\(\int \frac {x^4}{\text {arcsinh}(a x)} \, dx\) [44]

Optimal result

Integrand size = 10, antiderivative size = 41 \[ \int \frac {x^4}{\text {arcsinh}(a x)} \, dx=\frac {\text {Chi}(\text {arcsinh}(a x))}{8 a^5}-\frac {3 \text {Chi}(3 \text {arcsinh}(a x))}{16 a^5}+\frac {\text {Chi}(5 \text {arcsinh}(a x))}{16 a^5} \]

[Out]

1/8*Chi(arcsinh(a*x))/a^5-3/16*Chi(3*arcsinh(a*x))/a^5+1/16*Chi(5*arcsinh(a*x))/a^5

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5780, 5556, 3382} \[ \int \frac {x^4}{\text {arcsinh}(a x)} \, dx=\frac {\text {Chi}(\text {arcsinh}(a x))}{8 a^5}-\frac {3 \text {Chi}(3 \text {arcsinh}(a x))}{16 a^5}+\frac {\text {Chi}(5 \text {arcsinh}(a x))}{16 a^5} \]

[In]

Int[x^4/ArcSinh[a*x],x]

[Out]

CoshIntegral[ArcSinh[a*x]]/(8*a^5) - (3*CoshIntegral[3*ArcSinh[a*x]])/(16*a^5) + CoshIntegral[5*ArcSinh[a*x]]/
(16*a^5)

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 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 ^4(x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{a^5} \\ & = \frac {\text {Subst}\left (\int \left (\frac {\cosh (x)}{8 x}-\frac {3 \cosh (3 x)}{16 x}+\frac {\cosh (5 x)}{16 x}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a^5} \\ & = \frac {\text {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^5}+\frac {\text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{8 a^5}-\frac {3 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^5} \\ & = \frac {\text {Chi}(\text {arcsinh}(a x))}{8 a^5}-\frac {3 \text {Chi}(3 \text {arcsinh}(a x))}{16 a^5}+\frac {\text {Chi}(5 \text {arcsinh}(a x))}{16 a^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.76 \[ \int \frac {x^4}{\text {arcsinh}(a x)} \, dx=\frac {2 \text {Chi}(\text {arcsinh}(a x))-3 \text {Chi}(3 \text {arcsinh}(a x))+\text {Chi}(5 \text {arcsinh}(a x))}{16 a^5} \]

[In]

Integrate[x^4/ArcSinh[a*x],x]

[Out]

(2*CoshIntegral[ArcSinh[a*x]] - 3*CoshIntegral[3*ArcSinh[a*x]] + CoshIntegral[5*ArcSinh[a*x]])/(16*a^5)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.76

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{8}-\frac {3 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{16}+\frac {\operatorname {Chi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{16}}{a^{5}}\)	\(31\)
default	\(\frac {\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{8}-\frac {3 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{16}+\frac {\operatorname {Chi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{16}}{a^{5}}\)	\(31\)

[In]

int(x^4/arcsinh(a*x),x,method=_RETURNVERBOSE)

[Out]

1/a^5*(1/8*Chi(arcsinh(a*x))-3/16*Chi(3*arcsinh(a*x))+1/16*Chi(5*arcsinh(a*x)))

Fricas [F]

\[ \int \frac {x^4}{\text {arcsinh}(a x)} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )} \,d x } \]

[In]

integrate(x^4/arcsinh(a*x),x, algorithm="fricas")

[Out]

integral(x^4/arcsinh(a*x), x)

Sympy [F]

\[ \int \frac {x^4}{\text {arcsinh}(a x)} \, dx=\int \frac {x^{4}}{\operatorname {asinh}{\left (a x \right )}}\, dx \]

[In]

integrate(x**4/asinh(a*x),x)

[Out]

Integral(x**4/asinh(a*x), x)

Maxima [F]

\[ \int \frac {x^4}{\text {arcsinh}(a x)} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )} \,d x } \]

[In]

integrate(x^4/arcsinh(a*x),x, algorithm="maxima")

[Out]

integrate(x^4/arcsinh(a*x), x)

Giac [F]

\[ \int \frac {x^4}{\text {arcsinh}(a x)} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )} \,d x } \]

[In]

integrate(x^4/arcsinh(a*x),x, algorithm="giac")

[Out]

integrate(x^4/arcsinh(a*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\text {arcsinh}(a x)} \, dx=\int \frac {x^4}{\mathrm {asinh}\left (a\,x\right )} \,d x \]

[In]

int(x^4/asinh(a*x),x)

[Out]

int(x^4/asinh(a*x), x)