Optimal. Leaf size=67 \[ -\frac{e^a x \left (-b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-b x^n\right )}{2 n}-\frac{e^{-a} x \left (b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},b x^n\right )}{2 n} \]
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Rubi [A] time = 0.0215728, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5307, 2208} \[ -\frac{e^a x \left (-b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-b x^n\right )}{2 n}-\frac{e^{-a} x \left (b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},b x^n\right )}{2 n} \]
Antiderivative was successfully verified.
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Rule 5307
Rule 2208
Rubi steps
\begin{align*} \int \cosh \left (a+b x^n\right ) \, dx &=\frac{1}{2} \int e^{-a-b x^n} \, dx+\frac{1}{2} \int e^{a+b x^n} \, dx\\ &=-\frac{e^a x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac{1}{n},-b x^n\right )}{2 n}-\frac{e^{-a} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac{1}{n},b x^n\right )}{2 n}\\ \end{align*}
Mathematica [A] time = 0.0671503, size = 77, normalized size = 1.15 \[ -\frac{x \left (-b^2 x^{2 n}\right )^{-1/n} \left ((\cosh (a)-\sinh (a)) \left (-b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},b x^n\right )+(\sinh (a)+\cosh (a)) \left (b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},-b x^n\right )\right )}{2 n} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.055, size = 74, normalized size = 1.1 \begin{align*} x{\mbox{$_1$F$_2$}({\frac{1}{2\,n}};\,{\frac{1}{2}},1+{\frac{1}{2\,n}};\,{\frac{{x}^{2\,n}{b}^{2}}{4}})}\cosh \left ( a \right ) +{\frac{{x}^{n+1}b\sinh \left ( a \right ) }{n+1}{\mbox{$_1$F$_2$}({\frac{1}{2}}+{\frac{1}{2\,n}};\,{\frac{3}{2}},{\frac{3}{2}}+{\frac{1}{2\,n}};\,{\frac{{x}^{2\,n}{b}^{2}}{4}})}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13017, size = 82, normalized size = 1.22 \begin{align*} -\frac{x e^{\left (-a\right )} \Gamma \left (\frac{1}{n}, b x^{n}\right )}{2 \, \left (b x^{n}\right )^{\left (\frac{1}{n}\right )} n} - \frac{x e^{a} \Gamma \left (\frac{1}{n}, -b x^{n}\right )}{2 \, \left (-b x^{n}\right )^{\left (\frac{1}{n}\right )} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cosh \left (b x^{n} + a\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (a + b x^{n} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh \left (b x^{n} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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