Integrand size = 18, antiderivative size = 67 \[ \int x^{-1-n} \cosh ^2\left (a+b x^n\right ) \, dx=-\frac {x^{-n}}{2 n}-\frac {x^{-n} \cosh \left (2 \left (a+b x^n\right )\right )}{2 n}+\frac {b \text {Chi}\left (2 b x^n\right ) \sinh (2 a)}{n}+\frac {b \cosh (2 a) \text {Shi}\left (2 b x^n\right )}{n} \]
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Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5471, 5429, 3378, 3384, 3379, 3382} \[ \int x^{-1-n} \cosh ^2\left (a+b x^n\right ) \, dx=\frac {b \sinh (2 a) \text {Chi}\left (2 b x^n\right )}{n}+\frac {b \cosh (2 a) \text {Shi}\left (2 b x^n\right )}{n}-\frac {x^{-n} \cosh \left (2 \left (a+b x^n\right )\right )}{2 n}-\frac {x^{-n}}{2 n} \]
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5429
Rule 5471
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x^{-1-n}}{2}+\frac {1}{2} x^{-1-n} \cosh \left (2 a+2 b x^n\right )\right ) \, dx \\ & = -\frac {x^{-n}}{2 n}+\frac {1}{2} \int x^{-1-n} \cosh \left (2 a+2 b x^n\right ) \, dx \\ & = -\frac {x^{-n}}{2 n}+\frac {\text {Subst}\left (\int \frac {\cosh (2 a+2 b x)}{x^2} \, dx,x,x^n\right )}{2 n} \\ & = -\frac {x^{-n}}{2 n}-\frac {x^{-n} \cosh \left (2 \left (a+b x^n\right )\right )}{2 n}+\frac {b \text {Subst}\left (\int \frac {\sinh (2 a+2 b x)}{x} \, dx,x,x^n\right )}{n} \\ & = -\frac {x^{-n}}{2 n}-\frac {x^{-n} \cosh \left (2 \left (a+b x^n\right )\right )}{2 n}+\frac {(b \cosh (2 a)) \text {Subst}\left (\int \frac {\sinh (2 b x)}{x} \, dx,x,x^n\right )}{n}+\frac {(b \sinh (2 a)) \text {Subst}\left (\int \frac {\cosh (2 b x)}{x} \, dx,x,x^n\right )}{n} \\ & = -\frac {x^{-n}}{2 n}-\frac {x^{-n} \cosh \left (2 \left (a+b x^n\right )\right )}{2 n}+\frac {b \text {Chi}\left (2 b x^n\right ) \sinh (2 a)}{n}+\frac {b \cosh (2 a) \text {Shi}\left (2 b x^n\right )}{n} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.81 \[ \int x^{-1-n} \cosh ^2\left (a+b x^n\right ) \, dx=\frac {x^{-n} \left (-\cosh ^2\left (a+b x^n\right )+b x^n \text {Chi}\left (2 b x^n\right ) \sinh (2 a)+b x^n \cosh (2 a) \text {Shi}\left (2 b x^n\right )\right )}{n} \]
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Time = 1.35 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.12
method | result | size |
risch | \(\frac {\left (2 b \,{\mathrm e}^{-2 a} \operatorname {Ei}_{1}\left (2 b \,x^{n}\right ) x^{n}-2 b \,{\mathrm e}^{2 a} \operatorname {Ei}_{1}\left (-2 b \,x^{n}\right ) x^{n}-{\mathrm e}^{-2 a -2 b \,x^{n}}-{\mathrm e}^{2 a +2 b \,x^{n}}-2\right ) x^{-n}}{4 n}\) | \(75\) |
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Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (64) = 128\).
Time = 0.26 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.72 \[ \int x^{-1-n} \cosh ^2\left (a+b x^n\right ) \, dx=\frac {{\left ({\left (b \cosh \left (2 \, a\right ) + b \sinh \left (2 \, a\right )\right )} \cosh \left (n \log \left (x\right )\right ) + {\left (b \cosh \left (2 \, a\right ) + b \sinh \left (2 \, a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )} {\rm Ei}\left (2 \, b \cosh \left (n \log \left (x\right )\right ) + 2 \, b \sinh \left (n \log \left (x\right )\right )\right ) - {\left ({\left (b \cosh \left (2 \, a\right ) - b \sinh \left (2 \, a\right )\right )} \cosh \left (n \log \left (x\right )\right ) + {\left (b \cosh \left (2 \, a\right ) - b \sinh \left (2 \, a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )} {\rm Ei}\left (-2 \, b \cosh \left (n \log \left (x\right )\right ) - 2 \, b \sinh \left (n \log \left (x\right )\right )\right ) - \cosh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )^{2} - \sinh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )^{2} - 1}{2 \, {\left (n \cosh \left (n \log \left (x\right )\right ) + n \sinh \left (n \log \left (x\right )\right )\right )}} \]
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\[ \int x^{-1-n} \cosh ^2\left (a+b x^n\right ) \, dx=\int x^{- n - 1} \cosh ^{2}{\left (a + b x^{n} \right )}\, dx \]
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none
Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.70 \[ \int x^{-1-n} \cosh ^2\left (a+b x^n\right ) \, dx=-\frac {b e^{\left (-2 \, a\right )} \Gamma \left (-1, 2 \, b x^{n}\right )}{2 \, n} + \frac {b e^{\left (2 \, a\right )} \Gamma \left (-1, -2 \, b x^{n}\right )}{2 \, n} - \frac {1}{2 \, n x^{n}} \]
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\[ \int x^{-1-n} \cosh ^2\left (a+b x^n\right ) \, dx=\int { x^{-n - 1} \cosh \left (b x^{n} + a\right )^{2} \,d x } \]
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Timed out. \[ \int x^{-1-n} \cosh ^2\left (a+b x^n\right ) \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x^n\right )}^2}{x^{n+1}} \,d x \]
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