\(\int \frac {x^3}{\text {arcsinh}(a x)} \, dx\) [45]

Optimal result

Integrand size = 10, antiderivative size = 29 \[ \int \frac {x^3}{\text {arcsinh}(a x)} \, dx=-\frac {\text {Shi}(2 \text {arcsinh}(a x))}{4 a^4}+\frac {\text {Shi}(4 \text {arcsinh}(a x))}{8 a^4} \]

[Out]

-1/4*Shi(2*arcsinh(a*x))/a^4+1/8*Shi(4*arcsinh(a*x))/a^4

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5780, 5556, 3379} \[ \int \frac {x^3}{\text {arcsinh}(a x)} \, dx=\frac {\text {Shi}(4 \text {arcsinh}(a x))}{8 a^4}-\frac {\text {Shi}(2 \text {arcsinh}(a x))}{4 a^4} \]

[In]

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

[Out]

-1/4*SinhIntegral[2*ArcSinh[a*x]]/a^4 + SinhIntegral[4*ArcSinh[a*x]]/(8*a^4)

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 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]

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {x^3}{\text {arcsinh}(a x)} \, dx=\frac {-2 \text {Shi}(2 \text {arcsinh}(a x))+\text {Shi}(4 \text {arcsinh}(a x))}{8 a^4} \]

[In]

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

[Out]

(-2*SinhIntegral[2*ArcSinh[a*x]] + SinhIntegral[4*ArcSinh[a*x]])/(8*a^4)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {Shi}\left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{4}+\frac {\operatorname {Shi}\left (4 \,\operatorname {arcsinh}\left (a x \right )\right )}{8}}{a^{4}}\)	\(24\)
default	\(\frac {-\frac {\operatorname {Shi}\left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{4}+\frac {\operatorname {Shi}\left (4 \,\operatorname {arcsinh}\left (a x \right )\right )}{8}}{a^{4}}\)	\(24\)

[In]

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

[Out]

1/a^4*(-1/4*Shi(2*arcsinh(a*x))+1/8*Shi(4*arcsinh(a*x)))

Fricas [F]

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

[In]

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

[Out]

integral(x^3/arcsinh(a*x), x)

Sympy [F]

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

[In]

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

[Out]

Integral(x**3/asinh(a*x), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^3/arcsinh(a*x), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3}{\text {arcsinh}(a x)} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^3/asinh(a*x), x)