Integrand size = 14, antiderivative size = 37 \[ \int \frac {\cosh ^2\left (a+b x^2\right )}{x} \, dx=\frac {1}{4} \cosh (2 a) \text {Chi}\left (2 b x^2\right )+\frac {\log (x)}{2}+\frac {1}{4} \sinh (2 a) \text {Shi}\left (2 b x^2\right ) \]
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Time = 0.04 (sec) , antiderivative size = 37, 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 = {5449, 5427, 5425, 5424} \[ \int \frac {\cosh ^2\left (a+b x^2\right )}{x} \, dx=\frac {1}{4} \cosh (2 a) \text {Chi}\left (2 b x^2\right )+\frac {1}{4} \sinh (2 a) \text {Shi}\left (2 b x^2\right )+\frac {\log (x)}{2} \]
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Rule 5424
Rule 5425
Rule 5427
Rule 5449
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2 x}+\frac {\cosh \left (2 a+2 b x^2\right )}{2 x}\right ) \, dx \\ & = \frac {\log (x)}{2}+\frac {1}{2} \int \frac {\cosh \left (2 a+2 b x^2\right )}{x} \, dx \\ & = \frac {\log (x)}{2}+\frac {1}{2} \cosh (2 a) \int \frac {\cosh \left (2 b x^2\right )}{x} \, dx+\frac {1}{2} \sinh (2 a) \int \frac {\sinh \left (2 b x^2\right )}{x} \, dx \\ & = \frac {1}{4} \cosh (2 a) \text {Chi}\left (2 b x^2\right )+\frac {\log (x)}{2}+\frac {1}{4} \sinh (2 a) \text {Shi}\left (2 b x^2\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int \frac {\cosh ^2\left (a+b x^2\right )}{x} \, dx=\frac {1}{4} \left (\cosh (2 a) \text {Chi}\left (2 b x^2\right )+2 \log (x)+\sinh (2 a) \text {Shi}\left (2 b x^2\right )\right ) \]
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Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92
method | result | size |
risch | \(\frac {\ln \left (x \right )}{2}-\frac {{\mathrm e}^{-2 a} \operatorname {Ei}_{1}\left (2 b \,x^{2}\right )}{8}-\frac {{\mathrm e}^{2 a} \operatorname {Ei}_{1}\left (-2 b \,x^{2}\right )}{8}\) | \(34\) |
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Time = 0.24 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.32 \[ \int \frac {\cosh ^2\left (a+b x^2\right )}{x} \, dx=\frac {1}{8} \, {\left ({\rm Ei}\left (2 \, b x^{2}\right ) + {\rm Ei}\left (-2 \, b x^{2}\right )\right )} \cosh \left (2 \, a\right ) + \frac {1}{8} \, {\left ({\rm Ei}\left (2 \, b x^{2}\right ) - {\rm Ei}\left (-2 \, b x^{2}\right )\right )} \sinh \left (2 \, a\right ) + \frac {1}{2} \, \log \left (x\right ) \]
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\[ \int \frac {\cosh ^2\left (a+b x^2\right )}{x} \, dx=\int \frac {\cosh ^{2}{\left (a + b x^{2} \right )}}{x}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84 \[ \int \frac {\cosh ^2\left (a+b x^2\right )}{x} \, dx=\frac {1}{8} \, {\rm Ei}\left (2 \, b x^{2}\right ) e^{\left (2 \, a\right )} + \frac {1}{8} \, {\rm Ei}\left (-2 \, b x^{2}\right ) e^{\left (-2 \, a\right )} + \frac {1}{2} \, \log \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \frac {\cosh ^2\left (a+b x^2\right )}{x} \, dx=\frac {1}{8} \, {\rm Ei}\left (2 \, b x^{2}\right ) e^{\left (2 \, a\right )} + \frac {1}{8} \, {\rm Ei}\left (-2 \, b x^{2}\right ) e^{\left (-2 \, a\right )} + \frac {1}{4} \, \log \left (b x^{2}\right ) \]
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Timed out. \[ \int \frac {\cosh ^2\left (a+b x^2\right )}{x} \, dx=\int \frac {{\mathrm {cosh}\left (b\,x^2+a\right )}^2}{x} \,d x \]
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