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 = 0.02, 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 = {3554, 8}
\begin {gather*} \log (x)-\frac {\tanh \left (a d+b d \log \left (c x^n\right )\right )}{b d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3554
Rubi steps
\begin {align*} \int \frac {\tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac {\text {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 {\text {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 = 0.07, size = 51, normalized size = 1.82 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs.
\(2(28)=56\).
time = 2.08, size = 63, normalized size = 2.25
method | result | size |
derivativedivides | \(\frac {-\tanh \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-\frac {\ln \left (\tanh \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )+1\right )}{2}}{n b d}\) | \(63\) |
default | \(\frac {-\tanh \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-\frac {\ln \left (\tanh \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )+1\right )}{2}}{n b d}\) | \(63\) |
risch | \(\ln \left (x \right )+\frac {2}{d b n \left ({\mathrm e}^{d \left (-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )+i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )-i b \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )+2 b \ln \left (c \right )+2 b \ln \left (x^{n}\right )+2 a \right )}+1\right )}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 36, normalized size = 1.29 \begin {gather*} \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 \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 72 vs.
\(2 (28) = 56\).
time = 0.51, size = 72, normalized size = 2.57 \begin {gather*} \frac {{\left (b d n \log \left (x\right ) + 1\right )} \cosh \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) - \sinh \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )}{b d n \cosh \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs.
\(2 (22) = 44\).
time = 4.12, size = 70, normalized size = 2.50 \begin {gather*} - \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 37, normalized size = 1.32 \begin {gather*} \frac {2}{{\left (c^{2 \, b d} x^{2 \, b d n} e^{\left (2 \, a d\right )} + 1\right )} b d n} + \log \left (x\right ) \end {gather*}
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 \begin {gather*} \ln \left (x\right )+\frac {2}{b\,d\,n\,\left ({\mathrm {e}}^{2\,a\,d}\,{\left (c\,x^n\right )}^{2\,b\,d}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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