Integrand size = 16, antiderivative size = 78 \[ \int \frac {\sinh ^2(a+b x)}{c+d x} \, dx=\frac {\cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}-\frac {\log (c+d x)}{2 d}+\frac {\sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{2 d} \]
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Time = 0.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3393, 3384, 3379, 3382} \[ \int \frac {\sinh ^2(a+b x)}{c+d x} \, dx=\frac {\cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\frac {\sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}-\frac {\log (c+d x)}{2 d} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 3393
Rubi steps \begin{align*} \text {integral}& = -\int \left (\frac {1}{2 (c+d x)}-\frac {\cosh (2 a+2 b x)}{2 (c+d x)}\right ) \, dx \\ & = -\frac {\log (c+d x)}{2 d}+\frac {1}{2} \int \frac {\cosh (2 a+2 b x)}{c+d x} \, dx \\ & = -\frac {\log (c+d x)}{2 d}+\frac {1}{2} \cosh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\cosh \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx+\frac {1}{2} \sinh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sinh \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx \\ & = \frac {\cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}-\frac {\log (c+d x)}{2 d}+\frac {\sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{2 d} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.85 \[ \int \frac {\sinh ^2(a+b x)}{c+d x} \, dx=\frac {\cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b (c+d x)}{d}\right )-\log (c+d x)+\sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b (c+d x)}{d}\right )}{2 d} \]
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Time = 3.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.24
method | result | size |
risch | \(-\frac {\ln \left (d x +c \right )}{2 d}-\frac {{\mathrm e}^{-\frac {2 \left (a d -b c \right )}{d}} \operatorname {Ei}_{1}\left (2 b x +2 a -\frac {2 \left (a d -b c \right )}{d}\right )}{4 d}-\frac {{\mathrm e}^{\frac {2 a d -2 b c}{d}} \operatorname {Ei}_{1}\left (-2 b x -2 a -\frac {2 \left (-a d +b c \right )}{d}\right )}{4 d}\) | \(97\) |
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Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.33 \[ \int \frac {\sinh ^2(a+b x)}{c+d x} \, dx=\frac {{\left ({\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left ({\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - 2 \, \log \left (d x + c\right )}{4 \, d} \]
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\[ \int \frac {\sinh ^2(a+b x)}{c+d x} \, dx=\int \frac {\sinh ^{2}{\left (a + b x \right )}}{c + d x}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.92 \[ \int \frac {\sinh ^2(a+b x)}{c+d x} \, dx=-\frac {e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} E_{1}\left (\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{4 \, d} - \frac {e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} E_{1}\left (-\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{4 \, d} - \frac {\log \left (d x + c\right )}{2 \, d} \]
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Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.87 \[ \int \frac {\sinh ^2(a+b x)}{c+d x} \, dx=\frac {{\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} + {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} - 2 \, \log \left (d x + c\right )}{4 \, d} \]
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Timed out. \[ \int \frac {\sinh ^2(a+b x)}{c+d x} \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^2}{c+d\,x} \,d x \]
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