Integrand size = 20, antiderivative size = 44 \[ \int (e x)^{-1+n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\frac {a (e x)^n}{e n}+\frac {b x^{-n} (e x)^n \arctan \left (\sinh \left (c+d x^n\right )\right )}{d e n} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {14, 5548, 5544, 3855} \[ \int (e x)^{-1+n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\frac {a (e x)^n}{e n}+\frac {b x^{-n} (e x)^n \arctan \left (\sinh \left (c+d x^n\right )\right )}{d e n} \]
[In]
[Out]
Rule 14
Rule 3855
Rule 5544
Rule 5548
Rubi steps \begin{align*} \text {integral}& = \int \left (a (e x)^{-1+n}+b (e x)^{-1+n} \text {sech}\left (c+d x^n\right )\right ) \, dx \\ & = \frac {a (e x)^n}{e n}+b \int (e x)^{-1+n} \text {sech}\left (c+d x^n\right ) \, dx \\ & = \frac {a (e x)^n}{e n}+\frac {\left (b x^{-n} (e x)^n\right ) \int x^{-1+n} \text {sech}\left (c+d x^n\right ) \, dx}{e} \\ & = \frac {a (e x)^n}{e n}+\frac {\left (b x^{-n} (e x)^n\right ) \text {Subst}\left (\int \text {sech}(c+d x) \, dx,x,x^n\right )}{e n} \\ & = \frac {a (e x)^n}{e n}+\frac {b x^{-n} (e x)^n \arctan \left (\sinh \left (c+d x^n\right )\right )}{d e n} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.93 \[ \int (e x)^{-1+n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\frac {x^{-n} (e x)^n \left (a \left (c+d x^n\right )+b \arctan \left (\sinh \left (c+d x^n\right )\right )\right )}{d e n} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.72 (sec) , antiderivative size = 155, normalized size of antiderivative = 3.52
method | result | size |
risch | \(\frac {a x \,{\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right ) \pi +i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi +i \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi -i \operatorname {csgn}\left (i e x \right )^{3} \pi +2 \ln \left (e \right )+2 \ln \left (x \right )\right )}{2}}}{n}+\frac {2 \arctan \left ({\mathrm e}^{c +d \,x^{n}}\right ) e^{n} b \,{\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i e x \right ) \left (-1+n \right ) \left (\operatorname {csgn}\left (i e x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i e x \right )+\operatorname {csgn}\left (i e \right )\right )}{2}}}{d e n}\) | \(155\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (44) = 88\).
Time = 0.25 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.77 \[ \int (e x)^{-1+n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\frac {a d \cosh \left ({\left (n - 1\right )} \log \left (e\right )\right ) \cosh \left (n \log \left (x\right )\right ) + a d \cosh \left (n \log \left (x\right )\right ) \sinh \left ({\left (n - 1\right )} \log \left (e\right )\right ) + 2 \, {\left (b \cosh \left ({\left (n - 1\right )} \log \left (e\right )\right ) + b \sinh \left ({\left (n - 1\right )} \log \left (e\right )\right )\right )} \arctan \left (\cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right )\right ) + {\left (a d \cosh \left ({\left (n - 1\right )} \log \left (e\right )\right ) + a d \sinh \left ({\left (n - 1\right )} \log \left (e\right )\right )\right )} \sinh \left (n \log \left (x\right )\right )}{d n} \]
[In]
[Out]
\[ \int (e x)^{-1+n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\int \left (e x\right )^{n - 1} \left (a + b \operatorname {sech}{\left (c + d x^{n} \right )}\right )\, dx \]
[In]
[Out]
\[ \int (e x)^{-1+n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\int { {\left (b \operatorname {sech}\left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{n - 1} \,d x } \]
[In]
[Out]
\[ \int (e x)^{-1+n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\int { {\left (b \operatorname {sech}\left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{n - 1} \,d x } \]
[In]
[Out]
Time = 2.11 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.50 \[ \int (e x)^{-1+n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\frac {2\,\mathrm {atan}\left (\frac {b\,x\,{\mathrm {e}}^{d\,x^n}\,{\mathrm {e}}^c\,{\left (e\,x\right )}^{n-1}\,\sqrt {d^2\,n^2\,x^{2\,n}}}{d\,n\,x^n\,\sqrt {b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}\right )\,\sqrt {b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}{\sqrt {d^2\,n^2\,x^{2\,n}}}+\frac {a\,x\,{\left (e\,x\right )}^{n-1}}{n} \]
[In]
[Out]