3.45 \(\int \frac {x^3}{\sinh ^{-1}(a x)} \, dx\)

Optimal. Leaf size=29 \[ \frac {\text {Shi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^4}-\frac {\text {Shi}\left (2 \sinh ^{-1}(a x)\right )}{4 a^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5669, 5448, 3298} \[ \frac {\text {Shi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^4}-\frac {\text {Shi}\left (2 \sinh ^{-1}(a x)\right )}{4 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3298

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 5448

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 5669

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

Rubi steps

\begin {align*} \int \frac {x^3}{\sinh ^{-1}(a x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^3(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {\sinh (2 x)}{4 x}+\frac {\sinh (4 x)}{8 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^4}-\frac {\operatorname {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac {\text {Shi}\left (2 \sinh ^{-1}(a x)\right )}{4 a^4}+\frac {\text {Shi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 24, normalized size = 0.83 \[ \frac {\text {Shi}\left (4 \sinh ^{-1}(a x)\right )-2 \text {Shi}\left (2 \sinh ^{-1}(a x)\right )}{8 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3}}{\operatorname {arsinh}\left (a x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^3/arcsinh(a*x), x)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 0.16, size = 24, normalized size = 0.83 \[ \frac {-\frac {\Shi \left (2 \arcsinh \left (a x \right )\right )}{4}+\frac {\Shi \left (4 \arcsinh \left (a x \right )\right )}{8}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\operatorname {arsinh}\left (a x\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3/arcsinh(a*x), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^3}{\mathrm {asinh}\left (a\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3/asinh(a*x), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\operatorname {asinh}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/asinh(a*x), x)

________________________________________________________________________________________