Optimal. Leaf size=25 \[ \frac {\log \left (\sinh \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n} \]
[Out]
________________________________________________________________________________________
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} \[ \frac {\log \left (\sinh \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3475
Rubi steps
\begin {align*} \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac {\operatorname {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*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 40, normalized size = 1.60 \[ \frac {\log \left (\tanh \left (a d+b d \log \left (c x^n\right )\right )\right )+\log \left (\cosh \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{b d n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.41, size = 76, normalized size = 3.04 \[ -\frac {b d n \log \relax (x) - \log \left (\frac {2 \, \sinh \left (b d n \log \relax (x) + b d \log \relax (c) + a d\right )}{\cosh \left (b d n \log \relax (x) + b d \log \relax (c) + a d\right ) - \sinh \left (b d n \log \relax (x) + b d \log \relax (c) + a d\right )}\right )}{b d n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.31, size = 74, normalized size = 2.96 \[ \frac {\log \left (\sqrt {-2 \, x^{2 \, b d n} {\left | c \right |}^{2 \, b d} \cos \left (\pi b d \mathrm {sgn}\relax (c) - \pi b d\right ) e^{\left (2 \, a d\right )} + x^{4 \, b d n} {\left | c \right |}^{4 \, b d} e^{\left (4 \, a d\right )} + 1}\right )}{b d n} - \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 56, normalized size = 2.24 \[ -\frac {\ln \left (\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-1\right )}{2 b d n}-\frac {\ln \left (\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )+1\right )}{2 b d n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 24, normalized size = 0.96 \[ \frac {\log \left (\sinh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\right )}{b d n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.19, size = 34, normalized size = 1.36 \[ \frac {\ln \left ({\mathrm {e}}^{2\,a\,d}\,{\left (c\,x^n\right )}^{2\,b\,d}-1\right )}{b\,d\,n}-\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth {\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________