Integrand size = 14, antiderivative size = 67 \[ \int \frac {\cosh ^3\left (a+b x^n\right )}{x} \, dx=\frac {3 \cosh (a) \text {Chi}\left (b x^n\right )}{4 n}+\frac {\cosh (3 a) \text {Chi}\left (3 b x^n\right )}{4 n}+\frac {3 \sinh (a) \text {Shi}\left (b x^n\right )}{4 n}+\frac {\sinh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n} \]
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Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5471, 5427, 5425, 5424} \[ \int \frac {\cosh ^3\left (a+b x^n\right )}{x} \, dx=\frac {3 \cosh (a) \text {Chi}\left (b x^n\right )}{4 n}+\frac {\cosh (3 a) \text {Chi}\left (3 b x^n\right )}{4 n}+\frac {3 \sinh (a) \text {Shi}\left (b x^n\right )}{4 n}+\frac {\sinh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n} \]
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Rule 5424
Rule 5425
Rule 5427
Rule 5471
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 \cosh \left (a+b x^n\right )}{4 x}+\frac {\cosh \left (3 a+3 b x^n\right )}{4 x}\right ) \, dx \\ & = \frac {1}{4} \int \frac {\cosh \left (3 a+3 b x^n\right )}{x} \, dx+\frac {3}{4} \int \frac {\cosh \left (a+b x^n\right )}{x} \, dx \\ & = \frac {1}{4} (3 \cosh (a)) \int \frac {\cosh \left (b x^n\right )}{x} \, dx+\frac {1}{4} \cosh (3 a) \int \frac {\cosh \left (3 b x^n\right )}{x} \, dx+\frac {1}{4} (3 \sinh (a)) \int \frac {\sinh \left (b x^n\right )}{x} \, dx+\frac {1}{4} \sinh (3 a) \int \frac {\sinh \left (3 b x^n\right )}{x} \, dx \\ & = \frac {3 \cosh (a) \text {Chi}\left (b x^n\right )}{4 n}+\frac {\cosh (3 a) \text {Chi}\left (3 b x^n\right )}{4 n}+\frac {3 \sinh (a) \text {Shi}\left (b x^n\right )}{4 n}+\frac {\sinh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.78 \[ \int \frac {\cosh ^3\left (a+b x^n\right )}{x} \, dx=\frac {3 \cosh (a) \text {Chi}\left (b x^n\right )+\cosh (3 a) \text {Chi}\left (3 b x^n\right )+3 \sinh (a) \text {Shi}\left (b x^n\right )+\sinh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n} \]
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Time = 0.89 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00
method | result | size |
risch | \(-\frac {{\mathrm e}^{-3 a} \operatorname {Ei}_{1}\left (3 b \,x^{n}\right )}{8 n}-\frac {3 \,{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b \,x^{n}\right )}{8 n}-\frac {{\mathrm e}^{3 a} \operatorname {Ei}_{1}\left (-3 b \,x^{n}\right )}{8 n}-\frac {3 \,{\mathrm e}^{a} \operatorname {Ei}_{1}\left (-b \,x^{n}\right )}{8 n}\) | \(67\) |
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Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.70 \[ \int \frac {\cosh ^3\left (a+b x^n\right )}{x} \, dx=\frac {{\left (\cosh \left (3 \, a\right ) + \sinh \left (3 \, a\right )\right )} {\rm Ei}\left (3 \, b \cosh \left (n \log \left (x\right )\right ) + 3 \, b \sinh \left (n \log \left (x\right )\right )\right ) + 3 \, {\left (\cosh \left (a\right ) + \sinh \left (a\right )\right )} {\rm Ei}\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right )\right ) + 3 \, {\left (\cosh \left (a\right ) - \sinh \left (a\right )\right )} {\rm Ei}\left (-b \cosh \left (n \log \left (x\right )\right ) - b \sinh \left (n \log \left (x\right )\right )\right ) + {\left (\cosh \left (3 \, a\right ) - \sinh \left (3 \, a\right )\right )} {\rm Ei}\left (-3 \, b \cosh \left (n \log \left (x\right )\right ) - 3 \, b \sinh \left (n \log \left (x\right )\right )\right )}{8 \, n} \]
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\[ \int \frac {\cosh ^3\left (a+b x^n\right )}{x} \, dx=\int \frac {\cosh ^{3}{\left (a + b x^{n} \right )}}{x}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.93 \[ \int \frac {\cosh ^3\left (a+b x^n\right )}{x} \, dx=\frac {{\rm Ei}\left (3 \, b x^{n}\right ) e^{\left (3 \, a\right )}}{8 \, n} + \frac {3 \, {\rm Ei}\left (-b x^{n}\right ) e^{\left (-a\right )}}{8 \, n} + \frac {{\rm Ei}\left (-3 \, b x^{n}\right ) e^{\left (-3 \, a\right )}}{8 \, n} + \frac {3 \, {\rm Ei}\left (b x^{n}\right ) e^{a}}{8 \, n} \]
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\[ \int \frac {\cosh ^3\left (a+b x^n\right )}{x} \, dx=\int { \frac {\cosh \left (b x^{n} + a\right )^{3}}{x} \,d x } \]
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Timed out. \[ \int \frac {\cosh ^3\left (a+b x^n\right )}{x} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x^n\right )}^3}{x} \,d x \]
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