\(\int \frac {x^2 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx\) [290]

Optimal result

Integrand size = 23, antiderivative size = 87 \[ \int \frac {x^2 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {x \sqrt {1+a^2 x^2}}{4 a^2}-\frac {\text {arcsinh}(a x)}{4 a^3}-\frac {x^2 \text {arcsinh}(a x)}{2 a}+\frac {x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{2 a^2}-\frac {\text {arcsinh}(a x)^3}{6 a^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.83 \[ \int \frac {x^2 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {3 a x \sqrt {1+a^2 x^2}-3 \left (1+2 a^2 x^2\right ) \text {arcsinh}(a x)+6 a x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2-2 \text {arcsinh}(a x)^3}{12 a^3} \] Input:

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

Output:

(3*a*x*Sqrt[1 + a^2*x^2] - 3*(1 + 2*a^2*x^2)*ArcSinh[a*x] + 6*a*x*Sqrt[1 + 
 a^2*x^2]*ArcSinh[a*x]^2 - 2*ArcSinh[a*x]^3)/(12*a^3)
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.10, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6227, 6191, 262, 222, 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.

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

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 6198

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

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 = 1.14 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.79

method	result	size
default	\(-\frac {-6 \operatorname {arcsinh}\left (x a \right )^{2} \sqrt {a^{2} x^{2}+1}\, x a +6 \,\operatorname {arcsinh}\left (x a \right ) x^{2} a^{2}+2 \operatorname {arcsinh}\left (x a \right )^{3}-3 x a \sqrt {a^{2} x^{2}+1}+3 \,\operatorname {arcsinh}\left (x a \right )}{12 a^{3}}\)	\(69\)

Input:

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

Output:

-1/12*(-6*arcsinh(x*a)^2*(a^2*x^2+1)^(1/2)*x*a+6*arcsinh(x*a)*x^2*a^2+2*ar 
csinh(x*a)^3-3*x*a*(a^2*x^2+1)^(1/2)+3*arcsinh(x*a))/a^3
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.17 \[ \int \frac {x^2 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {6 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 2 \, \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} + 3 \, \sqrt {a^{2} x^{2} + 1} a x - 3 \, {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{12 \, a^{3}} \] Input:

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

Output:

1/12*(6*sqrt(a^2*x^2 + 1)*a*x*log(a*x + sqrt(a^2*x^2 + 1))^2 - 2*log(a*x + 
 sqrt(a^2*x^2 + 1))^3 + 3*sqrt(a^2*x^2 + 1)*a*x - 3*(2*a^2*x^2 + 1)*log(a* 
x + sqrt(a^2*x^2 + 1)))/a^3
 

Sympy [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90 \[ \int \frac {x^2 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} - \frac {x^{2} \operatorname {asinh}{\left (a x \right )}}{2 a} + \frac {x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{2 a^{2}} + \frac {x \sqrt {a^{2} x^{2} + 1}}{4 a^{2}} - \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{6 a^{3}} - \frac {\operatorname {asinh}{\left (a x \right )}}{4 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-x**2*asinh(a*x)/(2*a) + x*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/(2 
*a**2) + x*sqrt(a**2*x**2 + 1)/(4*a**2) - asinh(a*x)**3/(6*a**3) - asinh(a 
*x)/(4*a**3), Ne(a, 0)), (0, True))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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