3.225 \(\int \frac{x^4 \cosh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=243 \[ -\frac{x^3 \sqrt{1-a x} \sqrt{a x+1}}{32 a^2}-\frac{x^3 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^2}{4 a^2}-\frac{3 x^2 \sqrt{a x-1} \cosh ^{-1}(a x)}{8 a^3 \sqrt{1-a x}}-\frac{3 x \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^2}{8 a^4}-\frac{15 x \sqrt{1-a x} \sqrt{a x+1}}{64 a^4}+\frac{\sqrt{a x-1} \cosh ^{-1}(a x)^3}{8 a^5 \sqrt{1-a x}}+\frac{15 \sqrt{a x-1} \cosh ^{-1}(a x)}{64 a^5 \sqrt{1-a x}}-\frac{x^4 \sqrt{a x-1} \cosh ^{-1}(a x)}{8 a \sqrt{1-a x}} \]

[Out]

(-15*x*Sqrt[1 - a*x]*Sqrt[1 + a*x])/(64*a^4) - (x^3*Sqrt[1 - a*x]*Sqrt[1 + a*x])/(32*a^2) + (15*Sqrt[-1 + a*x]
*ArcCosh[a*x])/(64*a^5*Sqrt[1 - a*x]) - (3*x^2*Sqrt[-1 + a*x]*ArcCosh[a*x])/(8*a^3*Sqrt[1 - a*x]) - (x^4*Sqrt[
-1 + a*x]*ArcCosh[a*x])/(8*a*Sqrt[1 - a*x]) - (3*x*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)/(8*a^4) - (x^3*Sqrt[1 - a
^2*x^2]*ArcCosh[a*x]^2)/(4*a^2) + (Sqrt[-1 + a*x]*ArcCosh[a*x]^3)/(8*a^5*Sqrt[1 - a*x])

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Rubi [A]  time = 0.792808, antiderivative size = 329, normalized size of antiderivative = 1.35, number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5798, 5759, 5676, 5662, 90, 52, 100, 12} \[ -\frac{x^3 (1-a x) (a x+1)}{32 a^2 \sqrt{1-a^2 x^2}}-\frac{15 x (1-a x) (a x+1)}{64 a^4 \sqrt{1-a^2 x^2}}-\frac{x^4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{8 a \sqrt{1-a^2 x^2}}-\frac{x^3 (1-a x) (a x+1) \cosh ^{-1}(a x)^2}{4 a^2 \sqrt{1-a^2 x^2}}-\frac{3 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{8 a^3 \sqrt{1-a^2 x^2}}-\frac{3 x (1-a x) (a x+1) \cosh ^{-1}(a x)^2}{8 a^4 \sqrt{1-a^2 x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{8 a^5 \sqrt{1-a^2 x^2}}+\frac{15 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{64 a^5 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*ArcCosh[a*x]^2)/Sqrt[1 - a^2*x^2],x]

[Out]

(-15*x*(1 - a*x)*(1 + a*x))/(64*a^4*Sqrt[1 - a^2*x^2]) - (x^3*(1 - a*x)*(1 + a*x))/(32*a^2*Sqrt[1 - a^2*x^2])
+ (15*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(64*a^5*Sqrt[1 - a^2*x^2]) - (3*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a
*x]*ArcCosh[a*x])/(8*a^3*Sqrt[1 - a^2*x^2]) - (x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(8*a*Sqrt[1 - a^
2*x^2]) - (3*x*(1 - a*x)*(1 + a*x)*ArcCosh[a*x]^2)/(8*a^4*Sqrt[1 - a^2*x^2]) - (x^3*(1 - a*x)*(1 + a*x)*ArcCos
h[a*x]^2)/(4*a^2*Sqrt[1 - a^2*x^2]) + (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(8*a^5*Sqrt[1 - a^2*x^2])

Rule 5798

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist
[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^m*(1 + c*
x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[c^2*d + e, 0]
&&  !IntegerQ[p]

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rule 5662

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

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[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 52

