\(\int \frac {x}{\text {arcsinh}(a x)^4} \, dx\) [70]

Optimal result

Integrand size = 8, antiderivative size = 95 \[ \int \frac {x}{\text {arcsinh}(a x)^4} \, dx=-\frac {x \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^3}-\frac {1}{6 a^2 \text {arcsinh}(a x)^2}-\frac {x^2}{3 \text {arcsinh}(a x)^2}-\frac {2 x \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)}+\frac {2 \text {Chi}(2 \text {arcsinh}(a x))}{3 a^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5779, 5818, 5778, 3382, 5783} \[ \int \frac {x}{\text {arcsinh}(a x)^4} \, dx=\frac {2 \text {Chi}(2 \text {arcsinh}(a x))}{3 a^2}-\frac {2 x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)}-\frac {x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}-\frac {1}{6 a^2 \text {arcsinh}(a x)^2}-\frac {x^2}{3 \text {arcsinh}(a x)^2} \]

[In]

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

[Out]

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

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 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]

Rule 5779

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[c*((m + 1)/(b*(n + 1))), Int[x^(m + 1)*((a + b*ArcSinh[c*x])^(n +
1)/Sqrt[1 + c^2*x^2]), x], x] - Dist[m/(b*c*(n + 1)), Int[x^(m - 1)*((a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^
2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ
[e, c^2*d] && NeQ[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]

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.88 \[ \int \frac {x}{\text {arcsinh}(a x)^4} \, dx=-\frac {2 a x \sqrt {1+a^2 x^2}+\left (1+2 a^2 x^2\right ) \text {arcsinh}(a x)+4 a x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2-4 \text {arcsinh}(a x)^3 \text {Chi}(2 \text {arcsinh}(a x))}{6 a^2 \text {arcsinh}(a x)^3} \]

[In]

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

[Out]

-1/6*(2*a*x*Sqrt[1 + a^2*x^2] + (1 + 2*a^2*x^2)*ArcSinh[a*x] + 4*a*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2 - 4*ArcS
inh[a*x]^3*CoshIntegral[2*ArcSinh[a*x]])/(a^2*ArcSinh[a*x]^3)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.63

method	result	size
derivativedivides	\(\frac {-\frac {\sinh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{6 \operatorname {arcsinh}\left (a x \right )^{3}}-\frac {\cosh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{6 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{3 \,\operatorname {arcsinh}\left (a x \right )}+\frac {2 \,\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{3}}{a^{2}}\)	\(60\)
default	\(\frac {-\frac {\sinh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{6 \operatorname {arcsinh}\left (a x \right )^{3}}-\frac {\cosh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{6 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{3 \,\operatorname {arcsinh}\left (a x \right )}+\frac {2 \,\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{3}}{a^{2}}\)	\(60\)

[In]

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

[Out]

1/a^2*(-1/6/arcsinh(a*x)^3*sinh(2*arcsinh(a*x))-1/6/arcsinh(a*x)^2*cosh(2*arcsinh(a*x))-1/3/arcsinh(a*x)*sinh(
2*arcsinh(a*x))+2/3*Chi(2*arcsinh(a*x)))

Fricas [F]

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

[In]

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

[Out]

integral(x/arcsinh(a*x)^4, x)

Sympy [F]

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

[In]

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

[Out]

Integral(x/asinh(a*x)**4, x)

Maxima [F]

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

[In]

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

[Out]

