Integrand size = 10, antiderivative size = 85 \[ \int \frac {x^2}{\text {arccosh}(a x)^3} \, dx=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {x}{a^2 \text {arccosh}(a x)}-\frac {3 x^3}{2 \text {arccosh}(a x)}+\frac {\text {Shi}(\text {arccosh}(a x))}{8 a^3}+\frac {9 \text {Shi}(3 \text {arccosh}(a x))}{8 a^3} \]
[Out]
Time = 0.35 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5886, 5951, 5887, 5556, 3379, 5881} \[ \int \frac {x^2}{\text {arccosh}(a x)^3} \, dx=\frac {\text {Shi}(\text {arccosh}(a x))}{8 a^3}+\frac {9 \text {Shi}(3 \text {arccosh}(a x))}{8 a^3}+\frac {x}{a^2 \text {arccosh}(a x)}-\frac {3 x^3}{2 \text {arccosh}(a x)}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{2 a \text {arccosh}(a x)^2} \]
[In]
[Out]
Rule 3379
Rule 5556
Rule 5881
Rule 5886
Rule 5887
Rule 5951
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}-\frac {\int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2} \, dx}{a}+\frac {1}{2} (3 a) \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2} \, dx \\ & = -\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {x}{a^2 \text {arccosh}(a x)}-\frac {3 x^3}{2 \text {arccosh}(a x)}+\frac {9}{2} \int \frac {x^2}{\text {arccosh}(a x)} \, dx-\frac {\int \frac {1}{\text {arccosh}(a x)} \, dx}{a^2} \\ & = -\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {x}{a^2 \text {arccosh}(a x)}-\frac {3 x^3}{2 \text {arccosh}(a x)}-\frac {\text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{a^3}+\frac {9 \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{2 a^3} \\ & = -\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {x}{a^2 \text {arccosh}(a x)}-\frac {3 x^3}{2 \text {arccosh}(a x)}-\frac {\text {Shi}(\text {arccosh}(a x))}{a^3}+\frac {9 \text {Subst}\left (\int \left (\frac {\sinh (x)}{4 x}+\frac {\sinh (3 x)}{4 x}\right ) \, dx,x,\text {arccosh}(a x)\right )}{2 a^3} \\ & = -\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {x}{a^2 \text {arccosh}(a x)}-\frac {3 x^3}{2 \text {arccosh}(a x)}-\frac {\text {Shi}(\text {arccosh}(a x))}{a^3}+\frac {9 \text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{8 a^3}+\frac {9 \text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{8 a^3} \\ & = -\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {x}{a^2 \text {arccosh}(a x)}-\frac {3 x^3}{2 \text {arccosh}(a x)}+\frac {\text {Shi}(\text {arccosh}(a x))}{8 a^3}+\frac {9 \text {Shi}(3 \text {arccosh}(a x))}{8 a^3} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.81 \[ \int \frac {x^2}{\text {arccosh}(a x)^3} \, dx=\frac {-\frac {4 a x \left (a x \sqrt {-1+a x} \sqrt {1+a x}+\left (-2+3 a^2 x^2\right ) \text {arccosh}(a x)\right )}{\text {arccosh}(a x)^2}+\text {Shi}(\text {arccosh}(a x))+9 \text {Shi}(3 \text {arccosh}(a x))}{8 a^3} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.99
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{8 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {a x}{8 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Shi}\left (\operatorname {arccosh}\left (a x \right )\right )}{8}-\frac {\sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{8 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {3 \cosh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{8 \,\operatorname {arccosh}\left (a x \right )}+\frac {9 \,\operatorname {Shi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{8}}{a^{3}}\) | \(84\) |
default | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{8 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {a x}{8 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Shi}\left (\operatorname {arccosh}\left (a x \right )\right )}{8}-\frac {\sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{8 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {3 \cosh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{8 \,\operatorname {arccosh}\left (a x \right )}+\frac {9 \,\operatorname {Shi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{8}}{a^{3}}\) | \(84\) |
[In]
[Out]
\[ \int \frac {x^2}{\text {arccosh}(a x)^3} \, dx=\int { \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^2}{\text {arccosh}(a x)^3} \, dx=\int \frac {x^{2}}{\operatorname {acosh}^{3}{\left (a x \right )}}\, dx \]
[In]
[Out]
\[ \int \frac {x^2}{\text {arccosh}(a x)^3} \, dx=\int { \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^2}{\text {arccosh}(a x)^3} \, dx=\int { \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^2}{\text {arccosh}(a x)^3} \, dx=\int \frac {x^2}{{\mathrm {acosh}\left (a\,x\right )}^3} \,d x \]
[In]
[Out]