Integrand size = 16, antiderivative size = 73 \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b^2 n^2 x}-\frac {1}{b n x \left (a+b \log \left (c x^n\right )\right )} \]
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Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2343, 2347, 2209} \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b^2 n^2 x}-\frac {1}{b n x \left (a+b \log \left (c x^n\right )\right )} \]
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Rule 2209
Rule 2343
Rule 2347
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{b n 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}{b n} \\ & = -\frac {1}{b n 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 )}{b n^2 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 )}{b^2 n^2 x}-\frac {1}{b n x \left (a+b \log \left (c x^n\right )\right )} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {b n+e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.48 (sec) , antiderivative size = 347, normalized size of antiderivative = 4.75
method | result | size |
risch | \(-\frac {2}{x \left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 \ln \left (x^{n}\right ) b +2 a \right ) b n}+\frac {\left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 a}{2 b n}} \operatorname {Ei}_{1}\left (\ln \left (x \right )+\frac {-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+2 a}{2 b n}\right )}{b^{2} n^{2} x}\) | \(347\) |
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Time = 0.29 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.21 \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {{\left (b n x \log \left (x\right ) + b x \log \left (c\right ) + a x\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} \operatorname {log\_integral}\left (\frac {e^{\left (-\frac {b \log \left (c\right ) + a}{b n}\right )}}{x}\right ) + b n}{b^{3} n^{3} x \log \left (x\right ) + b^{3} n^{2} x \log \left (c\right ) + a b^{2} n^{2} x} \]
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\[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]
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\[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2}} \,d x } \]
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\[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{x^2\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]
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