Integrand size = 11, antiderivative size = 74 \[ \int \sinh \left (\frac {b x}{c+d x}\right ) \, dx=\frac {b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {b x}{c+d x}\right )}{d}-\frac {b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{d (c+d x)}\right )}{d^2} \]
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Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5726, 3378, 3384, 3379, 3382} \[ \int \sinh \left (\frac {b x}{c+d x}\right ) \, dx=\frac {b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{d (c+d x)}\right )}{d^2}-\frac {b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {b x}{c+d x}\right )}{d} \]
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5726
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\sinh \left (\frac {b}{d}-\frac {b c x}{d}\right )}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d} \\ & = \frac {(c+d x) \sinh \left (\frac {b x}{c+d x}\right )}{d}+\frac {(b c) \text {Subst}\left (\int \frac {\cosh \left (\frac {b}{d}-\frac {b c x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2} \\ & = \frac {(c+d x) \sinh \left (\frac {b x}{c+d x}\right )}{d}+\frac {\left (b c \cosh \left (\frac {b}{d}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {b c x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}-\frac {\left (b c \sinh \left (\frac {b}{d}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {b c x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2} \\ & = \frac {b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {b x}{c+d x}\right )}{d}-\frac {b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{d (c+d x)}\right )}{d^2} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.26 \[ \int \sinh \left (\frac {b x}{c+d x}\right ) \, dx=\frac {d e^{-\frac {b x}{c+d x}} \left (-1+e^{\frac {2 b x}{c+d x}}\right ) (c+d x)+2 b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{c d+d^2 x}\right )-2 b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{c d+d^2 x}\right )}{2 d^2} \]
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Time = 1.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.53
method | result | size |
risch | \(-\frac {{\mathrm e}^{-\frac {b x}{d x +c}} \left (d x +c \right )}{2 d}-\frac {b c \,{\mathrm e}^{-\frac {b}{d}} \operatorname {Ei}_{1}\left (-\frac {b c}{d \left (d x +c \right )}\right )}{2 d^{2}}+\frac {{\mathrm e}^{\frac {b x}{d x +c}} x}{2}+\frac {c \,{\mathrm e}^{\frac {b x}{d x +c}}}{2 d}-\frac {b c \,{\mathrm e}^{\frac {b}{d}} \operatorname {Ei}_{1}\left (\frac {b c}{d \left (d x +c \right )}\right )}{2 d^{2}}\) | \(113\) |
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Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (74) = 148\).
Time = 0.27 (sec) , antiderivative size = 253, normalized size of antiderivative = 3.42 \[ \int \sinh \left (\frac {b x}{c+d x}\right ) \, dx=-\frac {b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b}{d}\right ) \sinh \left (\frac {b x}{d x + c}\right )^{2} - {\left (b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{2} + b c {\rm Ei}\left (\frac {b c}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {b}{d}\right ) - 2 \, {\left (d^{2} x + c d\right )} \sinh \left (\frac {b x}{d x + c}\right ) - {\left (b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{2} - b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \sinh \left (\frac {b x}{d x + c}\right )^{2} - b c {\rm Ei}\left (\frac {b c}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {b}{d}\right )}{2 \, {\left (d^{2} \cosh \left (\frac {b x}{d x + c}\right )^{2} - d^{2} \sinh \left (\frac {b x}{d x + c}\right )^{2}\right )}} \]
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\[ \int \sinh \left (\frac {b x}{c+d x}\right ) \, dx=\int \sinh {\left (\frac {b x}{c + d x} \right )}\, dx \]
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\[ \int \sinh \left (\frac {b x}{c+d x}\right ) \, dx=\int { \sinh \left (\frac {b x}{d x + c}\right ) \,d x } \]
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\[ \int \sinh \left (\frac {b x}{c+d x}\right ) \, dx=\int { \sinh \left (\frac {b x}{d x + c}\right ) \,d x } \]
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Timed out. \[ \int \sinh \left (\frac {b x}{c+d x}\right ) \, dx=\int \mathrm {sinh}\left (\frac {b\,x}{c+d\,x}\right ) \,d x \]
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