\(\int \frac {x^2 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx\) [343]

Optimal result

Integrand size = 23, antiderivative size = 105 \[ \int \frac {x^2 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=-\frac {3 x^2}{8 a}+\frac {3 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{4 a^2}-\frac {3 \text {arcsinh}(a x)^2}{8 a^3}-\frac {3 x^2 \text {arcsinh}(a x)^2}{4 a}+\frac {x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{2 a^2}-\frac {\text {arcsinh}(a x)^4}{8 a^3} \] Output:

-3/8*x^2/a+3/4*x*(a^2*x^2+1)^(1/2)*arcsinh(a*x)/a^2-3/8*arcsinh(a*x)^2/a^3 
-3/4*x^2*arcsinh(a*x)^2/a+1/2*x*(a^2*x^2+1)^(1/2)*arcsinh(a*x)^3/a^2-1/8*a 
rcsinh(a*x)^4/a^3
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.79 \[ \int \frac {x^2 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=-\frac {3 a^2 x^2-6 a x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)+\left (3+6 a^2 x^2\right ) \text {arcsinh}(a x)^2-4 a x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3+\text {arcsinh}(a x)^4}{8 a^3} \] Input:

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

Output:

-1/8*(3*a^2*x^2 - 6*a*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + (3 + 6*a^2*x^2)*A 
rcSinh[a*x]^2 - 4*a*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3 + ArcSinh[a*x]^4)/a 
^3
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6227, 6191, 6198, 6227, 15, 6198}

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.

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

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 6198

\(\displaystyle -\frac {3 \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx\right )}{2 a}-\frac {\text {arcsinh}(a x)^4}{8 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 a^2}\)

\(\Big \downarrow \) 6227

\(\displaystyle -\frac {3 \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \left (-\frac {\int \frac {\text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2}-\frac {\int xdx}{2 a}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}\right )\right )}{2 a}-\frac {\text {arcsinh}(a x)^4}{8 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 a^2}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {3 \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \left (-\frac {\int \frac {\text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )\right )}{2 a}-\frac {\text {arcsinh}(a x)^4}{8 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 a^2}\)

\(\Big \downarrow \) 6198

\(\displaystyle -\frac {\text {arcsinh}(a x)^4}{8 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 a^2}-\frac {3 \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \left (-\frac {\text {arcsinh}(a x)^2}{4 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )\right )}{2 a}\)

Input:

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

Output:

(x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(2*a^2) - ArcSinh[a*x]^4/(8*a^3) - (3 
*((x^2*ArcSinh[a*x]^2)/2 - a*(-1/4*x^2/a + (x*Sqrt[1 + a^2*x^2]*ArcSinh[a* 
x])/(2*a^2) - ArcSinh[a*x]^2/(4*a^3))))/(2*a)
 

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 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 6198

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( 
a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c 
^2*d] && NeQ[n, -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 = 1.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.80

method	result	size
default	\(-\frac {-4 \operatorname {arcsinh}\left (x a \right )^{3} \sqrt {a^{2} x^{2}+1}\, x a +6 a^{2} x^{2} \operatorname {arcsinh}\left (x a \right )^{2}+\operatorname {arcsinh}\left (x a \right )^{4}-6 \,\operatorname {arcsinh}\left (x a \right ) \sqrt {a^{2} x^{2}+1}\, x a +3 a^{2} x^{2}+3 \operatorname {arcsinh}\left (x a \right )^{2}+3}{8 a^{3}}\)	\(84\)

Input:

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

Output:

-1/8*(-4*arcsinh(x*a)^3*(a^2*x^2+1)^(1/2)*x*a+6*a^2*x^2*arcsinh(x*a)^2+arc 
sinh(x*a)^4-6*arcsinh(x*a)*(a^2*x^2+1)^(1/2)*x*a+3*a^2*x^2+3*arcsinh(x*a)^ 
2+3)/a^3
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.22 \[ \int \frac {x^2 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\frac {4 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - 3 \, a^{2} x^{2} - \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4} + 6 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 3 \, {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{8 \, a^{3}} \] Input:

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

Output:

1/8*(4*sqrt(a^2*x^2 + 1)*a*x*log(a*x + sqrt(a^2*x^2 + 1))^3 - 3*a^2*x^2 - 
log(a*x + sqrt(a^2*x^2 + 1))^4 + 6*sqrt(a^2*x^2 + 1)*a*x*log(a*x + sqrt(a^ 
2*x^2 + 1)) - 3*(2*a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1))^2)/a^3
 

Sympy [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.95 \[ \int \frac {x^2 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} - \frac {3 x^{2} \operatorname {asinh}^{2}{\left (a x \right )}}{4 a} - \frac {3 x^{2}}{8 a} + \frac {x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{2 a^{2}} + \frac {3 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{4 a^{2}} - \frac {\operatorname {asinh}^{4}{\left (a x \right )}}{8 a^{3}} - \frac {3 \operatorname {asinh}^{2}{\left (a x \right )}}{8 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-3*x**2*asinh(a*x)**2/(4*a) - 3*x**2/(8*a) + x*sqrt(a**2*x**2 + 
 1)*asinh(a*x)**3/(2*a**2) + 3*x*sqrt(a**2*x**2 + 1)*asinh(a*x)/(4*a**2) - 
 asinh(a*x)**4/(8*a**3) - 3*asinh(a*x)**2/(8*a**3), Ne(a, 0)), (0, True))
 

Maxima [F]

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

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

Output:

integrate(x^2*arcsinh(a*x)^3/sqrt(a^2*x^2 + 1), x)
 

Giac [F]

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

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

Output:

integrate(x^2*arcsinh(a*x)^3/sqrt(a^2*x^2 + 1), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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