Optimal. Leaf size=28 \[ \log (x)-\frac {\coth \left (a d+b d \log \left (c x^n\right )\right )}{b d n} \]
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Rubi [A] time = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3473, 8} \[ \log (x)-\frac {\coth \left (a d+b d \log \left (c x^n\right )\right )}{b d n} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rubi steps
\begin {align*} \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \coth ^2(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\coth \left (a d+b d \log \left (c x^n\right )\right )}{b d n}+\frac {\operatorname {Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\coth \left (a d+b d \log \left (c x^n\right )\right )}{b d n}+\log (x)\\ \end {align*}
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Mathematica [C] time = 0.11, size = 49, normalized size = 1.75 \[ -\frac {\coth \left (a d+b d \log \left (c x^n\right )\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\tanh ^2\left (a d+b \log \left (c x^n\right ) d\right )\right )}{b d n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 72, normalized size = 2.57 \[ \frac {{\left (b d n \log \relax (x) + 1\right )} \sinh \left (b d n \log \relax (x) + b d \log \relax (c) + a d\right ) - \cosh \left (b d n \log \relax (x) + b d \log \relax (c) + a d\right )}{b d n \sinh \left (b d n \log \relax (x) + b d \log \relax (c) + a d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 37, normalized size = 1.32 \[ -\frac {2}{{\left (c^{2 \, b d} x^{2 \, b d n} e^{\left (2 \, a d\right )} - 1\right )} b d n} + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 80, normalized size = 2.86 \[ -\frac {\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{b d n}-\frac {\ln \left (\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-1\right )}{2 b d n}+\frac {\ln \left (\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )+1\right )}{2 b d n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 37, normalized size = 1.32 \[ -\frac {2}{b c^{2 \, b d} d n e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} - b d n} + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 34, normalized size = 1.21 \[ \ln \relax (x)-\frac {2}{b\,d\,n\,\left ({\mathrm {e}}^{2\,a\,d}\,{\left (c\,x^n\right )}^{2\,b\,d}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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