Optimal. Leaf size=25 \[ \frac {\log \left (\sinh \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n} \]
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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 = {3556}
\begin {gather*} \frac {\log \left (\sinh \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rubi steps
\begin {align*} \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \coth (d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\log \left (\sinh \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 = 0.05, size = 40, normalized size = 1.60 \begin {gather*} \frac {\log \left (\cosh \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )+\log \left (\tanh \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.29, size = 48, normalized size = 1.92
method | result | size |
derivativedivides | \(\frac {-\frac {\ln \left (\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-1\right )}{2}-\frac {\ln \left (\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )+1\right )}{2}}{n b d}\) | \(48\) |
default | \(\frac {-\frac {\ln \left (\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-1\right )}{2}-\frac {\ln \left (\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )+1\right )}{2}}{n b d}\) | \(48\) |
risch | \(\ln \left (x \right )-\frac {2 a}{n b}-\frac {2 \ln \left (c \right )}{n}-\frac {2 \ln \left (x^{n}\right )}{n}+\frac {i \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{n}-\frac {i \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )}{n}-\frac {i \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )}{n}+\frac {i \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )}{n}+\frac {\ln \left ({\mathrm e}^{d \left (-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )+i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )-i b \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )+2 b \ln \left (c \right )+2 b \ln \left (x^{n}\right )+2 a \right )}-1\right )}{b d n}\) | \(233\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 24, normalized size = 0.96 \begin {gather*} \frac {\log \left (\sinh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\right )}{b d n} \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. 76 vs.
\(2 (25) = 50\).
time = 0.37, size = 76, normalized size = 3.04 \begin {gather*} -\frac {b d n \log \left (x\right ) - \log \left (\frac {2 \, \sinh \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\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 )}\right )}{b d n} \end {gather*}
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 \begin {gather*} \int \frac {\coth {\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 74 vs.
\(2 (25) = 50\).
time = 0.49, size = 74, normalized size = 2.96 \begin {gather*} \frac {\log \left (\sqrt {-2 \, x^{2 \, b d n} {\left | c \right |}^{2 \, b d} \cos \left (\pi b d \mathrm {sgn}\left (c\right ) - \pi b d\right ) e^{\left (2 \, a d\right )} + x^{4 \, b d n} {\left | c \right |}^{4 \, b d} e^{\left (4 \, 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 = 1.19, size = 34, normalized size = 1.36 \begin {gather*} \frac {\ln \left ({\mathrm {e}}^{2\,a\,d}\,{\left (c\,x^n\right )}^{2\,b\,d}-1\right )}{b\,d\,n}-\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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