Optimal. Leaf size=28 \[ \log (x)-\frac {\tanh \left (a d+b d \log \left (c x^n\right )\right )}{b d n} \]
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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} \[ \log (x)-\frac {\tanh \left (a d+b d \log \left (c x^n\right )\right )}{b d n} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rubi steps
\begin {align*} \int \frac {\tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \tanh ^2(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\tanh \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}\\ &=\log (x)-\frac {\tanh \left (a d+b d \log \left (c x^n\right )\right )}{b d n}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 51, normalized size = 1.82 \[ \frac {\tanh ^{-1}\left (\tanh \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n}-\frac {\tanh \left (a d+b d \log \left (c x^n\right )\right )}{b d n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 72, normalized size = 2.57 \[ \frac {{\left (b d n \log \relax (x) + 1\right )} \cosh \left (b d n \log \relax (x) + b d \log \relax (c) + a d\right ) - \sinh \left (b d n \log \relax (x) + b d \log \relax (c) + a d\right )}{b d n \cosh \left (b d n \log \relax (x) + b d \log \relax (c) + a d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, 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 = 0.01, size = 80, normalized size = 2.86 \[ -\frac {\tanh \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{b d n}-\frac {\ln \left (\tanh \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-1\right )}{2 b d n}+\frac {\ln \left (\tanh \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )+1\right )}{2 b d n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 36, normalized size = 1.29 \[ \frac {2}{b c^{2 \, b d} d n e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + b d n} + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.06, 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 [B] time = 6.72, size = 70, normalized size = 2.50 \[ - \frac {\log {\left (\tanh {\left (a d + b d \log {\left (c x^{n} \right )} \right )} - 1 \right )}}{2 b d n} + \frac {\log {\left (\tanh {\left (a d + b d \log {\left (c x^{n} \right )} \right )} + 1 \right )}}{2 b d n} - \frac {\tanh {\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{b d n} \]
Verification of antiderivative is not currently implemented for this CAS.
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