Integrand size = 16, antiderivative size = 96 \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e n^2}-\frac {d+e x}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )} \]
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Time = 0.05 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2436, 2334, 2337, 2209} \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e n^2}-\frac {d+e x}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )} \]
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Rule 2209
Rule 2334
Rule 2337
Rule 2436
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx,x,d+e x\right )}{e} \\ & = -\frac {d+e x}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {\text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e n} \\ & = -\frac {d+e x}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {\left ((d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e n^2} \\ & = \frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e n^2}-\frac {d+e x}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.28 \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=-\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (b e^{\frac {a}{b n}} n \left (c (d+e x)^n\right )^{\frac {1}{n}}-\operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )\right )}{b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.98 (sec) , antiderivative size = 456, normalized size of antiderivative = 4.75
method | result | size |
risch | \(-\frac {2 \left (e x +d \right )}{\left (-i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )+i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b +i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b -i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b +2 b \ln \left (\left (e x +d \right )^{n}\right )+2 b \ln \left (c \right )+2 a \right ) b n e}-\frac {\left (e x +d \right ) \left (\left (e x +d \right )^{n}\right )^{-\frac {1}{n}} c^{-\frac {1}{n}} {\mathrm e}^{-\frac {-i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )+i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b +i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b -i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b +2 a}{2 b n}} \operatorname {Ei}_{1}\left (-\ln \left (e x +d \right )-\frac {-i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )+i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b +i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b -i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b +2 b \ln \left (c \right )+2 b \left (\ln \left (\left (e x +d \right )^{n}\right )-n \ln \left (e x +d \right )\right )+2 a}{2 b n}\right )}{b^{2} n^{2} e}\) | \(456\) |
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Time = 0.29 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=-\frac {{\left ({\left (b e n x + b d n\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} - {\left (b n \log \left (e x + d\right ) + b \log \left (c\right ) + a\right )} \operatorname {log\_integral}\left ({\left (e x + d\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )}\right )\right )} e^{\left (-\frac {b \log \left (c\right ) + a}{b n}\right )}}{b^{3} e n^{3} \log \left (e x + d\right ) + b^{3} e n^{2} \log \left (c\right ) + a b^{2} e n^{2}} \]
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\[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2}}\, dx \]
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\[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (95) = 190\).
Time = 0.39 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.98 \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\frac {b n {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (e x + d\right )\right ) e^{\left (-\frac {a}{b n}\right )} \log \left (e x + d\right )}{{\left (b^{3} e n^{3} \log \left (e x + d\right ) + b^{3} e n^{2} \log \left (c\right ) + a b^{2} e n^{2}\right )} c^{\left (\frac {1}{n}\right )}} - \frac {{\left (e x + d\right )} b n}{b^{3} e n^{3} \log \left (e x + d\right ) + b^{3} e n^{2} \log \left (c\right ) + a b^{2} e n^{2}} + \frac {b {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (e x + d\right )\right ) e^{\left (-\frac {a}{b n}\right )} \log \left (c\right )}{{\left (b^{3} e n^{3} \log \left (e x + d\right ) + b^{3} e n^{2} \log \left (c\right ) + a b^{2} e n^{2}\right )} c^{\left (\frac {1}{n}\right )}} + \frac {a {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (e x + d\right )\right ) e^{\left (-\frac {a}{b n}\right )}}{{\left (b^{3} e n^{3} \log \left (e x + d\right ) + b^{3} e n^{2} \log \left (c\right ) + a b^{2} e n^{2}\right )} c^{\left (\frac {1}{n}\right )}} \]
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Timed out. \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {1}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2} \,d x \]
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