\(\int \frac {x}{\text {arcsinh}(a x)^3} \, dx\) [63]
Optimal result
Integrand size = 8, antiderivative size = 63 \[
\int \frac {x}{\text {arcsinh}(a x)^3} \, dx=-\frac {x \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {1}{2 a^2 \text {arcsinh}(a x)}-\frac {x^2}{\text {arcsinh}(a x)}+\frac {\text {Shi}(2 \text {arcsinh}(a x))}{a^2}
\]
[Out]
-1/2/a^2/arcsinh(a*x)-x^2/arcsinh(a*x)+Shi(2*arcsinh(a*x))/a^2-1/2*x*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)^2
Rubi [A] (verified)
Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of
steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5779, 5818, 5780, 5556, 12,
3379, 5783} \[
\int \frac {x}{\text {arcsinh}(a x)^3} \, dx=\frac {\text {Shi}(2 \text {arcsinh}(a x))}{a^2}-\frac {x \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}-\frac {1}{2 a^2 \text {arcsinh}(a x)}-\frac {x^2}{\text {arcsinh}(a x)}
\]
[In]
Int[x/ArcSinh[a*x]^3,x]
[Out]
-1/2*(x*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x]^2) - 1/(2*a^2*ArcSinh[a*x]) - x^2/ArcSinh[a*x] + SinhIntegral[2*Arc
Sinh[a*x]]/a^2
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 3379
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - 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 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 5783
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ
[e, c^2*d] && NeQ[n, -1]
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 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}+\frac {\int \frac {1}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2} \, dx}{2 a}+a \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2} \, dx \\ & = -\frac {x \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {1}{2 a^2 \text {arcsinh}(a x)}-\frac {x^2}{\text {arcsinh}(a x)}+2 \int \frac {x}{\text {arcsinh}(a x)} \, dx \\ & = -\frac {x \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {1}{2 a^2 \text {arcsinh}(a x)}-\frac {x^2}{\text {arcsinh}(a x)}+\frac {2 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{a^2} \\ & = -\frac {x \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {1}{2 a^2 \text {arcsinh}(a x)}-\frac {x^2}{\text {arcsinh}(a x)}+\frac {2 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\text {arcsinh}(a x)\right )}{a^2} \\ & = -\frac {x \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {1}{2 a^2 \text {arcsinh}(a x)}-\frac {x^2}{\text {arcsinh}(a x)}+\frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{a^2} \\ & = -\frac {x \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {1}{2 a^2 \text {arcsinh}(a x)}-\frac {x^2}{\text {arcsinh}(a x)}+\frac {\text {Shi}(2 \text {arcsinh}(a x))}{a^2} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.98
\[
\int \frac {x}{\text {arcsinh}(a x)^3} \, dx=-\frac {x \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}+\frac {-1-2 a^2 x^2}{2 a^2 \text {arcsinh}(a x)}+\frac {\text {Shi}(2 \text {arcsinh}(a x))}{a^2}
\]
[In]
Integrate[x/ArcSinh[a*x]^3,x]
[Out]
-1/2*(x*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x]^2) + (-1 - 2*a^2*x^2)/(2*a^2*ArcSinh[a*x]) + SinhIntegral[2*ArcSinh
