Optimal. Leaf size=43 \[ \frac {\cosh (2 a) \text {Chi}\left (2 b x^n\right )}{2 n}+\frac {\sinh (2 a) \text {Shi}\left (2 b x^n\right )}{2 n}+\frac {\log (x)}{2} \]
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Rubi [A] time = 0.06, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5363, 5319, 5317, 5316} \[ \frac {\cosh (2 a) \text {Chi}\left (2 b x^n\right )}{2 n}+\frac {\sinh (2 a) \text {Shi}\left (2 b x^n\right )}{2 n}+\frac {\log (x)}{2} \]
Antiderivative was successfully verified.
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Rule 5316
Rule 5317
Rule 5319
Rule 5363
Rubi steps
\begin {align*} \int \frac {\cosh ^2\left (a+b x^n\right )}{x} \, dx &=\int \left (\frac {1}{2 x}+\frac {\cosh \left (2 a+2 b x^n\right )}{2 x}\right ) \, dx\\ &=\frac {\log (x)}{2}+\frac {1}{2} \int \frac {\cosh \left (2 a+2 b x^n\right )}{x} \, dx\\ &=\frac {\log (x)}{2}+\frac {1}{2} \cosh (2 a) \int \frac {\cosh \left (2 b x^n\right )}{x} \, dx+\frac {1}{2} \sinh (2 a) \int \frac {\sinh \left (2 b x^n\right )}{x} \, dx\\ &=\frac {\cosh (2 a) \text {Chi}\left (2 b x^n\right )}{2 n}+\frac {\log (x)}{2}+\frac {\sinh (2 a) \text {Shi}\left (2 b x^n\right )}{2 n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 36, normalized size = 0.84 \[ \frac {\cosh (2 a) \text {Chi}\left (2 b x^n\right )+\sinh (2 a) \text {Shi}\left (2 b x^n\right )+n \log (x)}{2 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 69, normalized size = 1.60 \[ \frac {{\left (\cosh \left (2 \, a\right ) + \sinh \left (2 \, a\right )\right )} {\rm Ei}\left (2 \, b \cosh \left (n \log \relax (x)\right ) + 2 \, b \sinh \left (n \log \relax (x)\right )\right ) + {\left (\cosh \left (2 \, a\right ) - \sinh \left (2 \, a\right )\right )} {\rm Ei}\left (-2 \, b \cosh \left (n \log \relax (x)\right ) - 2 \, b \sinh \left (n \log \relax (x)\right )\right ) + 2 \, n \log \relax (x)}{4 \, n} \]
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 \[ \int \frac {\cosh \left (b x^{n} + a\right )^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 40, normalized size = 0.93 \[ \frac {\ln \relax (x )}{2}-\frac {{\mathrm e}^{-2 a} \Ei \left (1, 2 b \,x^{n}\right )}{4 n}-\frac {{\mathrm e}^{2 a} \Ei \left (1, -2 b \,x^{n}\right )}{4 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 37, normalized size = 0.86 \[ \frac {{\rm Ei}\left (2 \, b x^{n}\right ) e^{\left (2 \, a\right )}}{4 \, n} + \frac {{\rm Ei}\left (-2 \, b x^{n}\right ) e^{\left (-2 \, a\right )}}{4 \, n} + \frac {1}{2} \, \log \relax (x) \]
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 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x^n\right )}^2}{x} \,d x \]
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 \[ \int \frac {\cosh ^{2}{\left (a + b x^{n} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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