\(\int \frac {x^4}{\text {arcsinh}(a x)^2} \, dx\) [53]

Optimal result

Integrand size = 10, antiderivative size = 68 \[ \int \frac {x^4}{\text {arcsinh}(a x)^2} \, dx=-\frac {x^4 \sqrt {1+a^2 x^2}}{a \text {arcsinh}(a x)}+\frac {\text {Shi}(\text {arcsinh}(a x))}{8 a^5}-\frac {9 \text {Shi}(3 \text {arcsinh}(a x))}{16 a^5}+\frac {5 \text {Shi}(5 \text {arcsinh}(a x))}{16 a^5} \]

[Out]

1/8*Shi(arcsinh(a*x))/a^5-9/16*Shi(3*arcsinh(a*x))/a^5+5/16*Shi(5*arcsinh(a*x))/a^5-x^4*(a^2*x^2+1)^(1/2)/a/ar
csinh(a*x)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5778, 3379} \[ \int \frac {x^4}{\text {arcsinh}(a x)^2} \, dx=\frac {\text {Shi}(\text {arcsinh}(a x))}{8 a^5}-\frac {9 \text {Shi}(3 \text {arcsinh}(a x))}{16 a^5}+\frac {5 \text {Shi}(5 \text {arcsinh}(a x))}{16 a^5}-\frac {x^4 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)} \]

[In]

Int[x^4/ArcSinh[a*x]^2,x]

[Out]

-((x^4*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x])) + SinhIntegral[ArcSinh[a*x]]/(8*a^5) - (9*SinhIntegral[3*ArcSinh[a
*x]])/(16*a^5) + (5*SinhIntegral[5*ArcSinh[a*x]])/(16*a^5)

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 5778

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[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), Si
nh[-a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}
, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^4 \sqrt {1+a^2 x^2}}{a \text {arcsinh}(a x)}+\frac {\text {Subst}\left (\int \left (\frac {\sinh (x)}{8 x}-\frac {9 \sinh (3 x)}{16 x}+\frac {5 \sinh (5 x)}{16 x}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a^5} \\ & = -\frac {x^4 \sqrt {1+a^2 x^2}}{a \text {arcsinh}(a x)}+\frac {\text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{8 a^5}+\frac {5 \text {Subst}\left (\int \frac {\sinh (5 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^5}-\frac {9 \text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^5} \\ & = -\frac {x^4 \sqrt {1+a^2 x^2}}{a \text {arcsinh}(a x)}+\frac {\text {Shi}(\text {arcsinh}(a x))}{8 a^5}-\frac {9 \text {Shi}(3 \text {arcsinh}(a x))}{16 a^5}+\frac {5 \text {Shi}(5 \text {arcsinh}(a x))}{16 a^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.88 \[ \int \frac {x^4}{\text {arcsinh}(a x)^2} \, dx=\frac {-\frac {16 a^4 x^4 \sqrt {1+a^2 x^2}}{\text {arcsinh}(a x)}+2 \text {Shi}(\text {arcsinh}(a x))-9 \text {Shi}(3 \text {arcsinh}(a x))+5 \text {Shi}(5 \text {arcsinh}(a x))}{16 a^5} \]

[In]

Integrate[x^4/ArcSinh[a*x]^2,x]

[Out]

((-16*a^4*x^4*Sqrt[1 + a^2*x^2])/ArcSinh[a*x] + 2*SinhIntegral[ArcSinh[a*x]] - 9*SinhIntegral[3*ArcSinh[a*x]]
+ 5*SinhIntegral[5*ArcSinh[a*x]])/(16*a^5)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.18

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {a^{2} x^{2}+1}}{8 \,\operatorname {arcsinh}\left (a x \right )}+\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{8}+\frac {3 \cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{16 \,\operatorname {arcsinh}\left (a x \right )}-\frac {9 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{16}-\frac {\cosh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{16 \,\operatorname {arcsinh}\left (a x \right )}+\frac {5 \,\operatorname {Shi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{16}}{a^{5}}\)	\(80\)
default	\(\frac {-\frac {\sqrt {a^{2} x^{2}+1}}{8 \,\operatorname {arcsinh}\left (a x \right )}+\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{8}+\frac {3 \cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{16 \,\operatorname {arcsinh}\left (a x \right )}-\frac {9 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{16}-\frac {\cosh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{16 \,\operatorname {arcsinh}\left (a x \right )}+\frac {5 \,\operatorname {Shi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{16}}{a^{5}}\)	\(80\)

[In]

int(x^4/arcsinh(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/a^5*(-1/8/arcsinh(a*x)*(a^2*x^2+1)^(1/2)+1/8*Shi(arcsinh(a*x))+3/16/arcsinh(a*x)*cosh(3*arcsinh(a*x))-9/16*S
hi(3*arcsinh(a*x))-1/16/arcsinh(a*x)*cosh(5*arcsinh(a*x))+5/16*Shi(5*arcsinh(a*x)))

Fricas [F]

\[ \int \frac {x^4}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^4/arcsinh(a*x)^2,x, algorithm="fricas")

[Out]

integral(x^4/arcsinh(a*x)^2, x)

Sympy [F]

\[ \int \frac {x^4}{\text {arcsinh}(a x)^2} \, dx=\int \frac {x^{4}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx \]

[In]

integrate(x**4/asinh(a*x)**2,x)

[Out]

Integral(x**4/asinh(a*x)**2, x)

Maxima [F]

\[ \int \frac {x^4}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^4/arcsinh(a*x)^2,x, algorithm="maxima")

[Out]

-(a^3*x^7 + a*x^5 + (a^2*x^6 + x^4)*sqrt(a^2*x^2 + 1))/((a^3*x^2 + sqrt(a^2*x^2 + 1)*a^2*x + a)*log(a*x + sqrt
(a^2*x^2 + 1))) + integrate((5*a^5*x^8 + 10*a^3*x^6 + 5*a*x^4 + (5*a^3*x^6 + 3*a*x^4)*(a^2*x^2 + 1) + (10*a^4*
x^7 + 13*a^2*x^5 + 4*x^3)*sqrt(a^2*x^2 + 1))/((a^5*x^4 + (a^2*x^2 + 1)*a^3*x^2 + 2*a^3*x^2 + 2*(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]

\[ \int \frac {x^4}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^4/arcsinh(a*x)^2,x, algorithm="giac")

[Out]

integrate(x^4/arcsinh(a*x)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\text {arcsinh}(a x)^2} \, dx=\int \frac {x^4}{{\mathrm {asinh}\left (a\,x\right )}^2} \,d x \]

[In]

int(x^4/asinh(a*x)^2,x)

[Out]

int(x^4/asinh(a*x)^2, x)