Integrand size = 20, antiderivative size = 45 \[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=\frac {a (e x)^n}{e n}-\frac {b x^{-n} (e x)^n \text {arctanh}\left (\cosh \left (c+d x^n\right )\right )}{d e n} \]
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Time = 0.04 (sec) , antiderivative size = 45, 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, 5549, 5545, 3855} \[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=\frac {a (e x)^n}{e n}-\frac {b x^{-n} (e x)^n \text {arctanh}\left (\cosh \left (c+d x^n\right )\right )}{d e n} \]
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Rule 14
Rule 3855
Rule 5545
Rule 5549
Rubi steps \begin{align*} \text {integral}& = \int \left (a (e x)^{-1+n}+b (e x)^{-1+n} \text {csch}\left (c+d x^n\right )\right ) \, dx \\ & = \frac {a (e x)^n}{e n}+b \int (e x)^{-1+n} \text {csch}\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 {csch}\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 {csch}(c+d x) \, dx,x,x^n\right )}{e n} \\ & = \frac {a (e x)^n}{e n}-\frac {b x^{-n} (e x)^n \text {arctanh}\left (\cosh \left (c+d x^n\right )\right )}{d e n} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.36 \[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=\frac {x^{-n} (e x)^n \left (a \left (c+d x^n\right )-b \log \left (\cosh \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )+b \log \left (\sinh \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )\right )}{d e n} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.79 (sec) , antiderivative size = 155, normalized size of antiderivative = 3.44
method | result | size |
risch | \(\frac {a x \,{\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right ) \pi +i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi +i \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi -i \operatorname {csgn}\left (i e x \right )^{3} \pi +2 \ln \left (e \right )+2 \ln \left (x \right )\right )}{2}}}{n}-\frac {2 \,\operatorname {arctanh}\left ({\mathrm e}^{c +d \,x^{n}}\right ) e^{n} b \,{\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i e x \right ) \left (-1+n \right ) \left (\operatorname {csgn}\left (i e x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i e x \right )+\operatorname {csgn}\left (i e \right )\right )}{2}}}{d e n}\) | \(155\) |
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Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (45) = 90\).
Time = 0.28 (sec) , antiderivative size = 180, normalized size of antiderivative = 4.00 \[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=\frac {a d \cosh \left ({\left (n - 1\right )} \log \left (e\right )\right ) \cosh \left (n \log \left (x\right )\right ) + a d \cosh \left (n \log \left (x\right )\right ) \sinh \left ({\left (n - 1\right )} \log \left (e\right )\right ) - {\left (b \cosh \left ({\left (n - 1\right )} \log \left (e\right )\right ) + b \sinh \left ({\left (n - 1\right )} \log \left (e\right )\right )\right )} \log \left (\cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + 1\right ) + {\left (b \cosh \left ({\left (n - 1\right )} \log \left (e\right )\right ) + b \sinh \left ({\left (n - 1\right )} \log \left (e\right )\right )\right )} \log \left (\cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) - 1\right ) + {\left (a d \cosh \left ({\left (n - 1\right )} \log \left (e\right )\right ) + a d \sinh \left ({\left (n - 1\right )} \log \left (e\right )\right )\right )} \sinh \left (n \log \left (x\right )\right )}{d n} \]
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\[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=\int \left (e x\right )^{n - 1} \left (a + b \operatorname {csch}{\left (c + d x^{n} \right )}\right )\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.67 \[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=-b {\left (\frac {e^{n - 1} \log \left ({\left (e^{\left (d x^{n} + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{d n} - \frac {e^{n - 1} \log \left ({\left (e^{\left (d x^{n} + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{d n}\right )} + \frac {\left (e x\right )^{n} a}{e n} \]
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\[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=\int { {\left (b \operatorname {csch}\left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{n - 1} \,d x } \]
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Time = 5.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.49 \[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=\frac {a\,x\,{\left (e\,x\right )}^{n-1}}{n}-\frac {2\,\mathrm {atan}\left (\frac {b\,x\,{\mathrm {e}}^{d\,x^n}\,{\mathrm {e}}^c\,{\left (e\,x\right )}^{n-1}\,\sqrt {-d^2\,n^2\,x^{2\,n}}}{d\,n\,x^n\,\sqrt {b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}\right )\,\sqrt {b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}{\sqrt {-d^2\,n^2\,x^{2\,n}}} \]
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