Integrand size = 10, antiderivative size = 62 \[ \int x \text {Chi}(b x) \sinh (b x) \, dx=-\frac {x}{2 b}+\frac {x \cosh (b x) \text {Chi}(b x)}{b}-\frac {\cosh (b x) \sinh (b x)}{2 b^2}-\frac {\text {Chi}(b x) \sinh (b x)}{b^2}+\frac {\text {Shi}(2 b x)}{2 b^2} \]
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Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6684, 12, 2715, 8, 6676, 5556, 3379} \[ \int x \text {Chi}(b x) \sinh (b x) \, dx=-\frac {\text {Chi}(b x) \sinh (b x)}{b^2}+\frac {\text {Shi}(2 b x)}{2 b^2}-\frac {\sinh (b x) \cosh (b x)}{2 b^2}+\frac {x \text {Chi}(b x) \cosh (b x)}{b}-\frac {x}{2 b} \]
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Rule 8
Rule 12
Rule 2715
Rule 3379
Rule 5556
Rule 6676
Rule 6684
Rubi steps \begin{align*} \text {integral}& = \frac {x \cosh (b x) \text {Chi}(b x)}{b}-\frac {\int \cosh (b x) \text {Chi}(b x) \, dx}{b}-\int \frac {\cosh ^2(b x)}{b} \, dx \\ & = \frac {x \cosh (b x) \text {Chi}(b x)}{b}-\frac {\text {Chi}(b x) \sinh (b x)}{b^2}-\frac {\int \cosh ^2(b x) \, dx}{b}+\frac {\int \frac {\cosh (b x) \sinh (b x)}{b x} \, dx}{b} \\ & = \frac {x \cosh (b x) \text {Chi}(b x)}{b}-\frac {\cosh (b x) \sinh (b x)}{2 b^2}-\frac {\text {Chi}(b x) \sinh (b x)}{b^2}+\frac {\int \frac {\cosh (b x) \sinh (b x)}{x} \, dx}{b^2}-\frac {\int 1 \, dx}{2 b} \\ & = -\frac {x}{2 b}+\frac {x \cosh (b x) \text {Chi}(b x)}{b}-\frac {\cosh (b x) \sinh (b x)}{2 b^2}-\frac {\text {Chi}(b x) \sinh (b x)}{b^2}+\frac {\int \frac {\sinh (2 b x)}{2 x} \, dx}{b^2} \\ & = -\frac {x}{2 b}+\frac {x \cosh (b x) \text {Chi}(b x)}{b}-\frac {\cosh (b x) \sinh (b x)}{2 b^2}-\frac {\text {Chi}(b x) \sinh (b x)}{b^2}+\frac {\int \frac {\sinh (2 b x)}{x} \, dx}{2 b^2} \\ & = -\frac {x}{2 b}+\frac {x \cosh (b x) \text {Chi}(b x)}{b}-\frac {\cosh (b x) \sinh (b x)}{2 b^2}-\frac {\text {Chi}(b x) \sinh (b x)}{b^2}+\frac {\text {Shi}(2 b x)}{2 b^2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.71 \[ \int x \text {Chi}(b x) \sinh (b x) \, dx=-\frac {2 b x+\text {Chi}(b x) (-4 b x \cosh (b x)+4 \sinh (b x))+\sinh (2 b x)-2 \text {Shi}(2 b x)}{4 b^2} \]
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Time = 0.71 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(\frac {\operatorname {Chi}\left (b x \right ) \left (b x \cosh \left (b x \right )-\sinh \left (b x \right )\right )-\frac {\cosh \left (b x \right ) \sinh \left (b x \right )}{2}-\frac {b x}{2}+\frac {\operatorname {Shi}\left (2 b x \right )}{2}}{b^{2}}\) | \(46\) |
default | \(\frac {\operatorname {Chi}\left (b x \right ) \left (b x \cosh \left (b x \right )-\sinh \left (b x \right )\right )-\frac {\cosh \left (b x \right ) \sinh \left (b x \right )}{2}-\frac {b x}{2}+\frac {\operatorname {Shi}\left (2 b x \right )}{2}}{b^{2}}\) | \(46\) |
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\[ \int x \text {Chi}(b x) \sinh (b x) \, dx=\int { x {\rm Chi}\left (b x\right ) \sinh \left (b x\right ) \,d x } \]
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\[ \int x \text {Chi}(b x) \sinh (b x) \, dx=\int x \sinh {\left (b x \right )} \operatorname {Chi}\left (b x\right )\, dx \]
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\[ \int x \text {Chi}(b x) \sinh (b x) \, dx=\int { x {\rm Chi}\left (b x\right ) \sinh \left (b x\right ) \,d x } \]
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\[ \int x \text {Chi}(b x) \sinh (b x) \, dx=\int { x {\rm Chi}\left (b x\right ) \sinh \left (b x\right ) \,d x } \]
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Timed out. \[ \int x \text {Chi}(b x) \sinh (b x) \, dx=\int x\,\mathrm {coshint}\left (b\,x\right )\,\mathrm {sinh}\left (b\,x\right ) \,d x \]
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