\(\int \frac {1}{\text {arcsinh}(a x)^4} \, dx\) [71]

Optimal result

Integrand size = 6, antiderivative size = 76 \[ \int \frac {1}{\text {arcsinh}(a x)^4} \, dx=-\frac {\sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^3}-\frac {x}{6 \text {arcsinh}(a x)^2}-\frac {\sqrt {1+a^2 x^2}}{6 a \text {arcsinh}(a x)}+\frac {\text {Shi}(\text {arcsinh}(a x))}{6 a} \]

[Out]

-1/6*x/arcsinh(a*x)^2+1/6*Shi(arcsinh(a*x))/a-1/3*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)^3-1/6*(a^2*x^2+1)^(1/2)/a/a
rcsinh(a*x)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5773, 5818, 5819, 3379} \[ \int \frac {1}{\text {arcsinh}(a x)^4} \, dx=-\frac {\sqrt {a^2 x^2+1}}{6 a \text {arcsinh}(a x)}-\frac {\sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}+\frac {\text {Shi}(\text {arcsinh}(a x))}{6 a}-\frac {x}{6 \text {arcsinh}(a x)^2} \]

[In]

Int[ArcSinh[a*x]^(-4),x]

[Out]

-1/3*Sqrt[1 + a^2*x^2]/(a*ArcSinh[a*x]^3) - x/(6*ArcSinh[a*x]^2) - Sqrt[1 + a^2*x^2]/(6*a*ArcSinh[a*x]) + Sinh
Integral[ArcSinh[a*x]]/(6*a)

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 5773

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1
)/(b*c*(n + 1))), x] - Dist[c/(b*(n + 1)), Int[x*((a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x] /; F
reeQ[{a, b, c}, x] && LtQ[n, -1]

Rule 5818

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] - Dist[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]

Rule 5819

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*
c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1),
x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m,
0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^3}+\frac {1}{3} a \int \frac {x}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3} \, dx \\ & = -\frac {\sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^3}-\frac {x}{6 \text {arcsinh}(a x)^2}+\frac {1}{6} \int \frac {1}{\text {arcsinh}(a x)^2} \, dx \\ & = -\frac {\sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^3}-\frac {x}{6 \text {arcsinh}(a x)^2}-\frac {\sqrt {1+a^2 x^2}}{6 a \text {arcsinh}(a x)}+\frac {1}{6} a \int \frac {x}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx \\ & = -\frac {\sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^3}-\frac {x}{6 \text {arcsinh}(a x)^2}-\frac {\sqrt {1+a^2 x^2}}{6 a \text {arcsinh}(a x)}+\frac {\text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{6 a} \\ & = -\frac {\sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^3}-\frac {x}{6 \text {arcsinh}(a x)^2}-\frac {\sqrt {1+a^2 x^2}}{6 a \text {arcsinh}(a x)}+\frac {\text {Shi}(\text {arcsinh}(a x))}{6 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\text {arcsinh}(a x)^4} \, dx=-\frac {2 \sqrt {1+a^2 x^2}+a x \text {arcsinh}(a x)+\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2-\text {arcsinh}(a x)^3 \text {Shi}(\text {arcsinh}(a x))}{6 a \text {arcsinh}(a x)^3} \]

[In]

Integrate[ArcSinh[a*x]^(-4),x]

[Out]

-1/6*(2*Sqrt[1 + a^2*x^2] + a*x*ArcSinh[a*x] + Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2 - ArcSinh[a*x]^3*SinhIntegral[
ArcSinh[a*x]])/(a*ArcSinh[a*x]^3)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {a^{2} x^{2}+1}}{3 \operatorname {arcsinh}\left (a x \right )^{3}}-\frac {a x}{6 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {\sqrt {a^{2} x^{2}+1}}{6 \,\operatorname {arcsinh}\left (a x \right )}+\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{6}}{a}\)	\(61\)
default	\(\frac {-\frac {\sqrt {a^{2} x^{2}+1}}{3 \operatorname {arcsinh}\left (a x \right )^{3}}-\frac {a x}{6 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {\sqrt {a^{2} x^{2}+1}}{6 \,\operatorname {arcsinh}\left (a x \right )}+\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{6}}{a}\)	\(61\)

