\(\int \frac {\text {arcsinh}(a x)}{x^2 \sqrt {1+a^2 x^2}} \, dx\) [118]

Optimal result

Integrand size = 21, antiderivative size = 27 \[ \int \frac {\text {arcsinh}(a x)}{x^2 \sqrt {1+a^2 x^2}} \, dx=-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{x}+a \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5800, 29} \[ \int \frac {\text {arcsinh}(a x)}{x^2 \sqrt {1+a^2 x^2}} \, dx=a \log (x)-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{x} \]

[In]

Int[ArcSinh[a*x]/(x^2*Sqrt[1 + a^2*x^2]),x]

[Out]

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

Rule 29

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

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\text {arcsinh}(a x)}{x^2 \sqrt {1+a^2 x^2}} \, dx=-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{x}+a \log (a x) \]

[In]

Integrate[ArcSinh[a*x]/(x^2*Sqrt[1 + a^2*x^2]),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(25)=50\).

Time = 0.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.07

method	result	size
default	\(-2 a \,\operatorname {arcsinh}\left (a x \right )+\frac {\left (a x -\sqrt {a^{2} x^{2}+1}\right ) \operatorname {arcsinh}\left (a x \right )}{x}+a \ln \left (\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}-1\right )\)	\(56\)

[In]

int(arcsinh(a*x)/x^2/(a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {\text {arcsinh}(a x)}{x^2 \sqrt {1+a^2 x^2}} \, dx=\frac {a x \log \left (x\right ) - \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{x} \]

[In]

integrate(arcsinh(a*x)/x^2/(a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\text {arcsinh}(a x)}{x^2 \sqrt {1+a^2 x^2}} \, dx=\int \frac {\operatorname {asinh}{\left (a x \right )}}{x^{2} \sqrt {a^{2} x^{2} + 1}}\, dx \]

[In]

integrate(asinh(a*x)/x**2/(a**2*x**2+1)**(1/2),x)

[Out]

Integral(asinh(a*x)/(x**2*sqrt(a**2*x**2 + 1)), x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {\text {arcsinh}(a x)}{x^2 \sqrt {1+a^2 x^2}} \, dx=a \log \left (x\right ) - \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )}{x} \]

[In]

integrate(arcsinh(a*x)/x^2/(a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (25) = 50\).

Time = 0.30 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.63 \[ \int \frac {\text {arcsinh}(a x)}{x^2 \sqrt {1+a^2 x^2}} \, dx=-a \log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right ) + a \log \left ({\left | x \right |}\right ) + \frac {2 \, {\left | a \right |} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1} \]

[In]

integrate(arcsinh(a*x)/x^2/(a^2*x^2+1)^(1/2),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) - sqr
t(a^2*x^2 + 1))^2 - 1)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}(a x)}{x^2 \sqrt {1+a^2 x^2}} \, dx=\int \frac {\mathrm {asinh}\left (a\,x\right )}{x^2\,\sqrt {a^2\,x^2+1}} \,d x \]

[In]

int(asinh(a*x)/(x^2*(a^2*x^2 + 1)^(1/2)),x)

[Out]

int(asinh(a*x)/(x^2*(a^2*x^2 + 1)^(1/2)), x)