-1/6*(2*a^12*x^12 + 10*a^10*x^10 + 20*a^8*x^8 + 20*a^6*x^6 + 10*a^4*x^4 + 2*a^2*x^2 + 2*(a^7*x^7 + a^5*x^5)*(a
^2*x^2 + 1)^(5/2) + 2*(5*a^8*x^8 + 9*a^6*x^6 + 4*a^4*x^4)*(a^2*x^2 + 1)^2 + (4*a^12*x^12 + 20*a^10*x^10 + 40*a
^8*x^8 + 40*a^6*x^6 + 20*a^4*x^4 + 4*a^2*x^2 + 4*(a^7*x^7 + a^5*x^5)*(a^2*x^2 + 1)^(5/2) + (20*a^8*x^8 + 36*a^
6*x^6 + 16*a^4*x^4 - 3*a^2*x^2 - 3)*(a^2*x^2 + 1)^2 + (40*a^9*x^9 + 104*a^7*x^7 + 88*a^5*x^5 + 21*a^3*x^3 - 3*
a*x)*(a^2*x^2 + 1)^(3/2) + (40*a^10*x^10 + 136*a^8*x^8 + 168*a^6*x^6 + 91*a^4*x^4 + 22*a^2*x^2 + 3)*(a^2*x^2 +
1) + (20*a^11*x^11 + 84*a^9*x^9 + 136*a^7*x^7 + 107*a^5*x^5 + 42*a^3*x^3 + 7*a*x)*sqrt(a^2*x^2 + 1))*log(a*x
+ sqrt(a^2*x^2 + 1))^2 + 4*(5*a^9*x^9 + 13*a^7*x^7 + 11*a^5*x^5 + 3*a^3*x^3)*(a^2*x^2 + 1)^(3/2) + 4*(5*a^10*x
^10 + 17*a^8*x^8 + 21*a^6*x^6 + 11*a^4*x^4 + 2*a^2*x^2)*(a^2*x^2 + 1) + (2*a^12*x^12 + 10*a^10*x^10 + 20*a^8*x
^8 + 20*a^6*x^6 + 10*a^4*x^4 + 2*a^2*x^2 + 2*(a^7*x^7 + a^5*x^5)*(a^2*x^2 + 1)^(5/2) + (10*a^8*x^8 + 18*a^6*x^
6 + 9*a^4*x^4 + a^2*x^2)*(a^2*x^2 + 1)^2 + (20*a^9*x^9 + 52*a^7*x^7 + 47*a^5*x^5 + 17*a^3*x^3 + 2*a*x)*(a^2*x^
2 + 1)^(3/2) + (20*a^10*x^10 + 68*a^8*x^8 + 87*a^6*x^6 + 51*a^4*x^4 + 13*a^2*x^2 + 1)*(a^2*x^2 + 1) + (10*a^11
*x^11 + 42*a^9*x^9 + 69*a^7*x^7 + 55*a^5*x^5 + 21*a^3*x^3 + 3*a*x)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 +
1)) + 2*(5*a^11*x^11 + 21*a^9*x^9 + 34*a^7*x^7 + 26*a^5*x^5 + 9*a^3*x^3 + a*x)*sqrt(a^2*x^2 + 1))/((a^12*x^10
+ 5*a^10*x^8 + (a^2*x^2 + 1)^(5/2)*a^7*x^5 + 10*a^8*x^6 + 10*a^6*x^4 + 5*a^4*x^2 + 5*(a^8*x^6 + a^6*x^4)*(a^2
*x^2 + 1)^2 + 10*(a^9*x^7 + 2*a^7*x^5 + a^5*x^3)*(a^2*x^2 + 1)^(3/2) + 10*(a^10*x^8 + 3*a^8*x^6 + 3*a^6*x^4 +
a^4*x^2)*(a^2*x^2 + 1) + a^2 + 5*(a^11*x^9 + 4*a^9*x^7 + 6*a^7*x^5 + 4*a^5*x^3 + a^3*x)*sqrt(a^2*x^2 + 1))*log
(a*x + sqrt(a^2*x^2 + 1))^3) + integrate(1/6*(8*a^13*x^13 + 48*a^11*x^11 + 120*a^9*x^9 + 8*(a^2*x^2 + 1)^3*a^7
*x^7 + 160*a^7*x^7 + 120*a^5*x^5 + 48*a^3*x^3 + (48*a^8*x^8 + 48*a^6*x^6 + 4*a^4*x^4 + 12*a^2*x^2 + 15)*(a^2*x
^2 + 1)^(5/2) + 8*(15*a^9*x^9 + 30*a^7*x^7 + 17*a^5*x^5 + 5*a^3*x^3 + 3*a*x)*(a^2*x^2 + 1)^2 + 2*(80*a^10*x^10
+ 240*a^8*x^8 + 252*a^6*x^6 + 104*a^4*x^4 + 3*a^2*x^2 - 9)*(a^2*x^2 + 1)^(3/2) + 8*(15*a^11*x^11 + 60*a^9*x^9
+ 92*a^7*x^7 + 63*a^5*x^5 + 15*a^3*x^3 - a*x)*(a^2*x^2 + 1) + 8*a*x + (48*a^12*x^12 + 240*a^10*x^10 + 484*a^8
*x^8 + 484*a^6*x^6 + 243*a^4*x^4 + 58*a^2*x^2 + 7)*sqrt(a^2*x^2 + 1))/((a^13*x^12 + 6*a^11*x^10 + 15*a^9*x^8 +
(a^2*x^2 + 1)^3*a^7*x^6 + 20*a^7*x^6 + 15*a^5*x^4 + 6*a^3*x^2 + 6*(a^8*x^7 + a^6*x^5)*(a^2*x^2 + 1)^(5/2) + 1
5*(a^9*x^8 + 2*a^7*x^6 + a^5*x^4)*(a^2*x^2 + 1)^2 + 20*(a^10*x^9 + 3*a^8*x^7 + 3*a^6*x^5 + a^4*x^3)*(a^2*x^2 +
1)^(3/2) + 15*(a^11*x^10 + 4*a^9*x^8 + 6*a^7*x^6 + 4*a^5*x^4 + a^3*x^2)*(a^2*x^2 + 1) + 6*(a^12*x^11 + 5*a^10
*x^9 + 10*a^8*x^7 + 10*a^6*x^5 + 5*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]

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

[In]

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

[Out]

integrate(x/arcsinh(a*x)^4, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x/asinh(a*x)^4, x)