[In]

int(1/arcsinh(a*x)^4,x,method=_RETURNVERBOSE)

[Out]

1/a*(-1/3/arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)-1/6*a*x/arcsinh(a*x)^2-1/6/arcsinh(a*x)*(a^2*x^2+1)^(1/2)+1/6*Shi(a
rcsinh(a*x)))

Fricas [F]

\[ \int \frac {1}{\text {arcsinh}(a x)^4} \, dx=\int { \frac {1}{\operatorname {arsinh}\left (a x\right )^{4}} \,d x } \]

[In]

integrate(1/arcsinh(a*x)^4,x, algorithm="fricas")

[Out]

integral(arcsinh(a*x)^(-4), x)

Sympy [F]

\[ \int \frac {1}{\text {arcsinh}(a x)^4} \, dx=\int \frac {1}{\operatorname {asinh}^{4}{\left (a x \right )}}\, dx \]

[In]

integrate(1/asinh(a*x)**4,x)

[Out]

Integral(asinh(a*x)**(-4), x)

Maxima [F]

\[ \int \frac {1}{\text {arcsinh}(a x)^4} \, dx=\int { \frac {1}{\operatorname {arsinh}\left (a x\right )^{4}} \,d x } \]

[In]

integrate(1/arcsinh(a*x)^4,x, algorithm="maxima")

[Out]

