Optimal. Leaf size=52 \[ -\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {b^2 n^2}{4 x^2} \]
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Rubi [A] time = 0.04, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2305, 2304} \[ -\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {b^2 n^2}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2305
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx &=-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+(b n) \int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx\\ &=-\frac {b^2 n^2}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 41, normalized size = 0.79 \[ -\frac {2 \left (a+b \log \left (c x^n\right )\right )^2+b n \left (2 a+2 b \log \left (c x^n\right )+b n\right )}{4 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 83, normalized size = 1.60 \[ -\frac {2 \, b^{2} n^{2} \log \relax (x)^{2} + b^{2} n^{2} + 2 \, b^{2} \log \relax (c)^{2} + 2 \, a b n + 2 \, a^{2} + 2 \, {\left (b^{2} n + 2 \, a b\right )} \log \relax (c) + 2 \, {\left (b^{2} n^{2} + 2 \, b^{2} n \log \relax (c) + 2 \, a b n\right )} \log \relax (x)}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 90, normalized size = 1.73 \[ -\frac {b^{2} n^{2} \log \relax (x)^{2}}{2 \, x^{2}} - \frac {{\left (b^{2} n^{2} + 2 \, b^{2} n \log \relax (c) + 2 \, a b n\right )} \log \relax (x)}{2 \, x^{2}} - \frac {b^{2} n^{2} + 2 \, b^{2} n \log \relax (c) + 2 \, b^{2} \log \relax (c)^{2} + 2 \, a b n + 4 \, a b \log \relax (c) + 2 \, a^{2}}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 703, normalized size = 13.52 \[ -\frac {b^{2} \ln \left (x^{n}\right )^{2}}{2 x^{2}}-\frac {\left (-i \pi \,b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi \,b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi \,b^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi \,b^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+b^{2} n +2 b^{2} \ln \relax (c )+2 a b \right ) \ln \left (x^{n}\right )}{2 x^{2}}-\frac {-\pi ^{2} b^{2} \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{4}-2 i \pi \,b^{2} n \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 i \pi \,b^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (c )-4 i \pi a b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+4 a^{2}+2 i \pi \,b^{2} n \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 i \pi \,b^{2} n \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi \,b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+4 i \pi \,b^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+4 i \pi a b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi a b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 b^{2} n^{2}-\pi ^{2} b^{2} \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}-\pi ^{2} b^{2} \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}+8 a b \ln \relax (c )+4 b^{2} n \ln \relax (c )+4 b^{2} \ln \relax (c )^{2}+4 a b n -\pi ^{2} b^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{6}-4 i \pi \,b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \relax (c )-4 i \pi a b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-2 i \pi \,b^{2} n \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{8 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 71, normalized size = 1.37 \[ -\frac {1}{4} \, b^{2} {\left (\frac {n^{2}}{x^{2}} + \frac {2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} - \frac {b^{2} \log \left (c x^{n}\right )^{2}}{2 \, x^{2}} - \frac {a b n}{2 \, x^{2}} - \frac {a b \log \left (c x^{n}\right )}{x^{2}} - \frac {a^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.43, size = 62, normalized size = 1.19 \[ -\frac {\frac {a^2}{2}+\frac {a\,b\,n}{2}+\frac {b^2\,n^2}{4}}{x^2}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {n\,b^2}{2}+a\,b\right )}{x^2}-\frac {b^2\,{\ln \left (c\,x^n\right )}^2}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.12, size = 128, normalized size = 2.46 \[ - \frac {a^{2}}{2 x^{2}} - \frac {a b n \log {\relax (x )}}{x^{2}} - \frac {a b n}{2 x^{2}} - \frac {a b \log {\relax (c )}}{x^{2}} - \frac {b^{2} n^{2} \log {\relax (x )}^{2}}{2 x^{2}} - \frac {b^{2} n^{2} \log {\relax (x )}}{2 x^{2}} - \frac {b^{2} n^{2}}{4 x^{2}} - \frac {b^{2} n \log {\relax (c )} \log {\relax (x )}}{x^{2}} - \frac {b^{2} n \log {\relax (c )}}{2 x^{2}} - \frac {b^{2} \log {\relax (c )}^{2}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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