\(\int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx\) [62]

Optimal result

Integrand size = 10, antiderivative size = 81 \[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=-\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {x}{a^2 \text {arcsinh}(a x)}-\frac {3 x^3}{2 \text {arcsinh}(a x)}-\frac {\text {Chi}(\text {arcsinh}(a x))}{8 a^3}+\frac {9 \text {Chi}(3 \text {arcsinh}(a x))}{8 a^3} \]

[Out]

-x/a^2/arcsinh(a*x)-3/2*x^3/arcsinh(a*x)-1/8*Chi(arcsinh(a*x))/a^3+9/8*Chi(3*arcsinh(a*x))/a^3-1/2*x^2*(a^2*x^
2+1)^(1/2)/a/arcsinh(a*x)^2

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 81, 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 = {5779, 5818, 5780, 5556, 3382, 5774} \[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=-\frac {\text {Chi}(\text {arcsinh}(a x))}{8 a^3}+\frac {9 \text {Chi}(3 \text {arcsinh}(a x))}{8 a^3}-\frac {x^2 \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}-\frac {x}{a^2 \text {arcsinh}(a x)}-\frac {3 x^3}{2 \text {arcsinh}(a x)} \]

[In]

Int[x^2/ArcSinh[a*x]^3,x]

[Out]

-1/2*(x^2*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x]^2) - x/(a^2*ArcSinh[a*x]) - (3*x^3)/(2*ArcSinh[a*x]) - CoshIntegr
al[ArcSinh[a*x]]/(8*a^3) + (9*CoshIntegral[3*ArcSinh[a*x]])/(8*a^3)

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5774

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cosh[-a/b + x/b], x], x
, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 5779

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSi
nh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Dist[c*((m + 1)/(b*(n + 1))), Int[x^(m + 1)*((a + b*ArcSinh[c*x])^(n +
1)/Sqrt[1 + c^2*x^2]), x], x] - Dist[m/(b*c*(n + 1)), Int[x^(m - 1)*((a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^
2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 5780

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Sinh
[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 5818

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp
[((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] - Dist[f*(m/
(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]], Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x]
/; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}+\frac {\int \frac {x}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2} \, dx}{a}+\frac {1}{2} (3 a) \int \frac {x^3}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2} \, dx \\ & = -\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {x}{a^2 \text {arcsinh}(a x)}-\frac {3 x^3}{2 \text {arcsinh}(a x)}+\frac {9}{2} \int \frac {x^2}{\text {arcsinh}(a x)} \, dx+\frac {\int \frac {1}{\text {arcsinh}(a x)} \, dx}{a^2} \\ & = -\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {x}{a^2 \text {arcsinh}(a x)}-\frac {3 x^3}{2 \text {arcsinh}(a x)}+\frac {\text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{a^3}+\frac {9 \text {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{2 a^3} \\ & = -\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {x}{a^2 \text {arcsinh}(a x)}-\frac {3 x^3}{2 \text {arcsinh}(a x)}+\frac {\text {Chi}(\text {arcsinh}(a x))}{a^3}+\frac {9 \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 x}+\frac {\cosh (3 x)}{4 x}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{2 a^3} \\ & = -\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {x}{a^2 \text {arcsinh}(a x)}-\frac {3 x^3}{2 \text {arcsinh}(a x)}+\frac {\text {Chi}(\text {arcsinh}(a x))}{a^3}-\frac {9 \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{8 a^3}+\frac {9 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{8 a^3} \\ & = -\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {x}{a^2 \text {arcsinh}(a x)}-\frac {3 x^3}{2 \text {arcsinh}(a x)}-\frac {\text {Chi}(\text {arcsinh}(a x))}{8 a^3}+\frac {9 \text {Chi}(3 \text {arcsinh}(a x))}{8 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.79 \[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=-\frac {\frac {4 a x \left (a x \sqrt {1+a^2 x^2}+\left (2+3 a^2 x^2\right ) \text {arcsinh}(a x)\right )}{\text {arcsinh}(a x)^2}+\text {Chi}(\text {arcsinh}(a x))-9 \text {Chi}(3 \text {arcsinh}(a x))}{8 a^3} \]

[In]

Integrate[x^2/ArcSinh[a*x]^3,x]

[Out]

-1/8*((4*a*x*(a*x*Sqrt[1 + a^2*x^2] + (2 + 3*a^2*x^2)*ArcSinh[a*x]))/ArcSinh[a*x]^2 + CoshIntegral[ArcSinh[a*x
]] - 9*CoshIntegral[3*ArcSinh[a*x]])/a^3

