\(\int \frac {x^3 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx\) [283]

Optimal result

Integrand size = 23, antiderivative size = 122 \[ \int \frac {x^3 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=-\frac {14 \sqrt {1+a^2 x^2}}{9 a^4}+\frac {2 \left (1+a^2 x^2\right )^{3/2}}{27 a^4}+\frac {4 x \text {arcsinh}(a x)}{3 a^3}-\frac {2 x^3 \text {arcsinh}(a x)}{9 a}-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a^2} \]

[Out]

2/27*(a^2*x^2+1)^(3/2)/a^4+4/3*x*arcsinh(a*x)/a^3-2/9*x^3*arcsinh(a*x)/a-14/9*(a^2*x^2+1)^(1/2)/a^4-2/3*arcsin
h(a*x)^2*(a^2*x^2+1)^(1/2)/a^4+1/3*x^2*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)/a^2

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5812, 5798, 5772, 267, 5776, 272, 45} \[ \int \frac {x^3 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {4 x \text {arcsinh}(a x)}{3 a^3}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{3 a^4}+\frac {2 \left (a^2 x^2+1\right )^{3/2}}{27 a^4}-\frac {14 \sqrt {a^2 x^2+1}}{9 a^4}-\frac {2 x^3 \text {arcsinh}(a x)}{9 a} \]

[In]

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

[Out]

(-14*Sqrt[1 + a^2*x^2])/(9*a^4) + (2*(1 + a^2*x^2)^(3/2))/(27*a^4) + (4*x*ArcSinh[a*x])/(3*a^3) - (2*x^3*ArcSi
nh[a*x])/(9*a) - (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(3*a^4) + (x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(3*a^2)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 5772

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 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 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)^2}{3 a^2}-\frac {2 \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{3 a^2}-\frac {2 \int x^2 \text {arcsinh}(a x) \, dx}{3 a} \\ & = -\frac {2 x^3 \text {arcsinh}(a x)}{9 a}-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a^2}+\frac {2}{9} \int \frac {x^3}{\sqrt {1+a^2 x^2}} \, dx+\frac {4 \int \text {arcsinh}(a x) \, dx}{3 a^3} \\ & = \frac {4 x \text {arcsinh}(a x)}{3 a^3}-\frac {2 x^3 \text {arcsinh}(a x)}{9 a}-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a^2}+\frac {1}{9} \text {Subst}\left (\int \frac {x}{\sqrt {1+a^2 x}} \, dx,x,x^2\right )-\frac {4 \int \frac {x}{\sqrt {1+a^2 x^2}} \, dx}{3 a^2} \\ & = -\frac {4 \sqrt {1+a^2 x^2}}{3 a^4}+\frac {4 x \text {arcsinh}(a x)}{3 a^3}-\frac {2 x^3 \text {arcsinh}(a x)}{9 a}-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a^2}+\frac {1}{9} \text {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {1+a^2 x}}+\frac {\sqrt {1+a^2 x}}{a^2}\right ) \, dx,x,x^2\right ) \\ & = -\frac {14 \sqrt {1+a^2 x^2}}{9 a^4}+\frac {2 \left (1+a^2 x^2\right )^{3/2}}{27 a^4}+\frac {4 x \text {arcsinh}(a x)}{3 a^3}-\frac {2 x^3 \text {arcsinh}(a x)}{9 a}-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.65 \[ \int \frac {x^3 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {2 \left (-20+a^2 x^2\right ) \sqrt {1+a^2 x^2}-6 a x \left (-6+a^2 x^2\right ) \text {arcsinh}(a x)+9 \left (-2+a^2 x^2\right ) \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{27 a^4} \]

[In]

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

[Out]

(2*(-20 + a^2*x^2)*Sqrt[1 + a^2*x^2] - 6*a*x*(-6 + a^2*x^2)*ArcSinh[a*x] + 9*(-2 + a^2*x^2)*Sqrt[1 + a^2*x^2]*
ArcSinh[a*x]^2)/(27*a^4)

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.93

method	result	size
default	\(\frac {9 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{2}-9 \operatorname {arcsinh}\left (a x \right )^{2} a^{2} x^{2}-6 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}+2 a^{4} x^{4}-38 a^{2} x^{2}-18 \operatorname {arcsinh}\left (a x \right )^{2}+36 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x -40}{27 a^{4} \sqrt {a^{2} x^{2}+1}}\)	\(113\)

[In]

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

[Out]

1/27/a^4/(a^2*x^2+1)^(1/2)*(9*a^4*x^4*arcsinh(a*x)^2-9*arcsinh(a*x)^2*a^2*x^2-6*a^3*x^3*arcsinh(a*x)*(a^2*x^2+
1)^(1/2)+2*a^4*x^4-38*a^2*x^2-18*arcsinh(a*x)^2+36*arcsinh(a*x)*(a^2*x^2+1)^(1/2)*a*x-40)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.80 \[ \int \frac {x^3 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {9 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 2\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 6 \, {\left (a^{3} x^{3} - 6 \, a x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) + 2 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 20\right )}}{27 \, a^{4}} \]

[In]

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

[Out]

1/27*(9*sqrt(a^2*x^2 + 1)*(a^2*x^2 - 2)*log(a*x + sqrt(a^2*x^2 + 1))^2 - 6*(a^3*x^3 - 6*a*x)*log(a*x + sqrt(a^
2*x^2 + 1)) + 2*sqrt(a^2*x^2 + 1)*(a^2*x^2 - 20))/a^4

Sympy [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.99 \[ \int \frac {x^3 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} - \frac {2 x^{3} \operatorname {asinh}{\left (a x \right )}}{9 a} + \frac {x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{3 a^{2}} + \frac {2 x^{2} \sqrt {a^{2} x^{2} + 1}}{27 a^{2}} + \frac {4 x \operatorname {asinh}{\left (a x \right )}}{3 a^{3}} - \frac {2 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{3 a^{4}} - \frac {40 \sqrt {a^{2} x^{2} + 1}}{27 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-2*x**3*asinh(a*x)/(9*a) + x**2*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/(3*a**2) + 2*x**2*sqrt(a**2*x**2
+ 1)/(27*a**2) + 4*x*asinh(a*x)/(3*a**3) - 2*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/(3*a**4) - 40*sqrt(a**2*x**2 +
1)/(27*a**4), Ne(a, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.83 \[ \int \frac {x^3 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {1}{3} \, {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arsinh}\left (a x\right )^{2} + \frac {2 \, {\left (\sqrt {a^{2} x^{2} + 1} x^{2} - \frac {20 \, \sqrt {a^{2} x^{2} + 1}}{a^{2}}\right )}}{27 \, a^{2}} - \frac {2 \, {\left (a^{2} x^{3} - 6 \, x\right )} \operatorname {arsinh}\left (a x\right )}{9 \, a^{3}} \]

[In]

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

[Out]

1/3*(sqrt(a^2*x^2 + 1)*x^2/a^2 - 2*sqrt(a^2*x^2 + 1)/a^4)*arcsinh(a*x)^2 + 2/27*(sqrt(a^2*x^2 + 1)*x^2 - 20*sq
rt(a^2*x^2 + 1)/a^2)/a^2 - 2/9*(a^2*x^3 - 6*x)*arcsinh(a*x)/a^3

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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