Integrand size = 12, antiderivative size = 44 \[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^2} \, dx=\frac {1}{2} b \text {Chi}(b x)^2+b \text {Chi}(2 b x)-\frac {\text {Chi}(b x) \sinh (b x)}{x}-\frac {\sinh (2 b x)}{2 x} \]
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Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6686, 6818, 12, 5556, 3378, 3382} \[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^2} \, dx=\frac {1}{2} b \text {Chi}(b x)^2+b \text {Chi}(2 b x)-\frac {\text {Chi}(b x) \sinh (b x)}{x}-\frac {\sinh (2 b x)}{2 x} \]
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Rule 12
Rule 3378
Rule 3382
Rule 5556
Rule 6686
Rule 6818
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Chi}(b x) \sinh (b x)}{x}+b \int \frac {\cosh (b x) \text {Chi}(b x)}{x} \, dx+b \int \frac {\cosh (b x) \sinh (b x)}{b x^2} \, dx \\ & = \frac {1}{2} b \text {Chi}(b x)^2-\frac {\text {Chi}(b x) \sinh (b x)}{x}+\int \frac {\cosh (b x) \sinh (b x)}{x^2} \, dx \\ & = \frac {1}{2} b \text {Chi}(b x)^2-\frac {\text {Chi}(b x) \sinh (b x)}{x}+\int \frac {\sinh (2 b x)}{2 x^2} \, dx \\ & = \frac {1}{2} b \text {Chi}(b x)^2-\frac {\text {Chi}(b x) \sinh (b x)}{x}+\frac {1}{2} \int \frac {\sinh (2 b x)}{x^2} \, dx \\ & = \frac {1}{2} b \text {Chi}(b x)^2-\frac {\text {Chi}(b x) \sinh (b x)}{x}-\frac {\sinh (2 b x)}{2 x}+b \int \frac {\cosh (2 b x)}{x} \, dx \\ & = \frac {1}{2} b \text {Chi}(b x)^2+b \text {Chi}(2 b x)-\frac {\text {Chi}(b x) \sinh (b x)}{x}-\frac {\sinh (2 b x)}{2 x} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^2} \, dx=\frac {1}{2} b \text {Chi}(b x)^2+b \text {Chi}(2 b x)-\frac {\text {Chi}(b x) \sinh (b x)}{x}-\frac {\sinh (2 b x)}{2 x} \]
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\[\int \frac {\operatorname {Chi}\left (b x \right ) \sinh \left (b x \right )}{x^{2}}d x\]
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\[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^2} \, dx=\int { \frac {{\rm Chi}\left (b x\right ) \sinh \left (b x\right )}{x^{2}} \,d x } \]
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\[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^2} \, dx=\int \frac {\sinh {\left (b x \right )} \operatorname {Chi}\left (b x\right )}{x^{2}}\, dx \]
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\[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^2} \, dx=\int { \frac {{\rm Chi}\left (b x\right ) \sinh \left (b x\right )}{x^{2}} \,d x } \]
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\[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^2} \, dx=\int { \frac {{\rm Chi}\left (b x\right ) \sinh \left (b x\right )}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^2} \, dx=\int \frac {\mathrm {coshint}\left (b\,x\right )\,\mathrm {sinh}\left (b\,x\right )}{x^2} \,d x \]
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