Integrand size = 10, antiderivative size = 84 \[ \int \frac {x}{\text {arcsinh}(a+b x)^2} \, dx=\frac {a \sqrt {1+(a+b x)^2}}{b^2 \text {arcsinh}(a+b x)}-\frac {(a+b x) \sqrt {1+(a+b x)^2}}{b^2 \text {arcsinh}(a+b x)}+\frac {\text {Chi}(2 \text {arcsinh}(a+b x))}{b^2}-\frac {a \text {Shi}(\text {arcsinh}(a+b x))}{b^2} \]
[Out]
Time = 0.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5859, 5829, 5773, 5819, 3379, 5778, 3382} \[ \int \frac {x}{\text {arcsinh}(a+b x)^2} \, dx=\frac {\text {Chi}(2 \text {arcsinh}(a+b x))}{b^2}-\frac {a \text {Shi}(\text {arcsinh}(a+b x))}{b^2}+\frac {a \sqrt {(a+b x)^2+1}}{b^2 \text {arcsinh}(a+b x)}-\frac {(a+b x) \sqrt {(a+b x)^2+1}}{b^2 \text {arcsinh}(a+b x)} \]
[In]
[Out]
Rule 3379
Rule 3382
Rule 5773
Rule 5778
Rule 5819
Rule 5829
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {-\frac {a}{b}+\frac {x}{b}}{\text {arcsinh}(x)^2} \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {a}{b \text {arcsinh}(x)^2}+\frac {x}{b \text {arcsinh}(x)^2}\right ) \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {x}{\text {arcsinh}(x)^2} \, dx,x,a+b x\right )}{b^2}-\frac {a \text {Subst}\left (\int \frac {1}{\text {arcsinh}(x)^2} \, dx,x,a+b x\right )}{b^2} \\ & = \frac {a \sqrt {1+(a+b x)^2}}{b^2 \text {arcsinh}(a+b x)}-\frac {(a+b x) \sqrt {1+(a+b x)^2}}{b^2 \text {arcsinh}(a+b x)}+\frac {\text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b^2}-\frac {a \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \text {arcsinh}(x)} \, dx,x,a+b x\right )}{b^2} \\ & = \frac {a \sqrt {1+(a+b x)^2}}{b^2 \text {arcsinh}(a+b x)}-\frac {(a+b x) \sqrt {1+(a+b x)^2}}{b^2 \text {arcsinh}(a+b x)}+\frac {\text {Chi}(2 \text {arcsinh}(a+b x))}{b^2}-\frac {a \text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b^2} \\ & = \frac {a \sqrt {1+(a+b x)^2}}{b^2 \text {arcsinh}(a+b x)}-\frac {(a+b x) \sqrt {1+(a+b x)^2}}{b^2 \text {arcsinh}(a+b x)}+\frac {\text {Chi}(2 \text {arcsinh}(a+b x))}{b^2}-\frac {a \text {Shi}(\text {arcsinh}(a+b x))}{b^2} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.74 \[ \int \frac {x}{\text {arcsinh}(a+b x)^2} \, dx=-\frac {b x \sqrt {1+(a+b x)^2}-\text {arcsinh}(a+b x) \text {Chi}(2 \text {arcsinh}(a+b x))+a \text {arcsinh}(a+b x) \text {Shi}(\text {arcsinh}(a+b x))}{b^2 \text {arcsinh}(a+b x)} \]
[In]
[Out]
Time = 0.17 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {-\frac {\sinh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{2 \,\operatorname {arcsinh}\left (b x +a \right )}+\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )-\frac {a \left (\operatorname {Shi}\left (\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}\right )}{\operatorname {arcsinh}\left (b x +a \right )}}{b^{2}}\) | \(73\) |
default | \(\frac {-\frac {\sinh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{2 \,\operatorname {arcsinh}\left (b x +a \right )}+\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )-\frac {a \left (\operatorname {Shi}\left (\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}\right )}{\operatorname {arcsinh}\left (b x +a \right )}}{b^{2}}\) | \(73\) |
[In]
[Out]
\[ \int \frac {x}{\text {arcsinh}(a+b x)^2} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {x}{\text {arcsinh}(a+b x)^2} \, dx=\int \frac {x}{\operatorname {asinh}^{2}{\left (a + b x \right )}}\, dx \]
[In]
[Out]
\[ \int \frac {x}{\text {arcsinh}(a+b x)^2} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {x}{\text {arcsinh}(a+b x)^2} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x}{\text {arcsinh}(a+b x)^2} \, dx=\int \frac {x}{{\mathrm {asinh}\left (a+b\,x\right )}^2} \,d x \]
[In]
[Out]