Integrand size = 12, antiderivative size = 58 \[ \int \frac {1}{a+b \text {arcsinh}(c+d x)} \, dx=\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{b d}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{b d} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5858, 5774, 3384, 3379, 3382} \[ \int \frac {1}{a+b \text {arcsinh}(c+d x)} \, dx=\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{b d}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{b d} \]
[In]
[Out]
Rule 3379
Rule 3382
Rule 3384
Rule 5774
Rule 5858
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{a+b \text {arcsinh}(x)} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b d} \\ & = \frac {\cosh \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b d}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b d} \\ & = \frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{b d}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{b d} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.84 \[ \int \frac {1}{a+b \text {arcsinh}(c+d x)} \, dx=\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )}{b d} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(\frac {-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (d x +c \right )+\frac {a}{b}\right )}{2 b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (d x +c \right )-\frac {a}{b}\right )}{2 b}}{d}\) | \(60\) |
default | \(\frac {-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (d x +c \right )+\frac {a}{b}\right )}{2 b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (d x +c \right )-\frac {a}{b}\right )}{2 b}}{d}\) | \(60\) |
[In]
[Out]
\[ \int \frac {1}{a+b \text {arcsinh}(c+d x)} \, dx=\int { \frac {1}{b \operatorname {arsinh}\left (d x + c\right ) + a} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{a+b \text {arcsinh}(c+d x)} \, dx=\int \frac {1}{a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{a+b \text {arcsinh}(c+d x)} \, dx=\int { \frac {1}{b \operatorname {arsinh}\left (d x + c\right ) + a} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{a+b \text {arcsinh}(c+d x)} \, dx=\int { \frac {1}{b \operatorname {arsinh}\left (d x + c\right ) + a} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{a+b \text {arcsinh}(c+d x)} \, dx=\int \frac {1}{a+b\,\mathrm {asinh}\left (c+d\,x\right )} \,d x \]
[In]
[Out]