Optimal. Leaf size=45 \[ \frac {b \sinh (a) \text {Chi}\left (b x^n\right )}{n}+\frac {b \cosh (a) \text {Shi}\left (b x^n\right )}{n}-\frac {x^{-n} \cosh \left (a+b x^n\right )}{n} \]
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Rubi [A] time = 0.09, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5321, 3297, 3303, 3298, 3301} \[ \frac {b \sinh (a) \text {Chi}\left (b x^n\right )}{n}+\frac {b \cosh (a) \text {Shi}\left (b x^n\right )}{n}-\frac {x^{-n} \cosh \left (a+b x^n\right )}{n} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5321
Rubi steps
\begin {align*} \int x^{-1-n} \cosh \left (a+b x^n\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh (a+b x)}{x^2} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-n} \cosh \left (a+b x^n\right )}{n}+\frac {b \operatorname {Subst}\left (\int \frac {\sinh (a+b x)}{x} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-n} \cosh \left (a+b x^n\right )}{n}+\frac {(b \cosh (a)) \operatorname {Subst}\left (\int \frac {\sinh (b x)}{x} \, dx,x,x^n\right )}{n}+\frac {(b \sinh (a)) \operatorname {Subst}\left (\int \frac {\cosh (b x)}{x} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-n} \cosh \left (a+b x^n\right )}{n}+\frac {b \text {Chi}\left (b x^n\right ) \sinh (a)}{n}+\frac {b \cosh (a) \text {Shi}\left (b x^n\right )}{n}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 46, normalized size = 1.02 \[ \frac {x^{-n} \left (b \sinh (a) x^n \text {Chi}\left (b x^n\right )+b \cosh (a) x^n \text {Shi}\left (b x^n\right )-\cosh \left (a+b x^n\right )\right )}{n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 140, normalized size = 3.11 \[ \frac {{\left ({\left (b \cosh \relax (a) + b \sinh \relax (a)\right )} \cosh \left (n \log \relax (x)\right ) + {\left (b \cosh \relax (a) + b \sinh \relax (a)\right )} \sinh \left (n \log \relax (x)\right )\right )} {\rm Ei}\left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right )\right ) - {\left ({\left (b \cosh \relax (a) - b \sinh \relax (a)\right )} \cosh \left (n \log \relax (x)\right ) + {\left (b \cosh \relax (a) - b \sinh \relax (a)\right )} \sinh \left (n \log \relax (x)\right )\right )} {\rm Ei}\left (-b \cosh \left (n \log \relax (x)\right ) - b \sinh \left (n \log \relax (x)\right )\right ) - 2 \, \cosh \left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a\right )}{2 \, {\left (n \cosh \left (n \log \relax (x)\right ) + n \sinh \left (n \log \relax (x)\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{-n - 1} \cosh \left (b x^{n} + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 74, normalized size = 1.64 \[ -\frac {{\mathrm e}^{-a -b \,x^{n}} x^{-n}}{2 n}+\frac {b \,{\mathrm e}^{-a} \Ei \left (1, b \,x^{n}\right )}{2 n}-\frac {{\mathrm e}^{a +b \,x^{n}} x^{-n}}{2 n}-\frac {b \,{\mathrm e}^{a} \Ei \left (1, -b \,x^{n}\right )}{2 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 34, normalized size = 0.76 \[ -\frac {b e^{\left (-a\right )} \Gamma \left (-1, b x^{n}\right )}{2 \, n} + \frac {b e^{a} \Gamma \left (-1, -b x^{n}\right )}{2 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {cosh}\left (a+b\,x^n\right )}{x^{n+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{- n - 1} \cosh {\left (a + b x^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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