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\frac {\frac {\sqrt {a^{2} x^{2}+1}}{8 \operatorname {arcsinh}\left (a x \right )^{2}}+\frac {a x}{8 \,\operatorname {arcsinh}\left (a x \right )}-\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{8}-\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {3 \sinh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8 \,\operatorname {arcsinh}\left (a x \right )}+\frac {9 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8}}{a^{3}}\)	\(81\)
default	\(\frac {\frac {\sqrt {a^{2} x^{2}+1}}{8 \operatorname {arcsinh}\left (a x \right )^{2}}+\frac {a x}{8 \,\operatorname {arcsinh}\left (a x \right )}-\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{8}-\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {3 \sinh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8 \,\operatorname {arcsinh}\left (a x \right )}+\frac {9 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8}}{a^{3}}\)	\(81\)

[In]

int(x^2/arcsinh(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/a^3*(1/8/arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)+1/8*a*x/arcsinh(a*x)-1/8*Chi(arcsinh(a*x))-1/8/arcsinh(a*x)^2*cosh
(3*arcsinh(a*x))-3/8/arcsinh(a*x)*sinh(3*arcsinh(a*x))+9/8*Chi(3*arcsinh(a*x)))

Fricas [F]

\[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{3}} \,d x } \]

[In]

integrate(x^2/arcsinh(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^2/arcsinh(a*x)^3, x)

Sympy [F]

\[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=\int \frac {x^{2}}{\operatorname {asinh}^{3}{\left (a x \right )}}\, dx \]

[In]

integrate(x**2/asinh(a*x)**3,x)

[Out]

Integral(x**2/asinh(a*x)**3, x)

Maxima [F]

\[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{3}} \,d x } \]

[In]

integrate(x^2/arcsinh(a*x)^3,x, algorithm="maxima")

[Out]

-1/2*(a^8*x^9 + 3*a^6*x^7 + 3*a^4*x^5 + a^2*x^3 + (a^5*x^6 + a^3*x^4)*(a^2*x^2 + 1)^(3/2) + (3*a^6*x^7 + 5*a^4
*x^5 + 2*a^2*x^3)*(a^2*x^2 + 1) + (3*a^8*x^9 + 9*a^6*x^7 + 9*a^4*x^5 + 3*a^2*x^3 + (3*a^5*x^6 + 4*a^3*x^4 + a*
x^2)*(a^2*x^2 + 1)^(3/2) + (9*a^6*x^7 + 17*a^4*x^5 + 10*a^2*x^3 + 2*x)*(a^2*x^2 + 1) + (9*a^7*x^8 + 22*a^5*x^6
+ 18*a^3*x^4 + 5*a*x^2)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1)) + (3*a^7*x^8 + 7*a^5*x^6 + 5*a^3*x^4
+ a*x^2)*sqrt(a^2*x^2 + 1))/((a^8*x^6 + 3*a^6*x^4 + (a^2*x^2 + 1)^(3/2)*a^5*x^3 + 3*a^4*x^2 + 3*(a^6*x^4 + a^4
*x^2)*(a^2*x^2 + 1) + a^2 + 3*(a^7*x^5 + 2*a^5*x^3 + a^3*x)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))^2)
+ integrate(1/2*(9*a^10*x^10 + 36*a^8*x^8 + 54*a^6*x^6 + 36*a^4*x^4 + 9*a^2*x^2 + (9*a^6*x^6 + 4*a^4*x^4 - a^
2*x^2)*(a^2*x^2 + 1)^2 + (36*a^7*x^7 + 48*a^5*x^5 + 13*a^3*x^3 - 2*a*x)*(a^2*x^2 + 1)^(3/2) + (54*a^8*x^8 + 12
0*a^6*x^6 + 83*a^4*x^4 + 19*a^2*x^2 + 2)*(a^2*x^2 + 1) + (36*a^9*x^9 + 112*a^7*x^7 + 123*a^5*x^5 + 57*a^3*x^3
+ 10*a*x)*sqrt(a^2*x^2 + 1))/((a^10*x^8 + 4*a^8*x^6 + (a^2*x^2 + 1)^2*a^6*x^4 + 6*a^6*x^4 + 4*a^4*x^2 + 4*(a^7
*x^5 + a^5*x^3)*(a^2*x^2 + 1)^(3/2) + 6*(a^8*x^6 + 2*a^6*x^4 + a^4*x^2)*(a^2*x^2 + 1) + a^2 + 4*(a^9*x^7 + 3*a
^7*x^5 + 3*a^5*x^3 + a^3*x)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))), x)

Giac [F]

\[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{3}} \,d x } \]

[In]

integrate(x^2/arcsinh(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^2/arcsinh(a*x)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=\int \frac {x^2}{{\mathrm {asinh}\left (a\,x\right )}^3} \,d x \]

[In]

int(x^2/asinh(a*x)^3,x)

[Out]

int(x^2/asinh(a*x)^3, x)