Optimal. Leaf size=67 \[ -\frac {e^a x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-b x^n\right )}{2 n}-\frac {e^{-a} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},b x^n\right )}{2 n} \]
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Rubi [A] time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 2208
Rule 5307
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*}
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Mathematica [A] time = 0.08, size = 77, normalized size = 1.15 \[ -\frac {x \left (-b^2 x^{2 n}\right )^{-1/n} \left ((\cosh (a)-\sinh (a)) \left (-b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},b x^n\right )+(\sinh (a)+\cosh (a)) \left (b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},-b x^n\right )\right )}{2 n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cosh \left (b x^{n} + a\right ), x\right ) \]
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 \[ \int \cosh \left (b x^{n} + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 74, normalized size = 1.10 \[ x \hypergeom \left (\left [\frac {1}{2 n}\right ], \left [\frac {1}{2}, 1+\frac {1}{2 n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \cosh \relax (a )+\frac {x^{n +1} b \hypergeom \left (\left [\frac {1}{2}+\frac {1}{2 n}\right ], \left [\frac {3}{2}, \frac {3}{2}+\frac {1}{2 n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \sinh \relax (a )}{n +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 61, normalized size = 0.91 \[ -\frac {x e^{\left (-a\right )} \Gamma \left (\frac {1}{n}, b x^{n}\right )}{2 \, \left (b x^{n}\right )^{\left (\frac {1}{n}\right )} n} - \frac {x e^{a} \Gamma \left (\frac {1}{n}, -b x^{n}\right )}{2 \, \left (-b x^{n}\right )^{\left (\frac {1}{n}\right )} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {cosh}\left (a+b\,x^n\right ) \,d x \]
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 \[ \int \cosh {\left (a + b x^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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