\(\int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\) [210]

Optimal result

Integrand size = 24, antiderivative size = 151 \[ \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {x \sqrt {1-a x} \sqrt {1+a x}}{4 a^2}-\frac {\sqrt {1-a x} \text {arccosh}(a x)}{4 a^3 \sqrt {-1+a x}}+\frac {x^2 \sqrt {1-a x} \text {arccosh}(a x)}{2 a \sqrt {-1+a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{2 a^2}-\frac {\sqrt {1-a x} \text {arccosh}(a x)^3}{6 a^3 \sqrt {-1+a x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.58 \[ \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {-((-1+a x) (1+a x))} \left (4 \text {arccosh}(a x)^3-6 \text {arccosh}(a x) \cosh (2 \text {arccosh}(a x))+\left (3+6 \text {arccosh}(a x)^2\right ) \sinh (2 \text {arccosh}(a x))\right )}{24 a^3 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \] Input:

Integrate[(x^2*ArcCosh[a*x]^2)/Sqrt[1 - a^2*x^2],x]
 

Output:

-1/24*(Sqrt[-((-1 + a*x)*(1 + a*x))]*(4*ArcCosh[a*x]^3 - 6*ArcCosh[a*x]*Co 
sh[2*ArcCosh[a*x]] + (3 + 6*ArcCosh[a*x]^2)*Sinh[2*ArcCosh[a*x]]))/(a^3*Sq 
rt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6353, 6298, 101, 43, 6307}

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.

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

\(\Big \downarrow \) 6353

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 43

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

\(\Big \downarrow \) 6307

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

Input:

Int[(x^2*ArcCosh[a*x]^2)/Sqrt[1 - a^2*x^2],x]
 

Output:

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

Defintions of rubi rules used

rule 43

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a 
*d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
 

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 6298

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* 
(n/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + 
 c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& NeQ[m, -1]
 

rule 6307

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d 
 + e*x^2])]*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x 
] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
 

rule 6353

Int[((a_.) + ArcCosh[(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*ArcCosh[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*ArcCosh[c*x])^n, x], x] 
- Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x) 
^p)]   Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*Ar 
cCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2* 
d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]
 

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.58

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{3}}{6 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (2 \operatorname {arccosh}\left (a x \right )^{2}-2 \,\operatorname {arccosh}\left (a x \right )+1\right )}{16 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (-2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+2 a^{3} x^{3}+\sqrt {a x -1}\, \sqrt {a x +1}-2 a x \right ) \left (2 \operatorname {arccosh}\left (a x \right )^{2}+2 \,\operatorname {arccosh}\left (a x \right )+1\right )}{16 a^{3} \left (a^{2} x^{2}-1\right )}\)	\(239\)

Input:

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

Output:

-1/6*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3/(a^2*x^2-1)*arccos 
h(a*x)^3-1/16*(-a^2*x^2+1)^(1/2)*(2*a^3*x^3-2*a*x+2*a^2*x^2*(a*x-1)^(1/2)* 
(a*x+1)^(1/2)-(a*x-1)^(1/2)*(a*x+1)^(1/2))*(2*arccosh(a*x)^2-2*arccosh(a*x 
)+1)/a^3/(a^2*x^2-1)-1/16*(-a^2*x^2+1)^(1/2)*(-2*a^2*x^2*(a*x-1)^(1/2)*(a* 
x+1)^(1/2)+2*a^3*x^3+(a*x-1)^(1/2)*(a*x+1)^(1/2)-2*a*x)*(2*arccosh(a*x)^2+ 
2*arccosh(a*x)+1)/a^3/(a^2*x^2-1)
 

Fricas [F]

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

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

Output:

integral(-sqrt(-a^2*x^2 + 1)*x^2*arccosh(a*x)^2/(a^2*x^2 - 1), x)
 

Sympy [F]

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

integrate(x**2*acosh(a*x)**2/(-a**2*x**2+1)**(1/2),x)
 

Output:

Integral(x**2*acosh(a*x)**2/sqrt(-(a*x - 1)*(a*x + 1)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int((x^2*acosh(a*x)^2)/(1 - a^2*x^2)^(1/2),x)
 

Output:

int((x^2*acosh(a*x)^2)/(1 - a^2*x^2)^(1/2), x)
 

Reduce [F]

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

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

Output:

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