3.1.63 \(\int \frac {x}{\text {arcsinh}(a x)^3} \, dx\) [63]

3.1.63.1 Optimal result

Integrand size = 8, antiderivative size = 63 \[ \int \frac {x}{\text {arcsinh}(a x)^3} \, dx=-\frac {x \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {1}{2 a^2 \text {arcsinh}(a x)}-\frac {x^2}{\text {arcsinh}(a x)}+\frac {\text {Shi}(2 \text {arcsinh}(a x))}{a^2} \]

output

-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
 

3.1.63.2 Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.98 \[ \int \frac {x}{\text {arcsinh}(a x)^3} \, dx=-\frac {x \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}+\frac {-1-2 a^2 x^2}{2 a^2 \text {arcsinh}(a x)}+\frac {\text {Shi}(2 \text {arcsinh}(a x))}{a^2} \]

input

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

output

-1/2*(x*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x]^2) + (-1 - 2*a^2*x^2)/(2*a^2*Ar 
cSinh[a*x]) + SinhIntegral[2*ArcSinh[a*x]]/a^2
 

3.1.63.3 Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {6194, 6198, 6233, 6195, 5971, 27, 3042, 26, 3779}

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.

\(\displaystyle \int \frac {x}{\text {arcsinh}(a x)^3} \, dx\)

\(\Big \downarrow \) 6194

\(\displaystyle \frac {\int \frac {1}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}dx}{2 a}+a \int \frac {x^2}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}dx-\frac {x \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}\)

\(\Big \downarrow \) 6198

\(\displaystyle a \int \frac {x^2}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}dx-\frac {x \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}-\frac {1}{2 a^2 \text {arcsinh}(a x)}\)

\(\Big \downarrow \) 6233

\(\displaystyle a \left (\frac {2 \int \frac {x}{\text {arcsinh}(a x)}dx}{a}-\frac {x^2}{a \text {arcsinh}(a x)}\right )-\frac {x \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}-\frac {1}{2 a^2 \text {arcsinh}(a x)}\)

\(\Big \downarrow \) 6195

\(\displaystyle a \left (\frac {2 \int \frac {a x \sqrt {a^2 x^2+1}}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a^3}-\frac {x^2}{a \text {arcsinh}(a x)}\right )-\frac {x \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}-\frac {1}{2 a^2 \text {arcsinh}(a x)}\)

\(\Big \downarrow \) 5971

\(\displaystyle a \left (\frac {2 \int \frac {\sinh (2 \text {arcsinh}(a x))}{2 \text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a^3}-\frac {x^2}{a \text {arcsinh}(a x)}\right )-\frac {x \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}-\frac {1}{2 a^2 \text {arcsinh}(a x)}\)

\(\Big \downarrow \) 27

\(\displaystyle a \left (\frac {\int \frac {\sinh (2 \text {arcsinh}(a x))}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a^3}-\frac {x^2}{a \text {arcsinh}(a x)}\right )-\frac {x \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}-\frac {1}{2 a^2 \text {arcsinh}(a x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle a \left (-\frac {x^2}{a \text {arcsinh}(a x)}+\frac {\int -\frac {i \sin (2 i \text {arcsinh}(a x))}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a^3}\right )-\frac {x \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}-\frac {1}{2 a^2 \text {arcsinh}(a x)}\)

\(\Big \downarrow \) 26

\(\displaystyle a \left (-\frac {x^2}{a \text {arcsinh}(a x)}-\frac {i \int \frac {\sin (2 i \text {arcsinh}(a x))}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a^3}\right )-\frac {x \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}-\frac {1}{2 a^2 \text {arcsinh}(a x)}\)

\(\Big \downarrow \) 3779

\(\displaystyle a \left (\frac {\text {Shi}(2 \text {arcsinh}(a x))}{a^3}-\frac {x^2}{a \text {arcsinh}(a x)}\right )-\frac {x \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}-\frac {1}{2 a^2 \text {arcsinh}(a x)}\)

input

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

output

-1/2*(x*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x]^2) - 1/(2*a^2*ArcSinh[a*x]) + a 
*(-(x^2/(a*ArcSinh[a*x])) + SinhIntegral[2*ArcSinh[a*x]]/a^3)
 

3.1.63.3.1 Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> 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 5971

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 6194

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] + (- 
Simp[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] - Simp[m/(b*c*(n + 1))   Int[x^(m - 1)*((a + b*A 
rcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && 
IGtQ[m, 0] && LtQ[n, -2]
 

rule 6195

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
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 6198

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[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 6233

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] - Simp[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]
 

3.1.63.4 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	\(\frac {-\frac {\sinh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{4 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{2 \,\operatorname {arcsinh}\left (a x \right )}+\operatorname {Shi}\left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{a^{2}}\)	\(43\)

input

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

output

1/a^2*(-1/4/arcsinh(a*x)^2*sinh(2*arcsinh(a*x))-1/2/arcsinh(a*x)*cosh(2*ar 
csinh(a*x))+Shi(2*arcsinh(a*x)))
 

3.1.63.5 Fricas [F]

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

input

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

output

integral(x/arcsinh(a*x)^3, x)
 

3.1.63.6 Sympy [F]

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

input

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

output

Integral(x/asinh(a*x)**3, x)
 

3.1.63.7 Maxima [F]

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

input

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

output

-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 + s 
qrt(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)
 

3.1.63.8 Giac [F]

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

input

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

output

integrate(x/arcsinh(a*x)^3, x)
 

3.1.63.9 Mupad [F(-1)]

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

input

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

output

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