\(\int \frac {\text {arcsinh}(a x)^2}{x^3} \, dx\) [19]

Optimal result

Integrand size = 10, antiderivative size = 43 \[ \int \frac {\text {arcsinh}(a x)^2}{x^3} \, dx=-\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{x}-\frac {\text {arcsinh}(a x)^2}{2 x^2}+a^2 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5776, 5800, 29} \[ \int \frac {\text {arcsinh}(a x)^2}{x^3} \, dx=-\frac {a \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{x}+a^2 \log (x)-\frac {\text {arcsinh}(a x)^2}{2 x^2} \]

[In]

Int[ArcSinh[a*x]^2/x^3,x]

[Out]

-((a*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/x) - ArcSinh[a*x]^2/(2*x^2) + a^2*Log[x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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)^2}{2 x^2}+a \int \frac {\text {arcsinh}(a x)}{x^2 \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{x}-\frac {\text {arcsinh}(a x)^2}{2 x^2}+a^2 \int \frac {1}{x} \, dx \\ & = -\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{x}-\frac {\text {arcsinh}(a x)^2}{2 x^2}+a^2 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)^2}{x^3} \, dx=-\frac {a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{x}-\frac {\text {arcsinh}(a x)^2}{2 x^2}+a^2 \log (x) \]

[In]

Integrate[ArcSinh[a*x]^2/x^3,x]

[Out]

-((a*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/x) - ArcSinh[a*x]^2/(2*x^2) + a^2*Log[x]

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.67

method	result	size
derivativedivides	\(a^{2} \left (-2 \,\operatorname {arcsinh}\left (a x \right )-\frac {\operatorname {arcsinh}\left (a x \right ) \left (-2 a^{2} x^{2}+2 a x \sqrt {a^{2} x^{2}+1}+\operatorname {arcsinh}\left (a x \right )\right )}{2 a^{2} x^{2}}+\ln \left (\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}-1\right )\right )\)	\(72\)
default	\(a^{2} \left (-2 \,\operatorname {arcsinh}\left (a x \right )-\frac {\operatorname {arcsinh}\left (a x \right ) \left (-2 a^{2} x^{2}+2 a x \sqrt {a^{2} x^{2}+1}+\operatorname {arcsinh}\left (a x \right )\right )}{2 a^{2} x^{2}}+\ln \left (\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}-1\right )\right )\)	\(72\)

[In]

int(arcsinh(a*x)^2/x^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.56 \[ \int \frac {\text {arcsinh}(a x)^2}{x^3} \, dx=\frac {2 \, a^{2} x^{2} \log \left (x\right ) - 2 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{2 \, x^{2}} \]

[In]

integrate(arcsinh(a*x)^2/x^3,x, algorithm="fricas")

[Out]

1/2*(2*a^2*x^2*log(x) - 2*sqrt(a^2*x^2 + 1)*a*x*log(a*x + sqrt(a^2*x^2 + 1)) - log(a*x + sqrt(a^2*x^2 + 1))^2)
/x^2

Sympy [F]

\[ \int \frac {\text {arcsinh}(a x)^2}{x^3} \, dx=\int \frac {\operatorname {asinh}^{2}{\left (a x \right )}}{x^{3}}\, dx \]

[In]

integrate(asinh(a*x)**2/x**3,x)

[Out]

Integral(asinh(a*x)**2/x**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91 \[ \int \frac {\text {arcsinh}(a x)^2}{x^3} \, dx=a^{2} \log \left (x\right ) - \frac {\sqrt {a^{2} x^{2} + 1} a \operatorname {arsinh}\left (a x\right )}{x} - \frac {\operatorname {arsinh}\left (a x\right )^{2}}{2 \, x^{2}} \]

[In]

integrate(arcsinh(a*x)^2/x^3,x, algorithm="maxima")

[Out]

a^2*log(x) - sqrt(a^2*x^2 + 1)*a*arcsinh(a*x)/x - 1/2*arcsinh(a*x)^2/x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (39) = 78\).

Time = 0.30 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.28 \[ \int \frac {\text {arcsinh}(a x)^2}{x^3} \, dx=-{\left (a \log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right ) - a \log \left ({\left | x \right |}\right ) - \frac {2 \, {\left | a \right |} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1}\right )} a - \frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{2 \, x^{2}} \]

[In]

integrate(arcsinh(a*x)^2/x^3,x, algorithm="giac")

[Out]

-(a*log(-x*abs(a) + sqrt(a^2*x^2 + 1)) - a*log(abs(x)) - 2*abs(a)*log(a*x + sqrt(a^2*x^2 + 1))/((x*abs(a) - sq
rt(a^2*x^2 + 1))^2 - 1))*a - 1/2*log(a*x + sqrt(a^2*x^2 + 1))^2/x^2

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}(a x)^2}{x^3} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^2}{x^3} \,d x \]

[In]

int(asinh(a*x)^2/x^3,x)

[Out]

int(asinh(a*x)^2/x^3, x)