Integrand size = 10, antiderivative size = 54 \[ \int \frac {x^2}{\text {arcsinh}(a x)^2} \, dx=-\frac {x^2 \sqrt {1+a^2 x^2}}{a \text {arcsinh}(a x)}-\frac {\text {Shi}(\text {arcsinh}(a x))}{4 a^3}+\frac {3 \text {Shi}(3 \text {arcsinh}(a x))}{4 a^3} \] Output:
-x^2*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)-1/4*Shi(arcsinh(a*x))/a^3+3/4*Shi(3* arcsinh(a*x))/a^3
Time = 0.22 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int \frac {x^2}{\text {arcsinh}(a x)^2} \, dx=-\frac {\frac {4 a^2 x^2 \sqrt {1+a^2 x^2}}{\text {arcsinh}(a x)}+\text {Shi}(\text {arcsinh}(a x))-3 \text {Shi}(3 \text {arcsinh}(a x))}{4 a^3} \] Input:
Integrate[x^2/ArcSinh[a*x]^2,x]
Output:
-1/4*((4*a^2*x^2*Sqrt[1 + a^2*x^2])/ArcSinh[a*x] + SinhIntegral[ArcSinh[a* x]] - 3*SinhIntegral[3*ArcSinh[a*x]])/a^3
Time = 0.32 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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.
\(\displaystyle \int \frac {x^2}{\text {arcsinh}(a x)^2} \, dx\) |
\(\Big \downarrow \) 6193 |
\(\displaystyle \frac {\int \left (\frac {3 \sinh (3 \text {arcsinh}(a x))}{4 \text {arcsinh}(a x)}-\frac {a x}{4 \text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)}{a^3}-\frac {x^2 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {3}{4} \text {Shi}(3 \text {arcsinh}(a x))-\frac {1}{4} \text {Shi}(\text {arcsinh}(a x))}{a^3}-\frac {x^2 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}\) |
Input:
Int[x^2/ArcSinh[a*x]^2,x]
Output:
-((x^2*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x])) + (-1/4*SinhIntegral[ArcSinh[a *x]] + (3*SinhIntegral[3*ArcSinh[a*x]])/4)/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]
Time = 0.48 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {\frac {\sqrt {a^{2} x^{2}+1}}{4 \,\operatorname {arcsinh}\left (x a \right )}-\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (x a \right )\right )}{4}-\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (x a \right )\right )}{4 \,\operatorname {arcsinh}\left (x a \right )}+\frac {3 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (x a \right )\right )}{4}}{a^{3}}\) | \(56\) |
default | \(\frac {\frac {\sqrt {a^{2} x^{2}+1}}{4 \,\operatorname {arcsinh}\left (x a \right )}-\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (x a \right )\right )}{4}-\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (x a \right )\right )}{4 \,\operatorname {arcsinh}\left (x a \right )}+\frac {3 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (x a \right )\right )}{4}}{a^{3}}\) | \(56\) |
Input:
int(x^2/arcsinh(x*a)^2,x,method=_RETURNVERBOSE)
Output:
1/a^3*(1/4/arcsinh(x*a)*(a^2*x^2+1)^(1/2)-1/4*Shi(arcsinh(x*a))-1/4/arcsin h(x*a)*cosh(3*arcsinh(x*a))+3/4*Shi(3*arcsinh(x*a)))
\[ \int \frac {x^2}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \] Input:
integrate(x^2/arcsinh(a*x)^2,x, algorithm="fricas")
Output:
integral(x^2/arcsinh(a*x)^2, x)
\[ \int \frac {x^2}{\text {arcsinh}(a x)^2} \, dx=\int \frac {x^{2}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx \] Input:
integrate(x**2/asinh(a*x)**2,x)
Output:
Integral(x**2/asinh(a*x)**2, x)
\[ \int \frac {x^2}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \] Input:
integrate(x^2/arcsinh(a*x)^2,x, algorithm="maxima")
Output:
-(a^3*x^5 + a*x^3 + (a^2*x^4 + x^2)*sqrt(a^2*x^2 + 1))/((a^3*x^2 + sqrt(a^ 2*x^2 + 1)*a^2*x + a)*log(a*x + sqrt(a^2*x^2 + 1))) + integrate((3*a^5*x^6 + 6*a^3*x^4 + 3*a*x^2 + (3*a^3*x^4 + a*x^2)*(a^2*x^2 + 1) + (6*a^4*x^5 + 7*a^2*x^3 + 2*x)*sqrt(a^2*x^2 + 1))/((a^5*x^4 + (a^2*x^2 + 1)*a^3*x^2 + 2* a^3*x^2 + 2*(a^4*x^3 + a^2*x)*sqrt(a^2*x^2 + 1) + a)*log(a*x + sqrt(a^2*x^ 2 + 1))), x)
\[ \int \frac {x^2}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \] Input:
integrate(x^2/arcsinh(a*x)^2,x, algorithm="giac")
Output:
integrate(x^2/arcsinh(a*x)^2, x)
Timed out. \[ \int \frac {x^2}{\text {arcsinh}(a x)^2} \, dx=\int \frac {x^2}{{\mathrm {asinh}\left (a\,x\right )}^2} \,d x \] Input:
int(x^2/asinh(a*x)^2,x)
Output:
int(x^2/asinh(a*x)^2, x)
\[ \int \frac {x^2}{\text {arcsinh}(a x)^2} \, dx=\int \frac {x^{2}}{\mathit {asinh} \left (a x \right )^{2}}d x \] Input:
int(x^2/asinh(a*x)^2,x)
Output:
int(x**2/asinh(a*x)**2,x)