[a*x]]/a^2
Maple [A] (verified)
Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.68
| | |
| method | result | size |
| | |
| derivativedivides |
\(\frac {-\frac {\sinh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{4 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{2 \,\operatorname {arcsinh}\left (a x \right )}+\operatorname {Shi}\left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{a^{2}}\) |
\(43\) |
| default |
\(\frac {-\frac {\sinh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{4 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{2 \,\operatorname {arcsinh}\left (a x \right )}+\operatorname {Shi}\left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{a^{2}}\) |
\(43\) |
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[In]
int(x/arcsinh(a*x)^3,x,method=_RETURNVERBOSE)
[Out]
1/a^2*(-1/4/arcsinh(a*x)^2*sinh(2*arcsinh(a*x))-1/2/arcsinh(a*x)*cosh(2*arcsinh(a*x))+Shi(2*arcsinh(a*x)))
Fricas [F]
\[
\int \frac {x}{\text {arcsinh}(a x)^3} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (a x\right )^{3}} \,d x }
\]
[In]
integrate(x/arcsinh(a*x)^3,x, algorithm="fricas")
[Out]
integral(x/arcsinh(a*x)^3, x)
Sympy [F]
\[
\int \frac {x}{\text {arcsinh}(a x)^3} \, dx=\int \frac {x}{\operatorname {asinh}^{3}{\left (a x \right )}}\, dx
\]
[In]
integrate(x/asinh(a*x)**3,x)
[Out]
Integral(x/asinh(a*x)**3, x)
Maxima [F]
\[
\int \frac {x}{\text {arcsinh}(a x)^3} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (a x\right )^{3}} \,d x }
\]
[In]
integrate(x/arcsinh(a*x)^3,x, algorithm="maxima")
[Out]
-1/2*(a^8*x^8 + 3*a^6*x^6 + 3*a^4*x^4 + a^2*x^2 + (a^5*x^5 + a^3*x^3)*(a^2*x^2 + 1)^(3/2) + (3*a^6*x^6 + 5*a^4
*x^4 + 2*a^2*x^2)*(a^2*x^2 + 1) + (2*a^8*x^8 + 6*a^6*x^6 + 6*a^4*x^4 + 2*a^2*x^2 + 2*(a^5*x^5 + a^3*x^3)*(a^2*
x^2 + 1)^(3/2) + (6*a^6*x^6 + 10*a^4*x^4 + 5*a^2*x^2 + 1)*(a^2*x^2 + 1) + (6*a^7*x^7 + 14*a^5*x^5 + 11*a^3*x^3
+ 3*a*x)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1)) + (3*a^7*x^7 + 7*a^5*x^5 + 5*a^3*x^3 + a*x)*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*
(4*a^9*x^9 + 16*a^7*x^7 + 4*(a^2*x^2 + 1)^2*a^5*x^5 + 24*a^5*x^5 + 16*a^3*x^3 + (16*a^6*x^6 + 16*a^4*x^4 - 3)*
(a^2*x^2 + 1)^(3/2) + 24*(a^7*x^7 + 2*a^5*x^5 + a^3*x^3)*(a^2*x^2 + 1) + 4*a*x + (16*a^8*x^8 + 48*a^6*x^6 + 48
*a^4*x^4 + 19*a^2*x^2 + 3)*sqrt(a^2*x^2 + 1))/((a^9*x^8 + 4*a^7*x^6 + (a^2*x^2 + 1)^2*a^5*x^4 + 6*a^5*x^4 + 4*
a^3*x^2 + 4*(a^6*x^5 + a^4*x^3)*(a^2*x^2 + 1)^(3/2) + 6*(a^7*x^6 + 2*a^5*x^4 + a^3*x^2)*(a^2*x^2 + 1) + 4*(a^8
*x^7 + 3*a^6*x^5 + 3*a^4*x^3 + a^2*x)*sqrt(a^2*x^2 + 1) + a)*log(a*x + sqrt(a^2*x^2 + 1))), x)
Giac [F]
\[
\int \frac {x}{\text {arcsinh}(a x)^3} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (a x\right )^{3}} \,d x }
\]
[In]
integrate(x/arcsinh(a*x)^3,x, algorithm="giac")
[Out]
integrate(x/arcsinh(a*x)^3, x)
Mupad [F(-1)]
Timed out. \[
\int \frac {x}{\text {arcsinh}(a x)^3} \, dx=\int \frac {x}{{\mathrm {asinh}\left (a\,x\right )}^3} \,d x
\]
[In]
int(x/asinh(a*x)^3,x)
[Out]
int(x/asinh(a*x)^3, x)