Integrand size = 12, antiderivative size = 43 \[ \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx=-2 a b n x+2 b^2 n^2 x-2 b^2 n x \log \left (c x^n\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \]
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Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2333, 2332} \[ \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx=x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x-2 b^2 n x \log \left (c x^n\right )+2 b^2 n^2 x \]
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Rule 2332
Rule 2333
Rubi steps \begin{align*} \text {integral}& = x \left (a+b \log \left (c x^n\right )\right )^2-(2 b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx \\ & = -2 a b n x+x \left (a+b \log \left (c x^n\right )\right )^2-\left (2 b^2 n\right ) \int \log \left (c x^n\right ) \, dx \\ & = -2 a b n x+2 b^2 n^2 x-2 b^2 n x \log \left (c x^n\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.77 \[ \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx=x \left (\left (a+b \log \left (c x^n\right )\right )^2-2 b n \left (a-b n+b \log \left (c x^n\right )\right )\right ) \]
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Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.33
method | result | size |
norman | \(\left (2 b^{2} n^{2}-2 a b n +a^{2}\right ) x +b^{2} x \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+\left (-2 b^{2} n +2 a b \right ) x \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )\) | \(57\) |
parallelrisch | \(x \,b^{2} \ln \left (c \,x^{n}\right )^{2}-2 b^{2} n x \ln \left (c \,x^{n}\right )+2 b^{2} n^{2} x +2 x a b \ln \left (c \,x^{n}\right )-2 a b n x +a^{2} x\) | \(59\) |
default | \(a^{2} x +b^{2} x \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+2 b^{2} n^{2} x -2 b^{2} n x \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )+2 x a b \ln \left (c \,x^{n}\right )-2 a b n x\) | \(63\) |
parts | \(a^{2} x +b^{2} x \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+2 b^{2} n^{2} x -2 b^{2} n x \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )+2 x a b \ln \left (c \,x^{n}\right )-2 a b n x\) | \(63\) |
risch | \(b^{2} x \ln \left (x^{n}\right )^{2}+x b \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 b n +2 a \right ) \ln \left (x^{n}\right )+\frac {x \left (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}\right )}{4}\) | \(684\) |
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Time = 0.31 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.98 \[ \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx=b^{2} n^{2} x \log \left (x\right )^{2} + b^{2} x \log \left (c\right )^{2} - 2 \, {\left (b^{2} n - a b\right )} x \log \left (c\right ) + {\left (2 \, b^{2} n^{2} - 2 \, a b n + a^{2}\right )} x + 2 \, {\left (b^{2} n x \log \left (c\right ) - {\left (b^{2} n^{2} - a b n\right )} x\right )} \log \left (x\right ) \]
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Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.51 \[ \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx=a^{2} x - 2 a b n x + 2 a b x \log {\left (c x^{n} \right )} + 2 b^{2} n^{2} x - 2 b^{2} n x \log {\left (c x^{n} \right )} + b^{2} x \log {\left (c x^{n} \right )}^{2} \]
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Time = 0.18 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.33 \[ \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx=b^{2} x \log \left (c x^{n}\right )^{2} - 2 \, a b n x + 2 \, a b x \log \left (c x^{n}\right ) + 2 \, {\left (n^{2} x - n x \log \left (c x^{n}\right )\right )} b^{2} + a^{2} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (43) = 86\).
Time = 0.36 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.05 \[ \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx=b^{2} n^{2} x \log \left (x\right )^{2} - 2 \, b^{2} n^{2} x \log \left (x\right ) + 2 \, b^{2} n x \log \left (c\right ) \log \left (x\right ) + 2 \, b^{2} n^{2} x - 2 \, b^{2} n x \log \left (c\right ) + b^{2} x \log \left (c\right )^{2} + 2 \, a b n x \log \left (x\right ) - 2 \, a b n x + 2 \, a b x \log \left (c\right ) + a^{2} x \]
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Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.14 \[ \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx=x\,\left (a^2-2\,a\,b\,n+2\,b^2\,n^2\right )+b^2\,x\,{\ln \left (c\,x^n\right )}^2+2\,b\,x\,\ln \left (c\,x^n\right )\,\left (a-b\,n\right ) \]
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