\(\int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx\) [113]

Optimal result

Integrand size = 21, antiderivative size = 70 \[ \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\frac {2 x}{3 a^3}-\frac {x^3}{9 a}-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{3 a^2} \]

[Out]

2/3*x/a^3-1/9*x^3/a-2/3*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a^4+1/3*x^2*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a^2

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5812, 5798, 8, 30} \[ \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\frac {2 x}{3 a^3}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{3 a^2}-\frac {2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{3 a^4}-\frac {x^3}{9 a} \]

[In]

Int[(x^3*ArcSinh[a*x])/Sqrt[1 + a^2*x^2],x]

[Out]

(2*x)/(3*a^3) - x^3/(9*a) - (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(3*a^4) + (x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/
(3*a^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

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

Rule 5798

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

Rule 5812

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

Rubi steps \begin{align*} \text {integral}& = \frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{3 a^2}-\frac {2 \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{3 a^2}-\frac {\int x^2 \, dx}{3 a} \\ & = -\frac {x^3}{9 a}-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{3 a^2}+\frac {2 \int 1 \, dx}{3 a^3} \\ & = \frac {2 x}{3 a^3}-\frac {x^3}{9 a}-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{3 a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.69 \[ \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\frac {6 a x-a^3 x^3+3 \left (-2+a^2 x^2\right ) \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{9 a^4} \]

[In]

Integrate[(x^3*ArcSinh[a*x])/Sqrt[1 + a^2*x^2],x]

[Out]

(6*a*x - a^3*x^3 + 3*(-2 + a^2*x^2)*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(9*a^4)

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.17

method	result	size
default	\(\frac {3 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )-3 a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )-a^{3} x^{3} \sqrt {a^{2} x^{2}+1}-6 \,\operatorname {arcsinh}\left (a x \right )+6 a x \sqrt {a^{2} x^{2}+1}}{9 a^{4} \sqrt {a^{2} x^{2}+1}}\)	\(82\)

[In]

int(x^3*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/9/a^4/(a^2*x^2+1)^(1/2)*(3*a^4*x^4*arcsinh(a*x)-3*a^2*x^2*arcsinh(a*x)-a^3*x^3*(a^2*x^2+1)^(1/2)-6*arcsinh(a
*x)+6*a*x*(a^2*x^2+1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.79 \[ \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {a^{3} x^{3} - 3 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 2\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 6 \, a x}{9 \, a^{4}} \]

[In]

integrate(x^3*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

-1/9*(a^3*x^3 - 3*sqrt(a^2*x^2 + 1)*(a^2*x^2 - 2)*log(a*x + sqrt(a^2*x^2 + 1)) - 6*a*x)/a^4

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.93 \[ \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} - \frac {x^{3}}{9 a} + \frac {x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{3 a^{2}} + \frac {2 x}{3 a^{3}} - \frac {2 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{3 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(x**3*asinh(a*x)/(a**2*x**2+1)**(1/2),x)

[Out]

Piecewise((-x**3/(9*a) + x**2*sqrt(a**2*x**2 + 1)*asinh(a*x)/(3*a**2) + 2*x/(3*a**3) - 2*sqrt(a**2*x**2 + 1)*a
sinh(a*x)/(3*a**4), Ne(a, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.84 \[ \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {1}{9} \, a {\left (\frac {x^{3}}{a^{2}} - \frac {6 \, x}{a^{4}}\right )} + \frac {1}{3} \, {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arsinh}\left (a x\right ) \]

[In]

integrate(x^3*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

-1/9*a*(x^3/a^2 - 6*x/a^4) + 1/3*(sqrt(a^2*x^2 + 1)*x^2/a^2 - 2*sqrt(a^2*x^2 + 1)/a^4)*arcsinh(a*x)

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^3*arcsinh(a*x)/(a^2*x^2+1)^(1/2),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. \[ \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^3\,\mathrm {asinh}\left (a\,x\right )}{\sqrt {a^2\,x^2+1}} \,d x \]

[In]

int((x^3*asinh(a*x))/(a^2*x^2 + 1)^(1/2),x)

[Out]

int((x^3*asinh(a*x))/(a^2*x^2 + 1)^(1/2), x)