Integrand size = 9, antiderivative size = 25 \[ \int \sinh (b x) \text {Shi}(b x) \, dx=\frac {\cosh (b x) \text {Shi}(b x)}{b}-\frac {\text {Shi}(2 b x)}{2 b} \]
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Time = 0.04 (sec) , antiderivative size = 25, 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 = {6675, 12, 5556, 3379} \[ \int \sinh (b x) \text {Shi}(b x) \, dx=\frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\text {Shi}(2 b x)}{2 b} \]
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Rule 12
Rule 3379
Rule 5556
Rule 6675
Rubi steps \begin{align*} \text {integral}& = \frac {\cosh (b x) \text {Shi}(b x)}{b}-\int \frac {\cosh (b x) \sinh (b x)}{b x} \, dx \\ & = \frac {\cosh (b x) \text {Shi}(b x)}{b}-\frac {\int \frac {\cosh (b x) \sinh (b x)}{x} \, dx}{b} \\ & = \frac {\cosh (b x) \text {Shi}(b x)}{b}-\frac {\int \frac {\sinh (2 b x)}{2 x} \, dx}{b} \\ & = \frac {\cosh (b x) \text {Shi}(b x)}{b}-\frac {\int \frac {\sinh (2 b x)}{x} \, dx}{2 b} \\ & = \frac {\cosh (b x) \text {Shi}(b x)}{b}-\frac {\text {Shi}(2 b x)}{2 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \sinh (b x) \text {Shi}(b x) \, dx=\frac {\cosh (b x) \text {Shi}(b x)}{b}-\frac {\text {Shi}(2 b x)}{2 b} \]
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Time = 0.37 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {\cosh \left (b x \right ) \operatorname {Shi}\left (b x \right )-\frac {\operatorname {Shi}\left (2 b x \right )}{2}}{b}\) | \(22\) |
default | \(\frac {\cosh \left (b x \right ) \operatorname {Shi}\left (b x \right )-\frac {\operatorname {Shi}\left (2 b x \right )}{2}}{b}\) | \(22\) |
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\[ \int \sinh (b x) \text {Shi}(b x) \, dx=\int { {\rm Shi}\left (b x\right ) \sinh \left (b x\right ) \,d x } \]
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\[ \int \sinh (b x) \text {Shi}(b x) \, dx=\int \sinh {\left (b x \right )} \operatorname {Shi}{\left (b x \right )}\, dx \]
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\[ \int \sinh (b x) \text {Shi}(b x) \, dx=\int { {\rm Shi}\left (b x\right ) \sinh \left (b x\right ) \,d x } \]
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\[ \int \sinh (b x) \text {Shi}(b x) \, dx=\int { {\rm Shi}\left (b x\right ) \sinh \left (b x\right ) \,d x } \]
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Timed out. \[ \int \sinh (b x) \text {Shi}(b x) \, dx=\int \mathrm {sinhint}\left (b\,x\right )\,\mathrm {sinh}\left (b\,x\right ) \,d x \]
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