Integrand size = 21, antiderivative size = 69 \[ \int \frac {c e+d e x}{a+b \text {arcsinh}(c+d x)} \, dx=-\frac {e \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{2 b d}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 b d} \]
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Time = 0.11 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5859, 12, 5780, 5556, 3384, 3379, 3382} \[ \int \frac {c e+d e x}{a+b \text {arcsinh}(c+d x)} \, dx=\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 b d}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 b d} \]
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Rule 12
Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5780
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e x}{a+b \text {arcsinh}(x)} \, dx,x,c+d x\right )}{d} \\ & = \frac {e \text {Subst}\left (\int \frac {x}{a+b \text {arcsinh}(x)} \, dx,x,c+d x\right )}{d} \\ & = -\frac {e \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b d} \\ & = -\frac {e \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b d} \\ & = -\frac {e \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{2 b d} \\ & = \frac {\left (e \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{2 b d}-\frac {\left (e \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{2 b d} \\ & = -\frac {e \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{2 b d}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 b d} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88 \[ \int \frac {c e+d e x}{a+b \text {arcsinh}(c+d x)} \, dx=-\frac {e \left (\text {Chi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c+d x)\right ) \sinh \left (\frac {2 a}{b}\right )-\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c+d x)\right )\right )}{2 b d} \]
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Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {2 a}{b}\right )}{4 b}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {2 a}{b}\right )}{4 b}}{d}\) | \(66\) |
default | \(\frac {\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {2 a}{b}\right )}{4 b}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {2 a}{b}\right )}{4 b}}{d}\) | \(66\) |
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\[ \int \frac {c e+d e x}{a+b \text {arcsinh}(c+d x)} \, dx=\int { \frac {d e x + c e}{b \operatorname {arsinh}\left (d x + c\right ) + a} \,d x } \]
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\[ \int \frac {c e+d e x}{a+b \text {arcsinh}(c+d x)} \, dx=e \left (\int \frac {c}{a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int \frac {d x}{a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx\right ) \]
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\[ \int \frac {c e+d e x}{a+b \text {arcsinh}(c+d x)} \, dx=\int { \frac {d e x + c e}{b \operatorname {arsinh}\left (d x + c\right ) + a} \,d x } \]
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\[ \int \frac {c e+d e x}{a+b \text {arcsinh}(c+d x)} \, dx=\int { \frac {d e x + c e}{b \operatorname {arsinh}\left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {c e+d e x}{a+b \text {arcsinh}(c+d x)} \, dx=\int \frac {c\,e+d\,e\,x}{a+b\,\mathrm {asinh}\left (c+d\,x\right )} \,d x \]
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