\(\int \frac {\text {arcsinh}(a x)^3}{x^3} \, dx\) [29]

Optimal result

Integrand size = 10, antiderivative size = 93 \[ \int \frac {\text {arcsinh}(a x)^3}{x^3} \, dx=-\frac {3}{2} a^2 \text {arcsinh}(a x)^2-\frac {3 a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 x}-\frac {\text {arcsinh}(a x)^3}{2 x^2}+3 a^2 \text {arcsinh}(a x) \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+\frac {3}{2} a^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right ) \]

[Out]

-3/2*a^2*arcsinh(a*x)^2-1/2*arcsinh(a*x)^3/x^2+3*a^2*arcsinh(a*x)*ln(1-(a*x+(a^2*x^2+1)^(1/2))^2)+3/2*a^2*poly
log(2,(a*x+(a^2*x^2+1)^(1/2))^2)-3/2*a*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)/x

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5776, 5800, 5775, 3797, 2221, 2317, 2438} \[ \int \frac {\text {arcsinh}(a x)^3}{x^3} \, dx=\frac {3}{2} a^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )-\frac {3 a \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{2 x}-\frac {3}{2} a^2 \text {arcsinh}(a x)^2+3 a^2 \text {arcsinh}(a x) \log \left (1-e^{2 \text {arcsinh}(a x)}\right )-\frac {\text {arcsinh}(a x)^3}{2 x^2} \]

[In]

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

[Out]

(-3*a^2*ArcSinh[a*x]^2)/2 - (3*a*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(2*x) - ArcSinh[a*x]^3/(2*x^2) + 3*a^2*ArcS
inh[a*x]*Log[1 - E^(2*ArcSinh[a*x])] + (3*a^2*PolyLog[2, E^(2*ArcSinh[a*x])])/2

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5775

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

Rule 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5800

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] - Dist[b*c*(n/(f*(m + 1)))*Simp[(
d + e*x^2)^p/(1 + c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]
/; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.86 \[ \int \frac {\text {arcsinh}(a x)^3}{x^3} \, dx=-\frac {\text {arcsinh}(a x)^3-3 a x \left (\text {arcsinh}(a x) \left (\left (a x-\sqrt {1+a^2 x^2}\right ) \text {arcsinh}(a x)+2 a x \log \left (1-e^{-2 \text {arcsinh}(a x)}\right )\right )-a x \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(a x)}\right )\right )}{2 x^2} \]

[In]

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

[Out]

-1/2*(ArcSinh[a*x]^3 - 3*a*x*(ArcSinh[a*x]*((a*x - Sqrt[1 + a^2*x^2])*ArcSinh[a*x] + 2*a*x*Log[1 - E^(-2*ArcSi
nh[a*x])]) - a*x*PolyLog[2, E^(-2*ArcSinh[a*x])]))/x^2

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.57

method	result	size
derivativedivides	\(a^{2} \left (-\frac {\operatorname {arcsinh}\left (a x \right )^{2} \left (3 a x \sqrt {a^{2} x^{2}+1}-3 a^{2} x^{2}+\operatorname {arcsinh}\left (a x \right )\right )}{2 a^{2} x^{2}}-3 \operatorname {arcsinh}\left (a x \right )^{2}+3 \,\operatorname {arcsinh}\left (a x \right ) \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )+3 \operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )+3 \,\operatorname {arcsinh}\left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+3 \operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\)	\(146\)
default	\(a^{2} \left (-\frac {\operatorname {arcsinh}\left (a x \right )^{2} \left (3 a x \sqrt {a^{2} x^{2}+1}-3 a^{2} x^{2}+\operatorname {arcsinh}\left (a x \right )\right )}{2 a^{2} x^{2}}-3 \operatorname {arcsinh}\left (a x \right )^{2}+3 \,\operatorname {arcsinh}\left (a x \right ) \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )+3 \operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )+3 \,\operatorname {arcsinh}\left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+3 \operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\)	\(146\)

[In]

int(arcsinh(a*x)^3/x^3,x,method=_RETURNVERBOSE)

[Out]

a^2*(-1/2*arcsinh(a*x)^2*(3*a*x*(a^2*x^2+1)^(1/2)-3*a^2*x^2+arcsinh(a*x))/a^2/x^2-3*arcsinh(a*x)^2+3*arcsinh(a
*x)*ln(1+a*x+(a^2*x^2+1)^(1/2))+3*polylog(2,-a*x-(a^2*x^2+1)^(1/2))+3*arcsinh(a*x)*ln(1-a*x-(a^2*x^2+1)^(1/2))
+3*polylog(2,a*x+(a^2*x^2+1)^(1/2)))

Fricas [F]

\[ \int \frac {\text {arcsinh}(a x)^3}{x^3} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{x^{3}} \,d x } \]

[In]

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

[Out]

integral(arcsinh(a*x)^3/x^3, x)

Sympy [F]

\[ \int \frac {\text {arcsinh}(a x)^3}{x^3} \, dx=\int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{x^{3}}\, dx \]

[In]

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

[Out]

Integral(asinh(a*x)**3/x**3, x)

Maxima [F]

\[ \int \frac {\text {arcsinh}(a x)^3}{x^3} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{x^{3}} \,d x } \]

[In]

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

[Out]

-1/2*log(a*x + sqrt(a^2*x^2 + 1))^3/x^2 + integrate(3/2*(a^3*x^2 + sqrt(a^2*x^2 + 1)*a^2*x + a)*log(a*x + sqrt
(a^2*x^2 + 1))^2/(a^3*x^5 + a*x^3 + (a^2*x^4 + x^2)*sqrt(a^2*x^2 + 1)), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\text {arcsinh}(a x)^3}{x^3} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}(a x)^3}{x^3} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{x^3} \,d x \]

[In]

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

[Out]

int(asinh(a*x)^3/x^3, x)