Integrand size = 14, antiderivative size = 52 \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{4} b^2 n^2 x^2-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \]
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Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2342, 2341} \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} b^2 n^2 x^2 \]
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Rule 2341
Rule 2342
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2-(b n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx \\ & = \frac {1}{4} b^2 n^2 x^2-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.79 \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{4} x^2 \left (b n \left (-2 a+b n-2 b \log \left (c x^n\right )\right )+2 \left (a+b \log \left (c x^n\right )\right )^2\right ) \]
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Time = 0.10 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.38
method | result | size |
parallelrisch | \(\frac {x^{2} b^{2} \ln \left (c \,x^{n}\right )^{2}}{2}-\frac {\ln \left (c \,x^{n}\right ) x^{2} b^{2} n}{2}+\frac {b^{2} n^{2} x^{2}}{4}+x^{2} a b \ln \left (c \,x^{n}\right )-\frac {a b n \,x^{2}}{2}+\frac {x^{2} a^{2}}{2}\) | \(72\) |
risch | \(\frac {b^{2} x^{2} \ln \left (x^{n}\right )^{2}}{2}+\frac {b \,x^{2} \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 )-b n +2 a \right ) \ln \left (x^{n}\right )}{2}+\frac {x^{2} \left (4 a^{2}+2 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}+2 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}+2 b^{2} n^{2}+8 \ln \left (c \right ) a b +4 \ln \left (c \right )^{2} b^{2}-4 b^{2} \ln \left (c \right ) n -4 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}-2 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}-2 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}\right )}{8}\) | \(692\) |
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Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (46) = 92\).
Time = 0.31 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.96 \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{2} \, b^{2} n^{2} x^{2} \log \left (x\right )^{2} + \frac {1}{2} \, b^{2} x^{2} \log \left (c\right )^{2} - \frac {1}{2} \, {\left (b^{2} n - 2 \, a b\right )} x^{2} \log \left (c\right ) + \frac {1}{4} \, {\left (b^{2} n^{2} - 2 \, a b n + 2 \, a^{2}\right )} x^{2} + \frac {1}{2} \, {\left (2 \, b^{2} n x^{2} \log \left (c\right ) - {\left (b^{2} n^{2} - 2 \, a b n\right )} x^{2}\right )} \log \left (x\right ) \]
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Time = 0.17 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.46 \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {a^{2} x^{2}}{2} - \frac {a b n x^{2}}{2} + a b x^{2} \log {\left (c x^{n} \right )} + \frac {b^{2} n^{2} x^{2}}{4} - \frac {b^{2} n x^{2} \log {\left (c x^{n} \right )}}{2} + \frac {b^{2} x^{2} \log {\left (c x^{n} \right )}^{2}}{2} \]
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none
Time = 0.19 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.35 \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{2} \, b^{2} x^{2} \log \left (c x^{n}\right )^{2} - \frac {1}{2} \, a b n x^{2} + a b x^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, a^{2} x^{2} + \frac {1}{4} \, {\left (n^{2} x^{2} - 2 \, n x^{2} \log \left (c x^{n}\right )\right )} b^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (46) = 92\).
Time = 0.34 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.08 \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{2} \, b^{2} n^{2} x^{2} \log \left (x\right )^{2} - \frac {1}{2} \, b^{2} n^{2} x^{2} \log \left (x\right ) + b^{2} n x^{2} \log \left (c\right ) \log \left (x\right ) + \frac {1}{4} \, b^{2} n^{2} x^{2} - \frac {1}{2} \, b^{2} n x^{2} \log \left (c\right ) + \frac {1}{2} \, b^{2} x^{2} \log \left (c\right )^{2} + a b n x^{2} \log \left (x\right ) - \frac {1}{2} \, a b n x^{2} + a b x^{2} \log \left (c\right ) + \frac {1}{2} \, a^{2} x^{2} \]
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Time = 0.30 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.15 \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx=x^2\,\left (\frac {a^2}{2}-\frac {a\,b\,n}{2}+\frac {b^2\,n^2}{4}\right )+x^2\,\ln \left (c\,x^n\right )\,\left (a\,b-\frac {b^2\,n}{2}\right )+\frac {b^2\,x^2\,{\ln \left (c\,x^n\right )}^2}{2} \]
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