-1/6*(2*a^11*x^11 + 10*a^9*x^9 + 20*a^7*x^7 + 20*a^5*x^5 + 10*a^3*x^3 + 2*(a^6*x^6 + a^4*x^4)*(a^2*x^2 + 1)^(5
/2) + 2*(5*a^7*x^7 + 9*a^5*x^5 + 4*a^3*x^3)*(a^2*x^2 + 1)^2 + (a^11*x^11 + 5*a^9*x^9 + 10*a^7*x^7 + 10*a^5*x^5
+ 5*a^3*x^3 + (a^6*x^6 + a^4*x^4 + 3*a^2*x^2 + 3)*(a^2*x^2 + 1)^(5/2) + (5*a^7*x^7 + 9*a^5*x^5 + 10*a^3*x^3 +
6*a*x)*(a^2*x^2 + 1)^2 + (10*a^8*x^8 + 26*a^6*x^6 + 22*a^4*x^4 + 3*a^2*x^2 - 3)*(a^2*x^2 + 1)^(3/2) + 2*(5*a^
9*x^9 + 17*a^7*x^7 + 18*a^5*x^5 + 5*a^3*x^3 - a*x)*(a^2*x^2 + 1) + a*x + (5*a^10*x^10 + 21*a^8*x^8 + 31*a^6*x^
6 + 20*a^4*x^4 + 6*a^2*x^2 + 1)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))^2 + 4*(5*a^8*x^8 + 13*a^6*x^6
+ 11*a^4*x^4 + 3*a^2*x^2)*(a^2*x^2 + 1)^(3/2) + 4*(5*a^9*x^9 + 17*a^7*x^7 + 21*a^5*x^5 + 11*a^3*x^3 + 2*a*x)*(
a^2*x^2 + 1) + 2*a*x + (a^11*x^11 + 5*a^9*x^9 + 10*a^7*x^7 + 10*a^5*x^5 + 5*a^3*x^3 + (a^6*x^6 - a^2*x^2)*(a^2
*x^2 + 1)^(5/2) + (5*a^7*x^7 + 5*a^5*x^5 - 2*a^3*x^3 - 2*a*x)*(a^2*x^2 + 1)^2 + (10*a^8*x^8 + 20*a^6*x^6 + 10*
a^4*x^4 - a^2*x^2 - 1)*(a^2*x^2 + 1)^(3/2) + 2*(5*a^9*x^9 + 15*a^7*x^7 + 16*a^5*x^5 + 7*a^3*x^3 + a*x)*(a^2*x^
2 + 1) + a*x + (5*a^10*x^10 + 20*a^8*x^8 + 31*a^6*x^6 + 23*a^4*x^4 + 8*a^2*x^2 + 1)*sqrt(a^2*x^2 + 1))*log(a*x
+ sqrt(a^2*x^2 + 1)) + 2*(5*a^10*x^10 + 21*a^8*x^8 + 34*a^6*x^6 + 26*a^4*x^4 + 9*a^2*x^2 + 1)*sqrt(a^2*x^2 +
1))/((a^11*x^10 + 5*a^9*x^8 + (a^2*x^2 + 1)^(5/2)*a^6*x^5 + 10*a^7*x^6 + 10*a^5*x^4 + 5*a^3*x^2 + 5*(a^7*x^6 +
a^5*x^4)*(a^2*x^2 + 1)^2 + 10*(a^8*x^7 + 2*a^6*x^5 + a^4*x^3)*(a^2*x^2 + 1)^(3/2) + 10*(a^9*x^8 + 3*a^7*x^6 +
3*a^5*x^4 + a^3*x^2)*(a^2*x^2 + 1) + 5*(a^10*x^9 + 4*a^8*x^7 + 6*a^6*x^5 + 4*a^4*x^3 + a^2*x)*sqrt(a^2*x^2 +
1) + a)*log(a*x + sqrt(a^2*x^2 + 1))^3) + integrate(1/6*(a^12*x^12 + 6*a^10*x^10 + 15*a^8*x^8 + 20*a^6*x^6 + 1
5*a^4*x^4 + (a^6*x^6 - a^4*x^4 - 9*a^2*x^2 - 15)*(a^2*x^2 + 1)^3 + 6*a^2*x^2 + (6*a^7*x^7 + a^5*x^5 - 31*a^3*x
^3 - 33*a*x)*(a^2*x^2 + 1)^(5/2) + (15*a^8*x^8 + 20*a^6*x^6 - 19*a^4*x^4 - 3*a^2*x^2 + 21)*(a^2*x^2 + 1)^2 + (
20*a^9*x^9 + 50*a^7*x^7 + 54*a^5*x^5 + 59*a^3*x^3 + 35*a*x)*(a^2*x^2 + 1)^(3/2) + (15*a^10*x^10 + 55*a^8*x^8 +
101*a^6*x^6 + 90*a^4*x^4 + 22*a^2*x^2 - 7)*(a^2*x^2 + 1) + (6*a^11*x^11 + 29*a^9*x^9 + 65*a^7*x^7 + 66*a^5*x^
5 + 23*a^3*x^3 - a*x)*sqrt(a^2*x^2 + 1) + 1)/((a^12*x^12 + 6*a^10*x^10 + 15*a^8*x^8 + (a^2*x^2 + 1)^3*a^6*x^6
+ 20*a^6*x^6 + 15*a^4*x^4 + 6*a^2*x^2 + 6*(a^7*x^7 + a^5*x^5)*(a^2*x^2 + 1)^(5/2) + 15*(a^8*x^8 + 2*a^6*x^6 +
a^4*x^4)*(a^2*x^2 + 1)^2 + 20*(a^9*x^9 + 3*a^7*x^7 + 3*a^5*x^5 + a^3*x^3)*(a^2*x^2 + 1)^(3/2) + 15*(a^10*x^10
+ 4*a^8*x^8 + 6*a^6*x^6 + 4*a^4*x^4 + a^2*x^2)*(a^2*x^2 + 1) + 6*(a^11*x^11 + 5*a^9*x^9 + 10*a^7*x^7 + 10*a^5*
x^5 + 5*a^3*x^3 + a*x)*sqrt(a^2*x^2 + 1) + 1)*log(a*x + sqrt(a^2*x^2 + 1))), x)

Giac [F]

\[ \int \frac {1}{\text {arcsinh}(a x)^4} \, dx=\int { \frac {1}{\operatorname {arsinh}\left (a x\right )^{4}} \,d x } \]

[In]

integrate(1/arcsinh(a*x)^4,x, algorithm="giac")

[Out]

integrate(arcsinh(a*x)^(-4), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\text {arcsinh}(a x)^4} \, dx=\int \frac {1}{{\mathrm {asinh}\left (a\,x\right )}^4} \,d x \]

[In]

int(1/asinh(a*x)^4,x)

[Out]

int(1/asinh(a*x)^4, x)