Integrand size = 34, antiderivative size = 15 \[ \int \frac {a x+2 b n \log \left (c x^n\right )}{a x^2+b x \log ^2\left (c x^n\right )} \, dx=\log \left (a x+b \log ^2\left (c x^n\right )\right ) \]
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Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2641, 2621} \[ \int \frac {a x+2 b n \log \left (c x^n\right )}{a x^2+b x \log ^2\left (c x^n\right )} \, dx=\log \left (a x+b \log ^2\left (c x^n\right )\right ) \]
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Rule 2621
Rule 2641
Rubi steps \begin{align*} \text {integral}& = \int \frac {a x+2 b n \log \left (c x^n\right )}{x \left (a x+b \log ^2\left (c x^n\right )\right )} \, dx \\ & = \log \left (a x+b \log ^2\left (c x^n\right )\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {a x+2 b n \log \left (c x^n\right )}{a x^2+b x \log ^2\left (c x^n\right )} \, dx=\log \left (a x+b \log ^2\left (c x^n\right )\right ) \]
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Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07
method | result | size |
parallelrisch | \(\ln \left (a x +b \ln \left (c \,x^{n}\right )^{2}\right )\) | \(16\) |
risch | \(\ln \left (\ln \left (x^{n}\right )^{2}+\left (-i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right )+i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i x^{n}\right )-i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 \ln \left (c \right )\right ) \ln \left (x^{n}\right )-\frac {b \,\pi ^{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 b \,\pi ^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+b \,\pi ^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}-2 b \,\pi ^{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 b \,\pi ^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{4}-2 b \,\pi ^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}+b \,\pi ^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}-2 b \,\pi ^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}+b \,\pi ^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{6}+4 i \pi \ln \left (c \right ) b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-4 i \pi \ln \left (c \right ) b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-4 i \pi \ln \left (c \right ) b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi \ln \left (c \right ) b \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-4 b \ln \left (c \right )^{2}-4 a x}{4 b}\right )\) | \(428\) |
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Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.87 \[ \int \frac {a x+2 b n \log \left (c x^n\right )}{a x^2+b x \log ^2\left (c x^n\right )} \, dx=\log \left (b n^{2} \log \left (x\right )^{2} + 2 \, b n \log \left (c\right ) \log \left (x\right ) + b \log \left (c\right )^{2} + a x\right ) \]
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Time = 0.73 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.60 \[ \int \frac {a x+2 b n \log \left (c x^n\right )}{a x^2+b x \log ^2\left (c x^n\right )} \, dx=\begin {cases} \log {\left (x + \frac {b \log {\left (c x^{n} \right )}^{2}}{a} \right )} & \text {for}\: a \neq 0 \\2 \log {\left (\log {\left (c x^{n} \right )} \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (15) = 30\).
Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.13 \[ \int \frac {a x+2 b n \log \left (c x^n\right )}{a x^2+b x \log ^2\left (c x^n\right )} \, dx=\log \left (\frac {b \log \left (c\right )^{2} + 2 \, b \log \left (c\right ) \log \left (x^{n}\right ) + b \log \left (x^{n}\right )^{2} + a x}{b}\right ) \]
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Time = 0.33 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.87 \[ \int \frac {a x+2 b n \log \left (c x^n\right )}{a x^2+b x \log ^2\left (c x^n\right )} \, dx=\log \left (b n^{2} \log \left (x\right )^{2} + 2 \, b n \log \left (c\right ) \log \left (x\right ) + b \log \left (c\right )^{2} + a x\right ) \]
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Time = 1.65 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int \frac {a x+2 b n \log \left (c x^n\right )}{a x^2+b x \log ^2\left (c x^n\right )} \, dx=\ln \left ({\ln \left (c\,x^n\right )}^2+\frac {a\,x}{b}\right ) \]
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