\(\int \frac {x^3 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx\) [289]

Optimal result

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

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

Mathematica [A] (verified)

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

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

Output:

(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)
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.26, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6227, 6191, 243, 53, 2009, 6213, 6187, 241}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6227

\(\displaystyle -\frac {2 \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {2 \int x^2 \text {arcsinh}(a x)dx}{3 a}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{3 a^2}\)

\(\Big \downarrow \) 6191

\(\displaystyle -\frac {2 \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {2 \left (\frac {1}{3} x^3 \text {arcsinh}(a x)-\frac {1}{3} a \int \frac {x^3}{\sqrt {a^2 x^2+1}}dx\right )}{3 a}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{3 a^2}\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {2 \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {2 \left (\frac {1}{3} x^3 \text {arcsinh}(a x)-\frac {1}{6} a \int \frac {x^2}{\sqrt {a^2 x^2+1}}dx^2\right )}{3 a}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{3 a^2}\)

\(\Big \downarrow \) 53

\(\displaystyle -\frac {2 \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {2 \left (\frac {1}{3} x^3 \text {arcsinh}(a x)-\frac {1}{6} a \int \left (\frac {\sqrt {a^2 x^2+1}}{a^2}-\frac {1}{a^2 \sqrt {a^2 x^2+1}}\right )dx^2\right )}{3 a}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{3 a^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {a^2 x^2+1}}dx}{3 a^2}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \left (\frac {1}{3} x^3 \text {arcsinh}(a x)-\frac {1}{6} a \left (\frac {2 \left (a^2 x^2+1\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {a^2 x^2+1}}{a^4}\right )\right )}{3 a}\)

\(\Big \downarrow \) 6213

\(\displaystyle -\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \int \text {arcsinh}(a x)dx}{a}\right )}{3 a^2}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \left (\frac {1}{3} x^3 \text {arcsinh}(a x)-\frac {1}{6} a \left (\frac {2 \left (a^2 x^2+1\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {a^2 x^2+1}}{a^4}\right )\right )}{3 a}\)

\(\Big \downarrow \) 6187

\(\displaystyle -\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \left (x \text {arcsinh}(a x)-a \int \frac {x}{\sqrt {a^2 x^2+1}}dx\right )}{a}\right )}{3 a^2}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \left (\frac {1}{3} x^3 \text {arcsinh}(a x)-\frac {1}{6} a \left (\frac {2 \left (a^2 x^2+1\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {a^2 x^2+1}}{a^4}\right )\right )}{3 a}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \left (x \text {arcsinh}(a x)-\frac {\sqrt {a^2 x^2+1}}{a}\right )}{a}\right )}{3 a^2}-\frac {2 \left (\frac {1}{3} x^3 \text {arcsinh}(a x)-\frac {1}{6} a \left (\frac {2 \left (a^2 x^2+1\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {a^2 x^2+1}}{a^4}\right )\right )}{3 a}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 53

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] && 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 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6187

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A 
rcSinh[c*x])^n, x] - Simp[b*c*n   Int[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 6191

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[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 6213

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] - Simp[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 6227

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] + (-Simp[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] 
 - Simp[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]
 

Maple [A] (verified)

Time = 1.69 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.93

method	result	size
default	\(\frac {9 x^{4} a^{4} \operatorname {arcsinh}\left (x a \right )^{2}-9 a^{2} x^{2} \operatorname {arcsinh}\left (x a \right )^{2}-6 x^{3} a^{3} \operatorname {arcsinh}\left (x a \right ) \sqrt {a^{2} x^{2}+1}+2 a^{4} x^{4}-38 a^{2} x^{2}-18 \operatorname {arcsinh}\left (x a \right )^{2}+36 \,\operatorname {arcsinh}\left (x a \right ) \sqrt {a^{2} x^{2}+1}\, x a -40}{27 a^{4} \sqrt {a^{2} x^{2}+1}}\)	\(113\)
orering	\(\frac {\left (19 a^{6} x^{6}-100 a^{4} x^{4}-380 a^{2} x^{2}-240\right ) \operatorname {arcsinh}\left (x a \right )^{2}}{27 a^{6} x^{2} \sqrt {a^{2} x^{2}+1}}-\frac {2 \left (a^{2} x^{2}+1\right ) \left (a^{4} x^{4}-12 a^{2} x^{2}-20\right ) \left (\frac {3 x^{2} \operatorname {arcsinh}\left (x a \right )^{2}}{\sqrt {a^{2} x^{2}+1}}+\frac {2 x^{3} \operatorname {arcsinh}\left (x a \right ) a}{a^{2} x^{2}+1}-\frac {x^{4} \operatorname {arcsinh}\left (x a \right )^{2} a^{2}}{\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{9 a^{6} x^{4}}+\frac {\left (a^{2} x^{2}-20\right ) \left (a^{2} x^{2}+1\right )^{2} \left (\frac {6 x \operatorname {arcsinh}\left (x a \right )^{2}}{\sqrt {a^{2} x^{2}+1}}+\frac {12 x^{2} \operatorname {arcsinh}\left (x a \right ) a}{a^{2} x^{2}+1}-\frac {7 x^{3} \operatorname {arcsinh}\left (x a \right )^{2} a^{2}}{\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {2 a^{2} x^{3}}{\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {6 x^{4} \operatorname {arcsinh}\left (x a \right ) a^{3}}{\left (a^{2} x^{2}+1\right )^{2}}+\frac {3 x^{5} \operatorname {arcsinh}\left (x a \right )^{2} a^{4}}{\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{27 a^{6} x^{3}}\)	\(318\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.80 \[ \int \frac {x^3 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {9 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 2\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 6 \, {\left (a^{3} x^{3} - 6 \, a x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) + 2 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 20\right )}}{27 \, a^{4}} \] Input:

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

Output:

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 = 0.45 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.99 \[ \int \frac {x^3 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} - \frac {2 x^{3} \operatorname {asinh}{\left (a x \right )}}{9 a} + \frac {x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{3 a^{2}} + \frac {2 x^{2} \sqrt {a^{2} x^{2} + 1}}{27 a^{2}} + \frac {4 x \operatorname {asinh}{\left (a x \right )}}{3 a^{3}} - \frac {2 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{3 a^{4}} - \frac {40 \sqrt {a^{2} x^{2} + 1}}{27 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

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

Output:

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)

Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.83 \[ \int \frac {x^3 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {1}{3} \, {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arsinh}\left (a x\right )^{2} + \frac {2 \, {\left (\sqrt {a^{2} x^{2} + 1} x^{2} - \frac {20 \, \sqrt {a^{2} x^{2} + 1}}{a^{2}}\right )}}{27 \, a^{2}} - \frac {2 \, {\left (a^{2} x^{3} - 6 \, x\right )} \operatorname {arsinh}\left (a x\right )}{9 \, a^{3}} \] Input:

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

Output:

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^3 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {\mathit {asinh} \left (a x \right )^{2} x^{3}}{\sqrt {a^{2} x^{2}+1}}d x \] Input:

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

Output:

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