Integrand size = 10, antiderivative size = 36 \[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=-\frac {a \text {Chi}\left (\frac {a}{c+d x}\right )}{d}+\frac {(c+d x) \sinh \left (\frac {a}{c+d x}\right )}{d} \]
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Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5418, 5410, 3378, 3382} \[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=\frac {(c+d x) \sinh \left (\frac {a}{c+d x}\right )}{d}-\frac {a \text {Chi}\left (\frac {a}{c+d x}\right )}{d} \]
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Rule 3378
Rule 3382
Rule 5410
Rule 5418
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \sinh \left (\frac {a}{x}\right ) \, dx,x,c+d x\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \frac {\sinh (a x)}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d} \\ & = \frac {(c+d x) \sinh \left (\frac {a}{c+d x}\right )}{d}-\frac {a \text {Subst}\left (\int \frac {\cosh (a x)}{x} \, dx,x,\frac {1}{c+d x}\right )}{d} \\ & = -\frac {a \text {Chi}\left (\frac {a}{c+d x}\right )}{d}+\frac {(c+d x) \sinh \left (\frac {a}{c+d x}\right )}{d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=-\frac {a \text {Chi}\left (\frac {a}{c+d x}\right )}{d}+\frac {(c+d x) \sinh \left (\frac {a}{c+d x}\right )}{d} \]
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Time = 1.64 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(-\frac {a \left (-\frac {\left (d x +c \right ) \sinh \left (\frac {a}{d x +c}\right )}{a}+\operatorname {Chi}\left (\frac {a}{d x +c}\right )\right )}{d}\) | \(38\) |
default | \(-\frac {a \left (-\frac {\left (d x +c \right ) \sinh \left (\frac {a}{d x +c}\right )}{a}+\operatorname {Chi}\left (\frac {a}{d x +c}\right )\right )}{d}\) | \(38\) |
risch | \(-\frac {{\mathrm e}^{-\frac {a}{d x +c}} x}{2}-\frac {{\mathrm e}^{-\frac {a}{d x +c}} c}{2 d}+\frac {a \,\operatorname {Ei}_{1}\left (\frac {a}{d x +c}\right )}{2 d}+\frac {{\mathrm e}^{\frac {a}{d x +c}} x}{2}+\frac {{\mathrm e}^{\frac {a}{d x +c}} c}{2 d}+\frac {a \,\operatorname {Ei}_{1}\left (-\frac {a}{d x +c}\right )}{2 d}\) | \(99\) |
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none
Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.33 \[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=-\frac {a {\rm Ei}\left (\frac {a}{d x + c}\right ) + a {\rm Ei}\left (-\frac {a}{d x + c}\right ) - 2 \, {\left (d x + c\right )} \sinh \left (\frac {a}{d x + c}\right )}{2 \, d} \]
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\[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=\int \sinh {\left (\frac {a}{c + d x} \right )}\, dx \]
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\[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=\int { \sinh \left (\frac {a}{d x + c}\right ) \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (36) = 72\).
Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.83 \[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=-\frac {{\left (\frac {a^{3} {\rm Ei}\left (\frac {a}{d x + c}\right )}{d x + c} - a^{2} e^{\left (\frac {a}{d x + c}\right )}\right )} {\left (d x + c\right )}}{2 \, a^{2} d} - \frac {{\left (\frac {a^{3} {\rm Ei}\left (-\frac {a}{d x + c}\right )}{d x + c} + a^{2} e^{\left (-\frac {a}{d x + c}\right )}\right )} {\left (d x + c\right )}}{2 \, a^{2} d} \]
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Timed out. \[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=\int \mathrm {sinh}\left (\frac {a}{c+d\,x}\right ) \,d x \]
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