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

Rule 100

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin{align*} \int \frac{x^4 \cosh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x^4 \cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{x^3 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{4 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (3 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x^2 \cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{4 a^2 \sqrt{1-a^2 x^2}}-\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int x^3 \cosh ^{-1}(a x) \, dx}{2 a \sqrt{1-a^2 x^2}}\\ &=-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{8 a \sqrt{1-a^2 x^2}}-\frac{3 x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{8 a^4 \sqrt{1-a^2 x^2}}-\frac{x^3 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{4 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x^4}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{8 \sqrt{1-a^2 x^2}}+\frac{\left (3 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{8 a^4 \sqrt{1-a^2 x^2}}-\frac{\left (3 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int x \cosh ^{-1}(a x) \, dx}{4 a^3 \sqrt{1-a^2 x^2}}\\ &=-\frac{x^3 (1-a x) (1+a x)}{32 a^2 \sqrt{1-a^2 x^2}}-\frac{3 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{8 a^3 \sqrt{1-a^2 x^2}}-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{8 a \sqrt{1-a^2 x^2}}-\frac{3 x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{8 a^4 \sqrt{1-a^2 x^2}}-\frac{x^3 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{4 a^2 \sqrt{1-a^2 x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{8 a^5 \sqrt{1-a^2 x^2}}+\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{3 x^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{32 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (3 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{8 a^2 \sqrt{1-a^2 x^2}}\\ &=-\frac{3 x (1-a x) (1+a x)}{16 a^4 \sqrt{1-a^2 x^2}}-\frac{x^3 (1-a x) (1+a x)}{32 a^2 \sqrt{1-a^2 x^2}}-\frac{3 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{8 a^3 \sqrt{1-a^2 x^2}}-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{8 a \sqrt{1-a^2 x^2}}-\frac{3 x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{8 a^4 \sqrt{1-a^2 x^2}}-\frac{x^3 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{4 a^2 \sqrt{1-a^2 x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{8 a^5 \sqrt{1-a^2 x^2}}+\frac{\left (3 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{1}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{16 a^4 \sqrt{1-a^2 x^2}}+\frac{\left (3 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{32 a^2 \sqrt{1-a^2 x^2}}\\ &=-\frac{15 x (1-a x) (1+a x)}{64 a^4 \sqrt{1-a^2 x^2}}-\frac{x^3 (1-a x) (1+a x)}{32 a^2 \sqrt{1-a^2 x^2}}+\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{16 a^5 \sqrt{1-a^2 x^2}}-\frac{3 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{8 a^3 \sqrt{1-a^2 x^2}}-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{8 a \sqrt{1-a^2 x^2}}-\frac{3 x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{8 a^4 \sqrt{1-a^2 x^2}}-\frac{x^3 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{4 a^2 \sqrt{1-a^2 x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{8 a^5 \sqrt{1-a^2 x^2}}+\frac{\left (3 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{1}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{64 a^4 \sqrt{1-a^2 x^2}}\\ &=-\frac{15 x (1-a x) (1+a x)}{64 a^4 \sqrt{1-a^2 x^2}}-\frac{x^3 (1-a x) (1+a x)}{32 a^2 \sqrt{1-a^2 x^2}}+\frac{15 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{64 a^5 \sqrt{1-a^2 x^2}}-\frac{3 x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{8 a^3 \sqrt{1-a^2 x^2}}-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{8 a \sqrt{1-a^2 x^2}}-\frac{3 x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{8 a^4 \sqrt{1-a^2 x^2}}-\frac{x^3 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{4 a^2 \sqrt{1-a^2 x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{8 a^5 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.266279, size = 116, normalized size = 0.48 \[ \frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \left (32 \cosh ^{-1}(a x)^3-4 \left (16 \cosh \left (2 \cosh ^{-1}(a x)\right )+\cosh \left (4 \cosh ^{-1}(a x)\right )\right ) \cosh ^{-1}(a x)+8 \cosh ^{-1}(a x)^2 \left (8 \sinh \left (2 \cosh ^{-1}(a x)\right )+\sinh \left (4 \cosh ^{-1}(a x)\right )\right )+32 \sinh \left (2 \cosh ^{-1}(a x)\right )+\sinh \left (4 \cosh ^{-1}(a x)\right )\right )}{256 a^5 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*ArcCosh[a*x]^2)/Sqrt[1 - a^2*x^2],x]

