Integrand size = 8, antiderivative size = 74 \[ \int x \text {Chi}(b x)^2 \, dx=\frac {\cosh (b x) \text {Chi}(b x)}{b^2}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {\text {Chi}(2 b x)}{2 b^2}-\frac {\log (x)}{2 b^2}-\frac {x \text {Chi}(b x) \sinh (b x)}{b}+\frac {\sinh ^2(b x)}{2 b^2} \]
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Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6672, 6678, 12, 2644, 30, 6682, 3393, 3382} \[ \int x \text {Chi}(b x)^2 \, dx=-\frac {\text {Chi}(2 b x)}{2 b^2}+\frac {\text {Chi}(b x) \cosh (b x)}{b^2}-\frac {\log (x)}{2 b^2}+\frac {\sinh ^2(b x)}{2 b^2}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {x \text {Chi}(b x) \sinh (b x)}{b} \]
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Rule 12
Rule 30
Rule 2644
Rule 3382
Rule 3393
Rule 6672
Rule 6678
Rule 6682
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \text {Chi}(b x)^2-\int x \cosh (b x) \text {Chi}(b x) \, dx \\ & = \frac {1}{2} x^2 \text {Chi}(b x)^2-\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 {1}{2} x^2 \text {Chi}(b x)^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 {1}{2} x^2 \text {Chi}(b x)^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 {1}{2} x^2 \text {Chi}(b x)^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 {1}{2} x^2 \text {Chi}(b x)^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 {1}{2} x^2 \text {Chi}(b x)^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 = 57, normalized size of antiderivative = 0.77 \[ \int x \text {Chi}(b x)^2 \, dx=\frac {\cosh (2 b x)+2 b^2 x^2 \text {Chi}(b x)^2-2 \text {Chi}(2 b x)-2 \log (x)+4 \text {Chi}(b x) (\cosh (b x)-b x \sinh (b x))}{4 b^2} \]
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Time = 0.54 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {\frac {b^{2} x^{2} \operatorname {Chi}\left (b x \right )^{2}}{2}-2 \,\operatorname {Chi}\left (b x \right ) \left (\frac {b x \sinh \left (b x \right )}{2}-\frac {\cosh \left (b x \right )}{2}\right )+\frac {\cosh \left (b x \right )^{2}}{2}-\frac {\ln \left (b x \right )}{2}-\frac {\operatorname {Chi}\left (2 b x \right )}{2}}{b^{2}}\) | \(62\) |
default | \(\frac {\frac {b^{2} x^{2} \operatorname {Chi}\left (b x \right )^{2}}{2}-2 \,\operatorname {Chi}\left (b x \right ) \left (\frac {b x \sinh \left (b x \right )}{2}-\frac {\cosh \left (b x \right )}{2}\right )+\frac {\cosh \left (b x \right )^{2}}{2}-\frac {\ln \left (b x \right )}{2}-\frac {\operatorname {Chi}\left (2 b x \right )}{2}}{b^{2}}\) | \(62\) |
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\[ \int x \text {Chi}(b x)^2 \, dx=\int { x {\rm Chi}\left (b x\right )^{2} \,d x } \]
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\[ \int x \text {Chi}(b x)^2 \, dx=\int x \operatorname {Chi}^{2}\left (b x\right )\, dx \]
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\[ \int x \text {Chi}(b x)^2 \, dx=\int { x {\rm Chi}\left (b x\right )^{2} \,d x } \]
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\[ \int x \text {Chi}(b x)^2 \, dx=\int { x {\rm Chi}\left (b x\right )^{2} \,d x } \]
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Timed out. \[ \int x \text {Chi}(b x)^2 \, dx=\int x\,{\mathrm {coshint}\left (b\,x\right )}^2 \,d x \]
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