Integrand size = 10, antiderivative size = 61 \[ \int x \cosh (b x) \text {Chi}(b x) \, dx=-\frac {\cosh (b x) \text {Chi}(b x)}{b^2}+\frac {\text {Chi}(2 b x)}{2 b^2}+\frac {\log (x)}{2 b^2}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}-\frac {\sinh ^2(b x)}{2 b^2} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 61, 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 = {6678, 12, 2644, 30, 6682, 3393, 3382} \[ \int x \cosh (b x) \text {Chi}(b x) \, dx=\frac {\text {Chi}(2 b x)}{2 b^2}-\frac {\text {Chi}(b x) \cosh (b x)}{b^2}+\frac {\log (x)}{2 b^2}-\frac {\sinh ^2(b x)}{2 b^2}+\frac {x \text {Chi}(b x) \sinh (b x)}{b} \]
[In]
[Out]
Rule 12
Rule 30
Rule 2644
Rule 3382
Rule 3393
Rule 6678
Rule 6682
Rubi steps \begin{align*} \text {integral}& = \frac {x \text {Chi}(b x) \sinh (b x)}{b}-\frac {\int \text {Chi}(b x) \sinh (b x) \, dx}{b}-\int \frac {\cosh (b x) \sinh (b x)}{b} \, dx \\ & = -\frac {\cosh (b x) \text {Chi}(b x)}{b^2}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}+\frac {\int \frac {\cosh ^2(b x)}{b x} \, dx}{b}-\frac {\int \cosh (b x) \sinh (b x) \, dx}{b} \\ & = -\frac {\cosh (b x) \text {Chi}(b x)}{b^2}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}+\frac {\int \frac {\cosh ^2(b x)}{x} \, dx}{b^2}+\frac {\text {Subst}(\int x \, dx,x,i \sinh (b x))}{b^2} \\ & = -\frac {\cosh (b x) \text {Chi}(b x)}{b^2}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}-\frac {\sinh ^2(b x)}{2 b^2}+\frac {\int \left (\frac {1}{2 x}+\frac {\cosh (2 b x)}{2 x}\right ) \, dx}{b^2} \\ & = -\frac {\cosh (b x) \text {Chi}(b x)}{b^2}+\frac {\log (x)}{2 b^2}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}-\frac {\sinh ^2(b x)}{2 b^2}+\frac {\int \frac {\cosh (2 b x)}{x} \, dx}{2 b^2} \\ & = -\frac {\cosh (b x) \text {Chi}(b x)}{b^2}+\frac {\text {Chi}(2 b x)}{2 b^2}+\frac {\log (x)}{2 b^2}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}-\frac {\sinh ^2(b x)}{2 b^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int x \cosh (b x) \text {Chi}(b x) \, dx=\frac {-\cosh (2 b x)+2 \text {Chi}(2 b x)+2 \log (x)+4 \text {Chi}(b x) (-\cosh (b x)+b x \sinh (b x))}{4 b^2} \]
[In]
[Out]
Time = 0.71 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {\operatorname {Chi}\left (b x \right ) \left (b x \sinh \left (b x \right )-\cosh \left (b x \right )\right )-\frac {\cosh \left (b x \right )^{2}}{2}+\frac {\ln \left (b x \right )}{2}+\frac {\operatorname {Chi}\left (2 b x \right )}{2}}{b^{2}}\) | \(46\) |
default | \(\frac {\operatorname {Chi}\left (b x \right ) \left (b x \sinh \left (b x \right )-\cosh \left (b x \right )\right )-\frac {\cosh \left (b x \right )^{2}}{2}+\frac {\ln \left (b x \right )}{2}+\frac {\operatorname {Chi}\left (2 b x \right )}{2}}{b^{2}}\) | \(46\) |
[In]
[Out]
\[ \int x \cosh (b x) \text {Chi}(b x) \, dx=\int { x {\rm Chi}\left (b x\right ) \cosh \left (b x\right ) \,d x } \]
[In]
[Out]
\[ \int x \cosh (b x) \text {Chi}(b x) \, dx=\int x \cosh {\left (b x \right )} \operatorname {Chi}\left (b x\right )\, dx \]
[In]
[Out]
\[ \int x \cosh (b x) \text {Chi}(b x) \, dx=\int { x {\rm Chi}\left (b x\right ) \cosh \left (b x\right ) \,d x } \]
[In]
[Out]
\[ \int x \cosh (b x) \text {Chi}(b x) \, dx=\int { x {\rm Chi}\left (b x\right ) \cosh \left (b x\right ) \,d x } \]
[In]
[Out]
Timed out. \[ \int x \cosh (b x) \text {Chi}(b x) \, dx=\int x\,\mathrm {coshint}\left (b\,x\right )\,\mathrm {cosh}\left (b\,x\right ) \,d x \]
[In]
[Out]