Optimal. Leaf size=19 \[ -\frac {\coth \left (a+b \log \left (c x^n\right )\right )}{b n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3767, 8} \[ -\frac {\coth \left (a+b \log \left (c x^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3767
Rubi steps
\begin {align*} \int \frac {\text {csch}^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \text {csch}^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {i \operatorname {Subst}\left (\int 1 \, dx,x,-i \coth \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=-\frac {\coth \left (a+b \log \left (c x^n\right )\right )}{b n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 19, normalized size = 1.00 \[ -\frac {\coth \left (a+b \log \left (c x^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.68, size = 71, normalized size = 3.74 \[ -\frac {2}{b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + 2 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + b n \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - b n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 28, normalized size = 1.47 \[ -\frac {2}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )} b n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.33, size = 20, normalized size = 1.05 \[ -\frac {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 29, normalized size = 1.53 \[ -\frac {2}{b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - b n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.44, size = 25, normalized size = 1.32 \[ \frac {2}{b\,n-b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________