Integrand size = 6, antiderivative size = 35 \[ \int x \text {Chi}(b x) \, dx=\frac {\cosh (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Chi}(b x)-\frac {x \sinh (b x)}{2 b} \]
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Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6668, 12, 3377, 2718} \[ \int x \text {Chi}(b x) \, dx=\frac {\cosh (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Chi}(b x)-\frac {x \sinh (b x)}{2 b} \]
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Rule 12
Rule 2718
Rule 3377
Rule 6668
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \text {Chi}(b x)-\frac {1}{2} b \int \frac {x \cosh (b x)}{b} \, dx \\ & = \frac {1}{2} x^2 \text {Chi}(b x)-\frac {1}{2} \int x \cosh (b x) \, dx \\ & = \frac {1}{2} x^2 \text {Chi}(b x)-\frac {x \sinh (b x)}{2 b}+\frac {\int \sinh (b x) \, dx}{2 b} \\ & = \frac {\cosh (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Chi}(b x)-\frac {x \sinh (b x)}{2 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int x \text {Chi}(b x) \, dx=\frac {\cosh (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Chi}(b x)-\frac {x \sinh (b x)}{2 b} \]
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Time = 0.34 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86
method | result | size |
parts | \(\frac {x^{2} \operatorname {Chi}\left (b x \right )}{2}-\frac {b x \sinh \left (b x \right )-\cosh \left (b x \right )}{2 b^{2}}\) | \(30\) |
derivativedivides | \(\frac {\frac {b^{2} x^{2} \operatorname {Chi}\left (b x \right )}{2}-\frac {b x \sinh \left (b x \right )}{2}+\frac {\cosh \left (b x \right )}{2}}{b^{2}}\) | \(32\) |
default | \(\frac {\frac {b^{2} x^{2} \operatorname {Chi}\left (b x \right )}{2}-\frac {b x \sinh \left (b x \right )}{2}+\frac {\cosh \left (b x \right )}{2}}{b^{2}}\) | \(32\) |
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\[ \int x \text {Chi}(b x) \, dx=\int { x {\rm Chi}\left (b x\right ) \,d x } \]
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Time = 1.00 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.51 \[ \int x \text {Chi}(b x) \, dx=- \frac {x^{2} \log {\left (b x \right )}}{2} + \frac {x^{2} \log {\left (b^{2} x^{2} \right )}}{4} + \frac {x^{2} \operatorname {Chi}\left (b x\right )}{2} - \frac {x \sinh {\left (b x \right )}}{2 b} + \frac {\cosh {\left (b x \right )}}{2 b^{2}} \]
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\[ \int x \text {Chi}(b x) \, dx=\int { x {\rm Chi}\left (b x\right ) \,d x } \]
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\[ \int x \text {Chi}(b x) \, dx=\int { x {\rm Chi}\left (b x\right ) \,d x } \]
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Timed out. \[ \int x \text {Chi}(b x) \, dx=\frac {\frac {\mathrm {cosh}\left (b\,x\right )}{2}-\frac {b\,x\,\mathrm {sinh}\left (b\,x\right )}{2}}{b^2}+\frac {x^2\,\mathrm {coshint}\left (b\,x\right )}{2} \]
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