Optimal. Leaf size=25 \[ \frac {\log \left (\sin \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n} \]
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Rubi [A] time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {3475} \[ \frac {\log \left (\sin \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n} \]
Antiderivative was successfully verified.
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Rule 3475
Rubi steps
\begin {align*} \int \frac {\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \cot (d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\log \left (\sin \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 40, normalized size = 1.60 \[ \frac {\log \left (\tan \left (a d+b d \log \left (c x^n\right )\right )\right )+\log \left (\cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{b d n} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.46, size = 35, normalized size = 1.40 \[ \frac {\log \left (-\frac {1}{2} \, \cos \left (2 \, b d n \log \relax (x) + 2 \, b d \log \relax (c) + 2 \, a d\right ) + \frac {1}{2}\right )}{2 \, b d n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 30, normalized size = 1.20 \[ -\frac {\ln \left (\cot ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )+1\right )}{2 n b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 24, normalized size = 0.96 \[ \frac {\log \left (\sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\right )}{b d n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.80, size = 37, normalized size = 1.48 \[ -\ln \relax (x)\,1{}\mathrm {i}+\frac {\ln \left ({\mathrm {e}}^{a\,d\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,d\,2{}\mathrm {i}}-1\right )}{b\,d\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.14, size = 46, normalized size = 1.84 \[ \begin {cases} \log {\relax (x )} \cot {\left (a d \right )} & \text {for}\: b = 0 \\\tilde {\infty } \log {\relax (x )} & \text {for}\: d = 0 \\\log {\relax (x )} \cot {\left (a d + b d \log {\relax (c )} \right )} & \text {for}\: n = 0 \\\frac {\log {\left (\sin {\left (a d + b d \log {\left (c x^{n} \right )} \right )} \right )}}{b d n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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