\(\int \frac {x^3 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx\) [342]

Optimal result

Integrand size = 23, antiderivative size = 153 \[ \int \frac {x^3 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\frac {40 x}{9 a^3}-\frac {2 x^3}{27 a}-\frac {40 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{9 a^4}+\frac {2 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{9 a^2}+\frac {2 x \text {arcsinh}(a x)^2}{a^3}-\frac {x^3 \text {arcsinh}(a x)^2}{3 a}-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{3 a^2} \] Output:

40/9*x/a^3-2/27*x^3/a-40/9*(a^2*x^2+1)^(1/2)*arcsinh(a*x)/a^4+2/9*x^2*(a^2 
*x^2+1)^(1/2)*arcsinh(a*x)/a^2+2*x*arcsinh(a*x)^2/a^3-1/3*x^3*arcsinh(a*x) 
^2/a-2/3*(a^2*x^2+1)^(1/2)*arcsinh(a*x)^3/a^4+1/3*x^2*(a^2*x^2+1)^(1/2)*ar 
csinh(a*x)^3/a^2
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.64 \[ \int \frac {x^3 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\frac {-2 a x \left (-60+a^2 x^2\right )+6 \left (-20+a^2 x^2\right ) \sqrt {1+a^2 x^2} \text {arcsinh}(a x)-9 a x \left (-6+a^2 x^2\right ) \text {arcsinh}(a x)^2+9 \left (-2+a^2 x^2\right ) \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{27 a^4} \] Input:

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

Output:

(-2*a*x*(-60 + a^2*x^2) + 6*(-20 + a^2*x^2)*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] 
 - 9*a*x*(-6 + a^2*x^2)*ArcSinh[a*x]^2 + 9*(-2 + a^2*x^2)*Sqrt[1 + a^2*x^2 
]*ArcSinh[a*x]^3)/(27*a^4)
 

Rubi [A] (verified)

Time = 1.32 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.33, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6227, 6191, 6213, 6187, 6213, 24, 6227, 15, 6213, 24}

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.

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

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 6187

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 15

\(\displaystyle -\frac {\frac {1}{3} x^3 \text {arcsinh}(a x)^2-\frac {2}{3} a \left (-\frac {2 \int \frac {x \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{3 a^2}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{3 a^2}-\frac {x^3}{9 a}\right )}{a}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{3 a^2}-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{a^2}-\frac {3 \left (x \text {arcsinh}(a x)^2-2 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{a^2}-\frac {x}{a}\right )\right )}{a}\right )}{3 a^2}\)

\(\Big \downarrow \) 6213

\(\displaystyle -\frac {\frac {1}{3} x^3 \text {arcsinh}(a x)^2-\frac {2}{3} a \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{a^2}-\frac {\int 1dx}{a}\right )}{3 a^2}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{3 a^2}-\frac {x^3}{9 a}\right )}{a}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{3 a^2}-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{a^2}-\frac {3 \left (x \text {arcsinh}(a x)^2-2 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{a^2}-\frac {x}{a}\right )\right )}{a}\right )}{3 a^2}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{3 a^2}-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{a^2}-\frac {3 \left (x \text {arcsinh}(a x)^2-2 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{a^2}-\frac {x}{a}\right )\right )}{a}\right )}{3 a^2}-\frac {\frac {1}{3} x^3 \text {arcsinh}(a x)^2-\frac {2}{3} a \left (\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{3 a^2}-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{a^2}-\frac {x}{a}\right )}{3 a^2}-\frac {x^3}{9 a}\right )}{a}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 15

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

rule 24

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

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 = 1.85 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.07

