Integrand size = 14, antiderivative size = 51 \[ \int \frac {\sinh (a+b x)}{c+d x} \, dx=\frac {\text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{d}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d} \]
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Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3384, 3379, 3382} \[ \int \frac {\sinh (a+b x)}{c+d x} \, dx=\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d} \]
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Rule 3379
Rule 3382
Rule 3384
Rubi steps \begin{align*} \text {integral}& = \cosh \left (a-\frac {b c}{d}\right ) \int \frac {\sinh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx+\sinh \left (a-\frac {b c}{d}\right ) \int \frac {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx \\ & = \frac {\text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{d}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.96 \[ \int \frac {\sinh (a+b x)}{c+d x} \, dx=\frac {\text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )+\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d} \]
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Time = 0.79 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.61
method | result | size |
risch | \(\frac {{\mathrm e}^{-\frac {a d -b c}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -b c}{d}\right )}{2 d}-\frac {{\mathrm e}^{\frac {a d -b c}{d}} \operatorname {Ei}_{1}\left (-b x -a -\frac {-a d +b c}{d}\right )}{2 d}\) | \(82\) |
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Time = 0.23 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.84 \[ \int \frac {\sinh (a+b x)}{c+d x} \, dx=\frac {{\left ({\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) + {\left ({\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right )}{2 \, d} \]
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\[ \int \frac {\sinh (a+b x)}{c+d x} \, dx=\int \frac {\sinh {\left (a + b x \right )}}{c + d x}\, dx \]
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Time = 0.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.12 \[ \int \frac {\sinh (a+b x)}{c+d x} \, dx=\frac {e^{\left (-a + \frac {b c}{d}\right )} E_{1}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{2 \, d} - \frac {e^{\left (a - \frac {b c}{d}\right )} E_{1}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{2 \, d} \]
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Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.12 \[ \int \frac {\sinh (a+b x)}{c+d x} \, dx=\frac {{\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} - {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{2 \, d} \]
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Timed out. \[ \int \frac {\sinh (a+b x)}{c+d x} \, dx=\int \frac {\mathrm {sinh}\left (a+b\,x\right )}{c+d\,x} \,d x \]
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