right )\right )^2}+\frac {1}{2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^2+b n \left (a-b n+b \log \left (c x^n\right )\right )}{2 b^3 n^3 x \left (a+b \log \left (c x^n\right )\right )^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.68 (sec) , antiderivative size = 449, normalized size of antiderivative = 4.40
method | result | size |
risch | \(\frac {-2 b n +2 a +2 b \ln \left (c \right )+2 \ln \left (x^{n}\right ) b -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}}{{\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}-\frac {c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{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}{2 b n}} \operatorname {Ei}_{1}\left (\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}{2 b n}\right )}{2 b^{3} n^{3} x}\) | \(449\) |
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Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (95) = 190\).
Time = 0.31 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.88 \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {b^{2} n^{2} \log \left (x\right ) - b^{2} n^{2} + b^{2} n \log \left (c\right ) + a b n + {\left (b^{2} n^{2} x \log \left (x\right )^{2} + b^{2} x \log \left (c\right )^{2} + 2 \, a b x \log \left (c\right ) + a^{2} x + 2 \, {\left (b^{2} n x \log \left (c\right ) + a b n x\right )} \log \left (x\right )\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} \operatorname {log\_integral}\left (\frac {e^{\left (-\frac {b \log \left (c\right ) + a}{b n}\right )}}{x}\right )}{2 \, {\left (b^{5} n^{5} x \log \left (x\right )^{2} + b^{5} n^{3} x \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} x \log \left (c\right ) + a^{2} b^{3} n^{3} x + 2 \, {\left (b^{5} n^{4} x \log \left (c\right ) + a b^{4} n^{4} x\right )} \log \left (x\right )\right )}} \]
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\[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {1}{x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )^{3}}\, dx \]
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\[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int { \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{2}} \,d x } \]
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\[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int { \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {1}{x^2\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \]
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