method	result	size
default	\(\frac {9 \operatorname {arcsinh}\left (x a \right )^{3} a^{4} x^{4}-9 \operatorname {arcsinh}\left (x a \right )^{3} a^{2} x^{2}-9 x^{3} a^{3} \operatorname {arcsinh}\left (x a \right )^{2} \sqrt {a^{2} x^{2}+1}+6 x^{4} a^{4} \operatorname {arcsinh}\left (x a \right )-114 \,\operatorname {arcsinh}\left (x a \right ) x^{2} a^{2}-2 x^{3} a^{3} \sqrt {a^{2} x^{2}+1}-18 \operatorname {arcsinh}\left (x a \right )^{3}+54 \operatorname {arcsinh}\left (x a \right )^{2} \sqrt {a^{2} x^{2}+1}\, x a -120 \,\operatorname {arcsinh}\left (x a \right )+120 x a \sqrt {a^{2} x^{2}+1}}{27 a^{4} \sqrt {a^{2} x^{2}+1}}\)	\(164\)
orering	\(\frac {5 \left (13 a^{6} x^{6}-144 a^{4} x^{4}-936 a^{2} x^{2}-864\right ) \operatorname {arcsinh}\left (x a \right )^{3}}{81 a^{6} x^{2} \sqrt {a^{2} x^{2}+1}}-\frac {\left (25 a^{6} x^{6}-578 a^{4} x^{4}-2940 a^{2} x^{2}-2520\right ) \left (\frac {3 x^{2} \operatorname {arcsinh}\left (x a \right )^{3}}{\sqrt {a^{2} x^{2}+1}}+\frac {3 x^{3} \operatorname {arcsinh}\left (x a \right )^{2} a}{a^{2} x^{2}+1}-\frac {x^{4} \operatorname {arcsinh}\left (x a \right )^{3} a^{2}}{\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{81 a^{6} x^{4}}+\frac {2 \left (a^{2} x^{2}+1\right ) \left (a^{4} x^{4}-38 a^{2} x^{2}-100\right ) \left (\frac {6 x \operatorname {arcsinh}\left (x a \right )^{3}}{\sqrt {a^{2} x^{2}+1}}+\frac {18 x^{2} \operatorname {arcsinh}\left (x a \right )^{2} a}{a^{2} x^{2}+1}-\frac {7 x^{3} \operatorname {arcsinh}\left (x a \right )^{3} a^{2}}{\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {6 x^{3} \operatorname {arcsinh}\left (x a \right ) a^{2}}{\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {9 x^{4} \operatorname {arcsinh}\left (x a \right )^{2} a^{3}}{\left (a^{2} x^{2}+1\right )^{2}}+\frac {3 x^{5} \operatorname {arcsinh}\left (x a \right )^{3} a^{4}}{\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{27 x^{3} a^{6}}-\frac {\left (a^{2} x^{2}-60\right ) \left (a^{2} x^{2}+1\right )^{2} \left (\frac {6 \operatorname {arcsinh}\left (x a \right )^{3}}{\sqrt {a^{2} x^{2}+1}}+\frac {54 x \operatorname {arcsinh}\left (x a \right )^{2} a}{a^{2} x^{2}+1}-\frac {27 x^{2} \operatorname {arcsinh}\left (x a \right )^{3} a^{2}}{\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {54 x^{2} \operatorname {arcsinh}\left (x a \right ) a^{2}}{\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {93 x^{3} \operatorname {arcsinh}\left (x a \right )^{2} a^{3}}{\left (a^{2} x^{2}+1\right )^{2}}+\frac {36 x^{4} \operatorname {arcsinh}\left (x a \right )^{3} a^{4}}{\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {6 x^{3} a^{3}}{\left (a^{2} x^{2}+1\right )^{2}}-\frac {36 x^{4} \operatorname {arcsinh}\left (x a \right ) a^{4}}{\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {45 x^{5} \operatorname {arcsinh}\left (x a \right )^{2} a^{5}}{\left (a^{2} x^{2}+1\right )^{3}}-\frac {15 x^{6} \operatorname {arcsinh}\left (x a \right )^{3} a^{6}}{\left (a^{2} x^{2}+1\right )^{\frac {7}{2}}}\right )}{81 x^{2} a^{6}}\)	\(593\)

Input:

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

Output:

1/27/a^4/(a^2*x^2+1)^(1/2)*(9*arcsinh(x*a)^3*a^4*x^4-9*arcsinh(x*a)^3*a^2* 
x^2-9*x^3*a^3*arcsinh(x*a)^2*(a^2*x^2+1)^(1/2)+6*x^4*a^4*arcsinh(x*a)-114* 
arcsinh(x*a)*x^2*a^2-2*x^3*a^3*(a^2*x^2+1)^(1/2)-18*arcsinh(x*a)^3+54*arcs 
inh(x*a)^2*(a^2*x^2+1)^(1/2)*x*a-120*arcsinh(x*a)+120*x*a*(a^2*x^2+1)^(1/2 
))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.84 \[ \int \frac {x^3 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=-\frac {2 \, a^{3} x^{3} - 9 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 2\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} + 9 \, {\left (a^{3} x^{3} - 6 \, a x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 6 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 20\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 120 \, a x}{27 \, a^{4}} \] Input:

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

Output:

-1/27*(2*a^3*x^3 - 9*sqrt(a^2*x^2 + 1)*(a^2*x^2 - 2)*log(a*x + sqrt(a^2*x^ 
2 + 1))^3 + 9*(a^3*x^3 - 6*a*x)*log(a*x + sqrt(a^2*x^2 + 1))^2 - 6*sqrt(a^ 
2*x^2 + 1)*(a^2*x^2 - 20)*log(a*x + sqrt(a^2*x^2 + 1)) - 120*a*x)/a^4
 

Sympy [A] (verification not implemented)

Time = 0.63 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.97 \[ \int \frac {x^3 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} - \frac {x^{3} \operatorname {asinh}^{2}{\left (a x \right )}}{3 a} - \frac {2 x^{3}}{27 a} + \frac {x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{3 a^{2}} + \frac {2 x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{9 a^{2}} + \frac {2 x \operatorname {asinh}^{2}{\left (a x \right )}}{a^{3}} + \frac {40 x}{9 a^{3}} - \frac {2 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{3 a^{4}} - \frac {40 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{9 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.83 \[ \int \frac {x^3 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\frac {1}{3} \, {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arsinh}\left (a x\right )^{3} + \frac {2}{27} \, a {\left (\frac {3 \, {\left (\sqrt {a^{2} x^{2} + 1} x^{2} - \frac {20 \, \sqrt {a^{2} x^{2} + 1}}{a^{2}}\right )} \operatorname {arsinh}\left (a x\right )}{a^{3}} - \frac {a^{2} x^{3} - 60 \, x}{a^{4}}\right )} - \frac {{\left (a^{2} x^{3} - 6 \, x\right )} \operatorname {arsinh}\left (a x\right )^{2}}{3 \, a^{3}} \] Input:

integrate(x^3*arcsinh(a*x)^3/(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)^3 + 
 2/27*a*(3*(sqrt(a^2*x^2 + 1)*x^2 - 20*sqrt(a^2*x^2 + 1)/a^2)*arcsinh(a*x) 
/a^3 - (a^2*x^3 - 60*x)/a^4) - 1/3*(a^2*x^3 - 6*x)*arcsinh(a*x)^2/a^3
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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