Integrand size = 9, antiderivative size = 34 \[ \int \cosh (b x) \text {Shi}(b x) \, dx=-\frac {\text {Chi}(2 b x)}{2 b}+\frac {\log (x)}{2 b}+\frac {\sinh (b x) \text {Shi}(b x)}{b} \]
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Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6681, 12, 3393, 3382} \[ \int \cosh (b x) \text {Shi}(b x) \, dx=-\frac {\text {Chi}(2 b x)}{2 b}+\frac {\text {Shi}(b x) \sinh (b x)}{b}+\frac {\log (x)}{2 b} \]
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Rule 12
Rule 3382
Rule 3393
Rule 6681
Rubi steps \begin{align*} \text {integral}& = \frac {\sinh (b x) \text {Shi}(b x)}{b}-\int \frac {\sinh ^2(b x)}{b x} \, dx \\ & = \frac {\sinh (b x) \text {Shi}(b x)}{b}-\frac {\int \frac {\sinh ^2(b x)}{x} \, dx}{b} \\ & = \frac {\sinh (b x) \text {Shi}(b x)}{b}+\frac {\int \left (\frac {1}{2 x}-\frac {\cosh (2 b x)}{2 x}\right ) \, dx}{b} \\ & = \frac {\log (x)}{2 b}+\frac {\sinh (b x) \text {Shi}(b x)}{b}-\frac {\int \frac {\cosh (2 b x)}{x} \, dx}{2 b} \\ & = -\frac {\text {Chi}(2 b x)}{2 b}+\frac {\log (x)}{2 b}+\frac {\sinh (b x) \text {Shi}(b x)}{b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \cosh (b x) \text {Shi}(b x) \, dx=-\frac {\text {Chi}(2 b x)}{2 b}+\frac {\log (b x)}{2 b}+\frac {\sinh (b x) \text {Shi}(b x)}{b} \]
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Time = 0.61 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\operatorname {Shi}\left (b x \right ) \sinh \left (b x \right )+\frac {\ln \left (b x \right )}{2}-\frac {\operatorname {Chi}\left (2 b x \right )}{2}}{b}\) | \(28\) |
default | \(\frac {\operatorname {Shi}\left (b x \right ) \sinh \left (b x \right )+\frac {\ln \left (b x \right )}{2}-\frac {\operatorname {Chi}\left (2 b x \right )}{2}}{b}\) | \(28\) |
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\[ \int \cosh (b x) \text {Shi}(b x) \, dx=\int { {\rm Shi}\left (b x\right ) \cosh \left (b x\right ) \,d x } \]
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\[ \int \cosh (b x) \text {Shi}(b x) \, dx=\int \cosh {\left (b x \right )} \operatorname {Shi}{\left (b x \right )}\, dx \]
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\[ \int \cosh (b x) \text {Shi}(b x) \, dx=\int { {\rm Shi}\left (b x\right ) \cosh \left (b x\right ) \,d x } \]
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\[ \int \cosh (b x) \text {Shi}(b x) \, dx=\int { {\rm Shi}\left (b x\right ) \cosh \left (b x\right ) \,d x } \]
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Timed out. \[ \int \cosh (b x) \text {Shi}(b x) \, dx=\int \mathrm {sinhint}\left (b\,x\right )\,\mathrm {cosh}\left (b\,x\right ) \,d x \]
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