Integrand size = 16, antiderivative size = 135 \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e n^3}-\frac {d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac {d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \]
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Time = 0.06 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2436, 2334, 2337, 2209} \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e n^3}-\frac {d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]
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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 )^3} \, dx,x,d+e x\right )}{e} \\ & = -\frac {d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {\text {Subst}\left (\int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx,x,d+e x\right )}{2 b e n} \\ & = -\frac {d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac {d+e x}{2 b^2 e n^2 \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 )}{2 b^2 e n^2} \\ & = -\frac {d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac {d+e x}{2 b^2 e n^2 \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 )}{2 b^2 e n^3} \\ & = \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 )}{2 b^3 e n^3}-\frac {d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac {d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=-\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (-\operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+b e^{\frac {a}{b n}} n \left (c (d+e x)^n\right )^{\frac {1}{n}} \left (a+b n+b \log \left (c (d+e x)^n\right )\right )\right )}{2 b^3 e n^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.14 (sec) , antiderivative size = 734, normalized size of antiderivative = 5.44
method | result | size |
risch | \(-\frac {2 b e n x +2 b d n +i \pi b d \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )-i \pi b d \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}-i \pi b e x \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+i \pi b e x \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b e x \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )+i \pi b e x \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+2 \ln \left (c \right ) b e x +2 b e x \ln \left (\left (e x +d \right )^{n}\right )+2 d b \ln \left (c \right )+2 a e x +2 b d \ln \left (\left (e x +d \right )^{n}\right )+2 a d}{{\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 )}^{2} b^{2} n^{2} e}-\frac {\left (e x +d \right ) c^{-\frac {1}{n}} \left (\left (e x +d \right )^{n}\right )^{-\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 )}{2 b^{3} n^{3} e}\) | \(734\) |
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Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (128) = 256\).
Time = 0.30 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.95 \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=-\frac {{\left ({\left (b^{2} d n^{2} + a b d n + {\left (b^{2} e n^{2} + a b e n\right )} x + {\left (b^{2} e n^{2} x + b^{2} d n^{2}\right )} \log \left (e x + d\right ) + {\left (b^{2} e n x + b^{2} d n\right )} \log \left (c\right )\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} - {\left (b^{2} n^{2} \log \left (e x + d\right )^{2} + b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2} + 2 \, {\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (e x + d\right )\right )} \operatorname {log\_integral}\left ({\left (e x + d\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )}\right )\right )} e^{\left (-\frac {b \log \left (c\right ) + a}{b n}\right )}}{2 \, {\left (b^{5} e n^{5} \log \left (e x + d\right )^{2} + b^{5} e n^{3} \log \left (c\right )^{2} + 2 \, a b^{4} e n^{3} \log \left (c\right ) + a^{2} b^{3} e n^{3} + 2 \, {\left (b^{5} e n^{4} \log \left (c\right ) + a b^{4} e n^{4}\right )} \log \left (e x + d\right )\right )}} \]
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\[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3}}\, dx \]
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\[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int { \frac {1}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 1218 vs. \(2 (128) = 256\).
Time = 0.41 (sec) , antiderivative size = 1218, normalized size of antiderivative = 9.02 \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int \frac {1}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3} \,d x \]
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