\(\int \frac {x}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx\) [384]

Optimal result

Integrand size = 21, antiderivative size = 9 \[ \int \frac {x}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\frac {\text {Shi}(\text {arcsinh}(a x))}{a^2} \]

[Out]

Shi(arcsinh(a*x))/a^2

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 9, 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 = {5819, 3379} \[ \int \frac {x}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\frac {\text {Shi}(\text {arcsinh}(a x))}{a^2} \]

[In]

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

[Out]

SinhIntegral[ArcSinh[a*x]]/a^2

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 5819

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{a^2} \\ & = \frac {\text {Shi}(\text {arcsinh}(a x))}{a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\frac {\text {Shi}(\text {arcsinh}(a x))}{a^2} \]

[In]

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

[Out]

SinhIntegral[ArcSinh[a*x]]/a^2

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.11

method	result	size
default	\(\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{a^{2}}\)	\(10\)

[In]

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

[Out]

Shi(arcsinh(a*x))/a^2

Fricas [F]

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

[In]

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

[Out]

integral(x/(sqrt(a^2*x^2 + 1)*arcsinh(a*x)), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(x/(sqrt(a^2*x^2 + 1)*arcsinh(a*x)), x)

Giac [F]

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

[In]

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

[Out]

integrate(x/(sqrt(a^2*x^2 + 1)*arcsinh(a*x)), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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