Integrand size = 23, antiderivative size = 135 \[ \int \frac {\text {arcsinh}(a x)^2}{x^3 \sqrt {1+a^2 x^2}} \, dx=-\frac {a \text {arcsinh}(a x)}{x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x^2}+a^2 \text {arcsinh}(a x)^2 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-a^2 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right )+a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-a^2 \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )+a^2 \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right ) \]
[Out]
Time = 0.20 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5809, 5816, 4267, 2611, 2320, 6724, 5776, 272, 65, 214} \[ \int \frac {\text {arcsinh}(a x)^2}{x^3 \sqrt {1+a^2 x^2}} \, dx=a^2 \text {arcsinh}(a x)^2 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-a^2 \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )+a^2 \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{2 x^2}-a^2 \text {arctanh}\left (\sqrt {a^2 x^2+1}\right )-\frac {a \text {arcsinh}(a x)}{x} \]
[In]
[Out]
Rule 65
Rule 214
Rule 272
Rule 2320
Rule 2611
Rule 4267
Rule 5776
Rule 5809
Rule 5816
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x^2}+a \int \frac {\text {arcsinh}(a x)}{x^2} \, dx-\frac {1}{2} a^2 \int \frac {\text {arcsinh}(a x)^2}{x \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {a \text {arcsinh}(a x)}{x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x^2}-\frac {1}{2} a^2 \text {Subst}\left (\int x^2 \text {csch}(x) \, dx,x,\text {arcsinh}(a x)\right )+a^2 \int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {a \text {arcsinh}(a x)}{x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x^2}+a^2 \text {arcsinh}(a x)^2 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+\frac {1}{2} a^2 \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )+a^2 \text {Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )-a^2 \text {Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -\frac {a \text {arcsinh}(a x)}{x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x^2}+a^2 \text {arcsinh}(a x)^2 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-a^2 \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )+a^2 \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )+\text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^2}\right ) \\ & = -\frac {a \text {arcsinh}(a x)}{x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x^2}+a^2 \text {arcsinh}(a x)^2 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-a^2 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right )+a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-a^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right )+a^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right ) \\ & = -\frac {a \text {arcsinh}(a x)}{x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x^2}+a^2 \text {arcsinh}(a x)^2 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-a^2 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right )+a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-a^2 \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )+a^2 \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right ) \\ \end{align*}
Time = 0.92 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.39 \[ \int \frac {\text {arcsinh}(a x)^2}{x^3 \sqrt {1+a^2 x^2}} \, dx=\frac {1}{8} a^2 \left (-4 \text {arcsinh}(a x) \coth \left (\frac {1}{2} \text {arcsinh}(a x)\right )-\text {arcsinh}(a x)^2 \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(a x)\right )-4 \text {arcsinh}(a x)^2 \log \left (1-e^{-\text {arcsinh}(a x)}\right )+4 \text {arcsinh}(a x)^2 \log \left (1+e^{-\text {arcsinh}(a x)}\right )+8 \log \left (\tanh \left (\frac {1}{2} \text {arcsinh}(a x)\right )\right )-8 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(a x)}\right )+8 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(a x)}\right )-8 \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(a x)}\right )+8 \operatorname {PolyLog}\left (3,e^{-\text {arcsinh}(a x)}\right )-\text {arcsinh}(a x)^2 \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(a x)\right )+4 \text {arcsinh}(a x) \tanh \left (\frac {1}{2} \text {arcsinh}(a x)\right )\right ) \]
[In]
[Out]
Time = 0.30 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.73
method | result | size |
default | \(-\frac {\operatorname {arcsinh}\left (a x \right ) \left (a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )+2 a x \sqrt {a^{2} x^{2}+1}+\operatorname {arcsinh}\left (a x \right )\right )}{2 \sqrt {a^{2} x^{2}+1}\, x^{2}}+\frac {a^{2} \operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )}{2}+a^{2} \operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-a^{2} \operatorname {polylog}\left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )-\frac {a^{2} \operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )}{2}-a^{2} \operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )+a^{2} \operatorname {polylog}\left (3, a x +\sqrt {a^{2} x^{2}+1}\right )-2 a^{2} \operatorname {arctanh}\left (a x +\sqrt {a^{2} x^{2}+1}\right )\) | \(233\) |
[In]
[Out]
\[ \int \frac {\text {arcsinh}(a x)^2}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{2}}{\sqrt {a^{2} x^{2} + 1} x^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {\text {arcsinh}(a x)^2}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int \frac {\operatorname {asinh}^{2}{\left (a x \right )}}{x^{3} \sqrt {a^{2} x^{2} + 1}}\, dx \]
[In]
[Out]
\[ \int \frac {\text {arcsinh}(a x)^2}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{2}}{\sqrt {a^{2} x^{2} + 1} x^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {\text {arcsinh}(a x)^2}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{2}}{\sqrt {a^{2} x^{2} + 1} x^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\text {arcsinh}(a x)^2}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^2}{x^3\,\sqrt {a^2\,x^2+1}} \,d x \]
[In]
[Out]