[Out]

(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(32*ArcCosh[a*x]^3 - 4*ArcCosh[a*x]*(16*Cosh[2*ArcCosh[a*x]] + Cosh[4*Ar
cCosh[a*x]]) + 32*Sinh[2*ArcCosh[a*x]] + Sinh[4*ArcCosh[a*x]] + 8*ArcCosh[a*x]^2*(8*Sinh[2*ArcCosh[a*x]] + Sin
h[4*ArcCosh[a*x]])))/(256*a^5*Sqrt[1 - a^2*x^2])

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Maple [B]  time = 0.28, size = 488, normalized size = 2. \begin{align*} -{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{8\,{a}^{5} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{8\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}-4\,{\rm arccosh} \left (ax\right )+1}{512\,{a}^{5} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 8\,{x}^{5}{a}^{5}-12\,{x}^{3}{a}^{3}+8\,\sqrt{ax+1}\sqrt{ax-1}{x}^{4}{a}^{4}+4\,ax-8\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}+\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{2\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}-2\,{\rm arccosh} \left (ax\right )+1}{16\,{a}^{5} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 2\,{x}^{3}{a}^{3}-2\,ax+2\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}-\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{2\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+2\,{\rm arccosh} \left (ax\right )+1}{16\,{a}^{5} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 2\,{x}^{3}{a}^{3}-2\,ax-2\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}+\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{8\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+4\,{\rm arccosh} \left (ax\right )+1}{512\,{a}^{5} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 8\,{x}^{5}{a}^{5}-12\,{x}^{3}{a}^{3}-8\,\sqrt{ax+1}\sqrt{ax-1}{x}^{4}{a}^{4}+4\,ax+8\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}-\sqrt{ax-1}\sqrt{ax+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^5/(a^2*x^2-1)*arccosh(a*x)^3-1/512*(-a^2*x^2+1)^(1/2)*(8
*x^5*a^5-12*x^3*a^3+8*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^4*a^4+4*a*x-8*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a^2+(a*x-1)^
(1/2)*(a*x+1)^(1/2))*(8*arccosh(a*x)^2-4*arccosh(a*x)+1)/a^5/(a^2*x^2-1)-1/16*(-a^2*x^2+1)^(1/2)*(2*x^3*a^3-2*
a*x+2*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a^2-(a*x-1)^(1/2)*(a*x+1)^(1/2))*(2*arccosh(a*x)^2-2*arccosh(a*x)+1)/a^5
/(a^2*x^2-1)-1/16*(-a^2*x^2+1)^(1/2)*(2*x^3*a^3-2*a*x-2*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a^2+(a*x-1)^(1/2)*(a*x
+1)^(1/2))*(2*arccosh(a*x)^2+2*arccosh(a*x)+1)/a^5/(a^2*x^2-1)-1/512*(-a^2*x^2+1)^(1/2)*(8*x^5*a^5-12*x^3*a^3-
8*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^4*a^4+4*a*x+8*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a^2-(a*x-1)^(1/2)*(a*x+1)^(1/2))
*(8*arccosh(a*x)^2+4*arccosh(a*x)+1)/a^5/(a^2*x^2-1)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1} x^{4} \operatorname{arcosh}\left (a x\right )^{2}}{a^{2} x^{2} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-a^2*x^2 + 1)*x^4*arccosh(a*x)^2/(a^2*x^2 - 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \operatorname{acosh}^{2}{\left (a x \right )}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**4*acosh(a*x)**2/sqrt(-(a*x - 1)*(a*x + 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \operatorname{arcosh}\left (a x\right )^{2}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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