Integrand size = 14, antiderivative size = 43 \[ \int \frac {\cosh ^2\left (a+b x^n\right )}{x} \, dx=\frac {\cosh (2 a) \text {Chi}\left (2 b x^n\right )}{2 n}+\frac {\log (x)}{2}+\frac {\sinh (2 a) \text {Shi}\left (2 b x^n\right )}{2 n} \]
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Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5471, 5427, 5425, 5424} \[ \int \frac {\cosh ^2\left (a+b x^n\right )}{x} \, dx=\frac {\cosh (2 a) \text {Chi}\left (2 b x^n\right )}{2 n}+\frac {\sinh (2 a) \text {Shi}\left (2 b x^n\right )}{2 n}+\frac {\log (x)}{2} \]
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Rule 5424
Rule 5425
Rule 5427
Rule 5471
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.84 \[ \int \frac {\cosh ^2\left (a+b x^n\right )}{x} \, dx=\frac {\cosh (2 a) \text {Chi}\left (2 b x^n\right )+n \log (x)+\sinh (2 a) \text {Shi}\left (2 b x^n\right )}{2 n} \]
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Time = 0.50 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.93
method | result | size |
risch | \(\frac {\ln \left (x \right )}{2}-\frac {{\mathrm e}^{-2 a} \operatorname {Ei}_{1}\left (2 b \,x^{n}\right )}{4 n}-\frac {{\mathrm e}^{2 a} \operatorname {Ei}_{1}\left (-2 b \,x^{n}\right )}{4 n}\) | \(40\) |
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Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.60 \[ \int \frac {\cosh ^2\left (a+b x^n\right )}{x} \, dx=\frac {{\left (\cosh \left (2 \, a\right ) + \sinh \left (2 \, a\right )\right )} {\rm Ei}\left (2 \, b \cosh \left (n \log \left (x\right )\right ) + 2 \, b \sinh \left (n \log \left (x\right )\right )\right ) + {\left (\cosh \left (2 \, a\right ) - \sinh \left (2 \, a\right )\right )} {\rm Ei}\left (-2 \, b \cosh \left (n \log \left (x\right )\right ) - 2 \, b \sinh \left (n \log \left (x\right )\right )\right ) + 2 \, n \log \left (x\right )}{4 \, n} \]
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\[ \int \frac {\cosh ^2\left (a+b x^n\right )}{x} \, dx=\int \frac {\cosh ^{2}{\left (a + b x^{n} \right )}}{x}\, dx \]
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Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {\cosh ^2\left (a+b x^n\right )}{x} \, dx=\frac {{\rm Ei}\left (2 \, b x^{n}\right ) e^{\left (2 \, a\right )}}{4 \, n} + \frac {{\rm Ei}\left (-2 \, b x^{n}\right ) e^{\left (-2 \, a\right )}}{4 \, n} + \frac {1}{2} \, \log \left (x\right ) \]
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\[ \int \frac {\cosh ^2\left (a+b x^n\right )}{x} \, dx=\int { \frac {\cosh \left (b x^{n} + a\right )^{2}}{x} \,d x } \]
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Timed out. \[ \int \frac {\cosh ^2\left (a+b x^n\right )}{x} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x^n\right )}^2}{x} \,d x \]
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