Integrand size = 16, antiderivative size = 46 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=-\frac {2 b^2 n^2}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \]
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Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2342, 2341} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {2 b^2 n^2}{x} \]
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Rule 2341
Rule 2342
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x}+(2 b n) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx \\ & = -\frac {2 b^2 n^2}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=-\frac {\left (a+b \log \left (c x^n\right )\right )^2+2 b n \left (a+b n+b \log \left (c x^n\right )\right )}{x} \]
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Time = 0.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.24
method | result | size |
parallelrisch | \(-\frac {b^{2} \ln \left (c \,x^{n}\right )^{2}+2 \ln \left (c \,x^{n}\right ) b^{2} n +2 b^{2} n^{2}+2 a b \ln \left (c \,x^{n}\right )+2 a b n +a^{2}}{x}\) | \(57\) |
risch | \(-\frac {b^{2} \ln \left (x^{n}\right )^{2}}{x}-\frac {\left (-i \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i \pi \,b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i \pi \,b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b^{2} \ln \left (c \right )+2 b^{2} n +2 a b \right ) \ln \left (x^{n}\right )}{x}-\frac {4 a^{2}-4 i \pi \,b^{2} n \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-4 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-4 i \pi a b \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-\pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-4 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-4 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-4 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-4 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-\pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-\pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+8 b^{2} n^{2}+8 \ln \left (c \right ) a b +4 \ln \left (c \right )^{2} b^{2}+8 b^{2} \ln \left (c \right ) n +8 a b n -\pi ^{2} b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{6}+4 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi a b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+4 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi \,b^{2} n \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+4 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{4 x}\) | \(704\) |
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Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.67 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=-\frac {b^{2} n^{2} \log \left (x\right )^{2} + 2 \, b^{2} n^{2} + b^{2} \log \left (c\right )^{2} + 2 \, a b n + a^{2} + 2 \, {\left (b^{2} n + a b\right )} \log \left (c\right ) + 2 \, {\left (b^{2} n^{2} + b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x\right )}{x} \]
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Time = 0.13 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=- \frac {a^{2}}{x} - \frac {2 a b n}{x} - \frac {2 a b \log {\left (c x^{n} \right )}}{x} - \frac {2 b^{2} n^{2}}{x} - \frac {2 b^{2} n \log {\left (c x^{n} \right )}}{x} - \frac {b^{2} \log {\left (c x^{n} \right )}^{2}}{x} \]
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Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.52 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=-2 \, b^{2} {\left (\frac {n^{2}}{x} + \frac {n \log \left (c x^{n}\right )}{x}\right )} - \frac {b^{2} \log \left (c x^{n}\right )^{2}}{x} - \frac {2 \, a b n}{x} - \frac {2 \, a b \log \left (c x^{n}\right )}{x} - \frac {a^{2}}{x} \]
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Time = 0.35 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.87 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=-\frac {b^{2} n^{2} \log \left (x\right )^{2}}{x} - \frac {2 \, {\left (b^{2} n^{2} + b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x\right )}{x} - \frac {2 \, b^{2} n^{2} + 2 \, b^{2} n \log \left (c\right ) + b^{2} \log \left (c\right )^{2} + 2 \, a b n + 2 \, a b \log \left (c\right ) + a^{2}}{x} \]
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Time = 0.32 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=-\frac {a^2+2\,a\,b\,n+2\,b^2\,n^2}{x}-\frac {b^2\,{\ln \left (c\,x^n\right )}^2}{x}-\frac {2\,b\,\ln \left (c\,x^n\right )\,\left (a+b\,n\right )}{x} \]
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