Integrand size = 16, antiderivative size = 121 \[ \int \frac {\sinh ^3(a+b x)}{c+d x} \, dx=\frac {\text {Chi}\left (\frac {3 b c}{d}+3 b x\right ) \sinh \left (3 a-\frac {3 b c}{d}\right )}{4 d}-\frac {3 \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{4 d}-\frac {3 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d}+\frac {\cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d} \]
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Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3393, 3384, 3379, 3382} \[ \int \frac {\sinh ^3(a+b x)}{c+d x} \, dx=\frac {\sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d}-\frac {3 \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{4 d}-\frac {3 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d}+\frac {\cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 3393
Rubi steps \begin{align*} \text {integral}& = i \int \left (\frac {3 i \sinh (a+b x)}{4 (c+d x)}-\frac {i \sinh (3 a+3 b x)}{4 (c+d x)}\right ) \, dx \\ & = \frac {1}{4} \int \frac {\sinh (3 a+3 b x)}{c+d x} \, dx-\frac {3}{4} \int \frac {\sinh (a+b x)}{c+d x} \, dx \\ & = \frac {1}{4} \cosh \left (3 a-\frac {3 b c}{d}\right ) \int \frac {\sinh \left (\frac {3 b c}{d}+3 b x\right )}{c+d x} \, dx-\frac {1}{4} \left (3 \cosh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx+\frac {1}{4} \sinh \left (3 a-\frac {3 b c}{d}\right ) \int \frac {\cosh \left (\frac {3 b c}{d}+3 b x\right )}{c+d x} \, dx-\frac {1}{4} \left (3 \sinh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx \\ & = \frac {\text {Chi}\left (\frac {3 b c}{d}+3 b x\right ) \sinh \left (3 a-\frac {3 b c}{d}\right )}{4 d}-\frac {3 \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{4 d}-\frac {3 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d}+\frac {\cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.84 \[ \int \frac {\sinh ^3(a+b x)}{c+d x} \, dx=\frac {\text {Chi}\left (\frac {3 b (c+d x)}{d}\right ) \sinh \left (3 a-\frac {3 b c}{d}\right )-3 \text {Chi}\left (b \left (\frac {c}{d}+x\right )\right ) \sinh \left (a-\frac {b c}{d}\right )-3 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (b \left (\frac {c}{d}+x\right )\right )+\cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b (c+d x)}{d}\right )}{4 d} \]
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Time = 2.22 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.37
method | result | size |
risch | \(\frac {{\mathrm e}^{-\frac {3 \left (a d -b c \right )}{d}} \operatorname {Ei}_{1}\left (3 b x +3 a -\frac {3 \left (a d -b c \right )}{d}\right )}{8 d}-\frac {3 \,{\mathrm e}^{-\frac {a d -b c}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -b c}{d}\right )}{8 d}+\frac {3 \,{\mathrm e}^{\frac {a d -b c}{d}} \operatorname {Ei}_{1}\left (-b x -a -\frac {-a d +b c}{d}\right )}{8 d}-\frac {{\mathrm e}^{\frac {3 a d -3 b c}{d}} \operatorname {Ei}_{1}\left (-3 b x -3 a -\frac {3 \left (-a d +b c \right )}{d}\right )}{8 d}\) | \(166\) |
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Time = 0.24 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.55 \[ \int \frac {\sinh ^3(a+b x)}{c+d x} \, dx=-\frac {3 \, {\left ({\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - {\left ({\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 3 \, {\left ({\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right ) - {\left ({\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{8 \, d} \]
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\[ \int \frac {\sinh ^3(a+b x)}{c+d x} \, dx=\int \frac {\sinh ^{3}{\left (a + b x \right )}}{c + d x}\, dx \]
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Time = 0.24 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.97 \[ \int \frac {\sinh ^3(a+b x)}{c+d x} \, dx=\frac {e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} E_{1}\left (\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac {3 \, e^{\left (-a + \frac {b c}{d}\right )} E_{1}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, d} + \frac {3 \, e^{\left (a - \frac {b c}{d}\right )} E_{1}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac {e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} E_{1}\left (-\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, d} \]
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Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.93 \[ \int \frac {\sinh ^3(a+b x)}{c+d x} \, dx=\frac {{\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} - 3 \, {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} + 3 \, {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} - {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )}}{8 \, d} \]
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Timed out. \[ \int \frac {\sinh ^3(a+b x)}{c+d x} \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^3}{c+d\,x} \,d x \]
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