Integrand size = 17, antiderivative size = 18 \[ \int \frac {\text {sech}^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\tanh \left (a+b \log \left (c x^n\right )\right )}{b n} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3852, 8} \[ \int \frac {\text {sech}^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\tanh \left (a+b \log \left (c x^n\right )\right )}{b n} \]
[In]
[Out]
Rule 8
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \text {sech}^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {i \text {Subst}\left (\int 1 \, dx,x,-i \tanh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \\ & = \frac {\tanh \left (a+b \log \left (c x^n\right )\right )}{b n} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\tanh \left (a+b \log \left (c x^n\right )\right )}{b n} \]
[In]
[Out]
Time = 2.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\) | \(19\) |
default | \(\frac {\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\) | \(19\) |
parallelrisch | \(\frac {2 \tanh \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{b n \left (1+{\tanh \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{2}\right )}\) | \(45\) |
risch | \(-\frac {2}{b n \left (\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{2 a} {\mathrm e}^{i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}+1\right )}\) | \(116\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (18) = 36\).
Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.89 \[ \int \frac {\text {sech}^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2}{b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + b n} \]
[In]
[Out]
\[ \int \frac {\text {sech}^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\operatorname {sech}^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.56 \[ \int \frac {\text {sech}^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2}{b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.56 \[ \int \frac {\text {sech}^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )} b n} \]
[In]
[Out]
Time = 2.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33 \[ \int \frac {\text {sech}^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2}{b\,n+b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}} \]
[In]
[Out]