Integrand size = 12, antiderivative size = 130 \[ \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}} \, dx=-\frac {2 x^2 \sqrt {1+a^2 x^2}}{a \sqrt {\text {arcsinh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{4 a^3}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{4 a^3}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{4 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{4 a^3} \] Output:
-2*x^2*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)^(1/2)+1/4*Pi^(1/2)*erf(arcsinh(a*x )^(1/2))/a^3-1/4*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*arcsinh(a*x)^(1/2))/a^3-1/4* Pi^(1/2)*erfi(arcsinh(a*x)^(1/2))/a^3+1/4*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*ar csinh(a*x)^(1/2))/a^3
Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.08 \[ \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}} \, dx=\frac {-e^{-3 \text {arcsinh}(a x)}+e^{-\text {arcsinh}(a x)}+e^{\text {arcsinh}(a x)}-e^{3 \text {arcsinh}(a x)}+\sqrt {3} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-3 \text {arcsinh}(a x)\right )-\sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-\text {arcsinh}(a x)\right )-\sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},\text {arcsinh}(a x)\right )+\sqrt {3} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},3 \text {arcsinh}(a x)\right )}{4 a^3 \sqrt {\text {arcsinh}(a x)}} \] Input:
Integrate[x^2/ArcSinh[a*x]^(3/2),x]
Output:
(-E^(-3*ArcSinh[a*x]) + E^(-ArcSinh[a*x]) + E^ArcSinh[a*x] - E^(3*ArcSinh[ a*x]) + Sqrt[3]*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -3*ArcSinh[a*x]] - Sqrt[-Ar cSinh[a*x]]*Gamma[1/2, -ArcSinh[a*x]] - Sqrt[ArcSinh[a*x]]*Gamma[1/2, ArcS inh[a*x]] + Sqrt[3]*Sqrt[ArcSinh[a*x]]*Gamma[1/2, 3*ArcSinh[a*x]])/(4*a^3* Sqrt[ArcSinh[a*x]])
Time = 0.31 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6193, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 6193 |
\(\displaystyle \frac {2 \int \left (\frac {3 \sinh (3 \text {arcsinh}(a x))}{4 \sqrt {\text {arcsinh}(a x)}}-\frac {a x}{4 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{a^3}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \left (\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\) |
Input:
Int[x^2/ArcSinh[a*x]^(3/2),x]
Output:
(-2*x^2*Sqrt[1 + a^2*x^2])/(a*Sqrt[ArcSinh[a*x]]) + (2*((Sqrt[Pi]*Erf[Sqrt [ArcSinh[a*x]]])/8 - (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/8 - (Sqr t[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/8 + (Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[ a*x]]])/8))/a^3
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Si mp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Sinh[- a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSi nh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, - 1]
\[\int \frac {x^{2}}{\operatorname {arcsinh}\left (x a \right )^{\frac {3}{2}}}d x\]
Input:
int(x^2/arcsinh(x*a)^(3/2),x)
Output:
int(x^2/arcsinh(x*a)^(3/2),x)
Exception generated. \[ \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^2/arcsinh(a*x)^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {x^{2}}{\operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \] Input:
integrate(x**2/asinh(a*x)**(3/2),x)
Output:
Integral(x**2/asinh(a*x)**(3/2), x)
\[ \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2/arcsinh(a*x)^(3/2),x, algorithm="maxima")
Output:
integrate(x^2/arcsinh(a*x)^(3/2), x)
\[ \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2/arcsinh(a*x)^(3/2),x, algorithm="giac")
Output:
integrate(x^2/arcsinh(a*x)^(3/2), x)
Timed out. \[ \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {x^2}{{\mathrm {asinh}\left (a\,x\right )}^{3/2}} \,d x \] Input:
int(x^2/asinh(a*x)^(3/2),x)
Output:
int(x^2/asinh(a*x)^(3/2), x)
\[ \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}} \, dx=\frac {6 \mathit {asinh} \left (a x \right ) \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {asinh} \left (a x \right )}\, x^{3}}{\mathit {asinh} \left (a x \right ) a^{2} x^{2}+\mathit {asinh} \left (a x \right )}d x \right ) a^{2}+4 \mathit {asinh} \left (a x \right ) \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {asinh} \left (a x \right )}\, x}{\mathit {asinh} \left (a x \right ) a^{2} x^{2}+\mathit {asinh} \left (a x \right )}d x \right )-2 \sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {asinh} \left (a x \right )}\, x^{2}}{\mathit {asinh} \left (a x \right ) a} \] Input:
int(x^2/asinh(a*x)^(3/2),x)
Output:
(2*(3*asinh(a*x)*int((sqrt(a**2*x**2 + 1)*sqrt(asinh(a*x))*x**3)/(asinh(a* x)*a**2*x**2 + asinh(a*x)),x)*a**2 + 2*asinh(a*x)*int((sqrt(a**2*x**2 + 1) *sqrt(asinh(a*x))*x)/(asinh(a*x)*a**2*x**2 + asinh(a*x)),x) - sqrt(a**2*x* *2 + 1)*sqrt(asinh(a*x))*x**2))/(asinh(a*x)*a)