\(\int \frac {\text {arcsinh}(a x)^2}{x^4} \, dx\) [20]

Optimal result

Integrand size = 10, antiderivative size = 99 \[ \int \frac {\text {arcsinh}(a x)^2}{x^4} \, dx=-\frac {a^2}{3 x}-\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{3 x^2}-\frac {\text {arcsinh}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+\frac {1}{3} a^3 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-\frac {1}{3} a^3 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5776, 5809, 5816, 4267, 2317, 2438, 30} \[ \int \frac {\text {arcsinh}(a x)^2}{x^4} \, dx=\frac {2}{3} a^3 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+\frac {1}{3} a^3 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-\frac {1}{3} a^3 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-\frac {a \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{3 x^2}-\frac {a^2}{3 x}-\frac {\text {arcsinh}(a x)^2}{3 x^3} \]

[In]

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

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 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 5809

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[c^2*((m + 2*p + 3)/(f^2*
(m + 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], 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, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]

Rule 5816

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

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

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.26 \[ \int \frac {\text {arcsinh}(a x)^2}{x^4} \, dx=-\frac {a^2 x^2+a x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)+\text {arcsinh}(a x)^2+a^3 x^3 \text {arcsinh}(a x) \log \left (1-e^{-\text {arcsinh}(a x)}\right )-a^3 x^3 \text {arcsinh}(a x) \log \left (1+e^{-\text {arcsinh}(a x)}\right )+a^3 x^3 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(a x)}\right )-a^3 x^3 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(a x)}\right )}{3 x^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.37

method	result	size
derivativedivides	\(a^{3} \left (-\frac {\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +\operatorname {arcsinh}\left (a x \right )^{2}+a^{2} x^{2}}{3 a^{3} x^{3}}+\frac {\operatorname {arcsinh}\left (a x \right ) \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )}{3}+\frac {\operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )}{3}-\frac {\operatorname {arcsinh}\left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )}{3}-\frac {\operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )}{3}\right )\)	\(136\)
default	\(a^{3} \left (-\frac {\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +\operatorname {arcsinh}\left (a x \right )^{2}+a^{2} x^{2}}{3 a^{3} x^{3}}+\frac {\operatorname {arcsinh}\left (a x \right ) \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )}{3}+\frac {\operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )}{3}-\frac {\operatorname {arcsinh}\left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )}{3}-\frac {\operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )}{3}\right )\)	\(136\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(arcsinh(a*x)^2/x^4, x)

Sympy [F]

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

[In]

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

[Out]

Integral(asinh(a*x)**2/x**4, x)

Maxima [F]

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

[In]

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

[Out]

-1/3*log(a*x + sqrt(a^2*x^2 + 1))^2/x^3 + integrate(2/3*(a^3*x^2 + sqrt(a^2*x^2 + 1)*a^2*x + a)*log(a*x + sqrt
(a^2*x^2 + 1))/(a^3*x^6 + a*x^4 + (a^2*x^5 + x^3)*sqrt(a^2*x^2 + 1)), x)

Giac [F]

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

[In]

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

[Out]

integrate(arcsinh(a*x)^2/x^4, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(asinh(a*x)^2/x^4, x)