\(\int \frac {x^4 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx\) [288]

Optimal result

Integrand size = 23, antiderivative size = 153 \[ \int \frac {x^4 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=-\frac {15 x \sqrt {1+a^2 x^2}}{64 a^4}+\frac {x^3 \sqrt {1+a^2 x^2}}{32 a^2}+\frac {15 \text {arcsinh}(a x)}{64 a^5}+\frac {3 x^2 \text {arcsinh}(a x)}{8 a^3}-\frac {x^4 \text {arcsinh}(a x)}{8 a}-\frac {3 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{8 a^4}+\frac {x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{4 a^2}+\frac {\text {arcsinh}(a x)^3}{8 a^5} \] Output:

-15/64*x*(a^2*x^2+1)^(1/2)/a^4+1/32*x^3*(a^2*x^2+1)^(1/2)/a^2+15/64*arcsin 
h(a*x)/a^5+3/8*x^2*arcsinh(a*x)/a^3-1/8*x^4*arcsinh(a*x)/a-3/8*x*(a^2*x^2+ 
1)^(1/2)*arcsinh(a*x)^2/a^4+1/4*x^3*(a^2*x^2+1)^(1/2)*arcsinh(a*x)^2/a^2+1 
/8*arcsinh(a*x)^3/a^5
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.64 \[ \int \frac {x^4 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {a x \sqrt {1+a^2 x^2} \left (-15+2 a^2 x^2\right )+\left (15+24 a^2 x^2-8 a^4 x^4\right ) \text {arcsinh}(a x)+8 a x \sqrt {1+a^2 x^2} \left (-3+2 a^2 x^2\right ) \text {arcsinh}(a x)^2+8 \text {arcsinh}(a x)^3}{64 a^5} \] Input:

Integrate[(x^4*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2],x]
 

Output:

(a*x*Sqrt[1 + a^2*x^2]*(-15 + 2*a^2*x^2) + (15 + 24*a^2*x^2 - 8*a^4*x^4)*A 
rcSinh[a*x] + 8*a*x*Sqrt[1 + a^2*x^2]*(-3 + 2*a^2*x^2)*ArcSinh[a*x]^2 + 8* 
ArcSinh[a*x]^3)/(64*a^5)
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.44, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6227, 6191, 262, 262, 222, 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.

\(\displaystyle \int \frac {x^4 \text {arcsinh}(a x)^2}{\sqrt {a^2 x^2+1}} \, dx\)

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 6198

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

Input:

Int[(x^4*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2],x]
 

Output:

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

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 = 1.59 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.82

method	result	size
default	\(\frac {16 x^{3} a^{3} \operatorname {arcsinh}\left (x a \right )^{2} \sqrt {a^{2} x^{2}+1}-8 x^{4} a^{4} \operatorname {arcsinh}\left (x a \right )+2 x^{3} a^{3} \sqrt {a^{2} x^{2}+1}-24 \operatorname {arcsinh}\left (x a \right )^{2} \sqrt {a^{2} x^{2}+1}\, x a +24 \,\operatorname {arcsinh}\left (x a \right ) x^{2} a^{2}+8 \operatorname {arcsinh}\left (x a \right )^{3}-15 x a \sqrt {a^{2} x^{2}+1}+15 \,\operatorname {arcsinh}\left (x a \right )}{64 a^{5}}\)	\(125\)

Input:

int(x^4*arcsinh(x*a)^2/(a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/64*(16*x^3*a^3*arcsinh(x*a)^2*(a^2*x^2+1)^(1/2)-8*x^4*a^4*arcsinh(x*a)+2 
*x^3*a^3*(a^2*x^2+1)^(1/2)-24*arcsinh(x*a)^2*(a^2*x^2+1)^(1/2)*x*a+24*arcs 
inh(x*a)*x^2*a^2+8*arcsinh(x*a)^3-15*x*a*(a^2*x^2+1)^(1/2)+15*arcsinh(x*a) 
)/a^5
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.86 \[ \int \frac {x^4 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {8 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 8 \, \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - {\left (8 \, a^{4} x^{4} - 24 \, a^{2} x^{2} - 15\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) + {\left (2 \, a^{3} x^{3} - 15 \, a x\right )} \sqrt {a^{2} x^{2} + 1}}{64 \, a^{5}} \] Input:

integrate(x^4*arcsinh(a*x)^2/(a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

1/64*(8*(2*a^3*x^3 - 3*a*x)*sqrt(a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1)) 
^2 + 8*log(a*x + sqrt(a^2*x^2 + 1))^3 - (8*a^4*x^4 - 24*a^2*x^2 - 15)*log( 
a*x + sqrt(a^2*x^2 + 1)) + (2*a^3*x^3 - 15*a*x)*sqrt(a^2*x^2 + 1))/a^5
 

Sympy [A] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95 \[ \int \frac {x^4 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} - \frac {x^{4} \operatorname {asinh}{\left (a x \right )}}{8 a} + \frac {x^{3} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{4 a^{2}} + \frac {x^{3} \sqrt {a^{2} x^{2} + 1}}{32 a^{2}} + \frac {3 x^{2} \operatorname {asinh}{\left (a x \right )}}{8 a^{3}} - \frac {3 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{8 a^{4}} - \frac {15 x \sqrt {a^{2} x^{2} + 1}}{64 a^{4}} + \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{8 a^{5}} + \frac {15 \operatorname {asinh}{\left (a x \right )}}{64 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

integrate(x**4*asinh(a*x)**2/(a**2*x**2+1)**(1/2),x)
 

Output:

Piecewise((-x**4*asinh(a*x)/(8*a) + x**3*sqrt(a**2*x**2 + 1)*asinh(a*x)**2 
/(4*a**2) + x**3*sqrt(a**2*x**2 + 1)/(32*a**2) + 3*x**2*asinh(a*x)/(8*a**3 
) - 3*x*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/(8*a**4) - 15*x*sqrt(a**2*x**2 + 
 1)/(64*a**4) + asinh(a*x)**3/(8*a**5) + 15*asinh(a*x)/(64*a**5), Ne(a, 0) 
), (0, True))
 

Maxima [F]

\[ \int \frac {x^4 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{4} \operatorname {arsinh}\left (a x\right )^{2}}{\sqrt {a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^4*arcsinh(a*x)^2/(a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^4*arcsinh(a*x)^2/sqrt(a^2*x^2 + 1), x)
 

Giac [F]

\[ \int \frac {x^4 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{4} \operatorname {arsinh}\left (a x\right )^{2}}{\sqrt {a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^4*arcsinh(a*x)^2/(a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^4*arcsinh(a*x)^2/sqrt(a^2*x^2 + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^4\,{\mathrm {asinh}\left (a\,x\right )}^2}{\sqrt {a^2\,x^2+1}} \,d x \] Input:

int((x^4*asinh(a*x)^2)/(a^2*x^2 + 1)^(1/2),x)
 

Output:

int((x^4*asinh(a*x)^2)/(a^2*x^2 + 1)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^4 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {\mathit {asinh} \left (a x \right )^{2} x^{4}}{\sqrt {a^{2} x^{2}+1}}d x \] Input:

int(x^4*asinh(a*x)^2/(a^2*x^2+1)^(1/2),x)
 

Output:

int((asinh(a*x)**2*x**4)/sqrt(a**2*x**2 + 1),x)