3.254 \(\int \frac{x^3 \cosh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=243 \[ -\frac{x^2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^3}{3 a^2}-\frac{2 x^2 \sqrt{1-a x} \sqrt{a x+1} \cosh ^{-1}(a x)}{9 a^2}-\frac{2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^3}{3 a^4}-\frac{40 x \sqrt{a x-1}}{9 a^3 \sqrt{1-a x}}-\frac{2 x \sqrt{a x-1} \cosh ^{-1}(a x)^2}{a^3 \sqrt{1-a x}}-\frac{40 \sqrt{1-a x} \sqrt{a x+1} \cosh ^{-1}(a x)}{9 a^4}-\frac{2 x^3 \sqrt{a x-1}}{27 a \sqrt{1-a x}}-\frac{x^3 \sqrt{a x-1} \cosh ^{-1}(a x)^2}{3 a \sqrt{1-a x}} \]

[Out]

(-40*x*Sqrt[-1 + a*x])/(9*a^3*Sqrt[1 - a*x]) - (2*x^3*Sqrt[-1 + a*x])/(27*a*Sqrt[1 - a*x]) - (40*Sqrt[1 - a*x]
*Sqrt[1 + a*x]*ArcCosh[a*x])/(9*a^4) - (2*x^2*Sqrt[1 - a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(9*a^2) - (2*x*Sqrt[-1
+ a*x]*ArcCosh[a*x]^2)/(a^3*Sqrt[1 - a*x]) - (x^3*Sqrt[-1 + a*x]*ArcCosh[a*x]^2)/(3*a*Sqrt[1 - a*x]) - (2*Sqr
t[1 - a^2*x^2]*ArcCosh[a*x]^3)/(3*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3)/(3*a^2)

________________________________________________________________________________________

Rubi [A]  time = 1.02626, antiderivative size = 329, normalized size of antiderivative = 1.35, number of steps used = 11, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {5798, 5759, 5718, 5654, 8, 5662, 30} \[ -\frac{2 x^3 \sqrt{a x-1} \sqrt{a x+1}}{27 a \sqrt{1-a^2 x^2}}-\frac{40 x \sqrt{a x-1} \sqrt{a x+1}}{9 a^3 \sqrt{1-a^2 x^2}}-\frac{x^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{3 a \sqrt{1-a^2 x^2}}-\frac{x^2 (1-a x) (a x+1) \cosh ^{-1}(a x)^3}{3 a^2 \sqrt{1-a^2 x^2}}-\frac{2 x^2 (1-a x) (a x+1) \cosh ^{-1}(a x)}{9 a^2 \sqrt{1-a^2 x^2}}-\frac{2 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{a^3 \sqrt{1-a^2 x^2}}-\frac{2 (1-a x) (a x+1) \cosh ^{-1}(a x)^3}{3 a^4 \sqrt{1-a^2 x^2}}-\frac{40 (1-a x) (a x+1) \cosh ^{-1}(a x)}{9 a^4 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*ArcCosh[a*x]^3)/Sqrt[1 - a^2*x^2],x]

[Out]

(-40*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(9*a^3*Sqrt[1 - a^2*x^2]) - (2*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a*Sq
rt[1 - a^2*x^2]) - (40*(1 - a*x)*(1 + a*x)*ArcCosh[a*x])/(9*a^4*Sqrt[1 - a^2*x^2]) - (2*x^2*(1 - a*x)*(1 + a*x
)*ArcCosh[a*x])/(9*a^2*Sqrt[1 - a^2*x^2]) - (2*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(a^3*Sqrt[1 - a^
2*x^2]) - (x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(3*a*Sqrt[1 - a^2*x^2]) - (2*(1 - a*x)*(1 + a*x)*A
rcCosh[a*x]^3)/(3*a^4*Sqrt[1 - a^2*x^2]) - (x^2*(1 - a*x)*(1 + a*x)*ArcCosh[a*x]^3)/(3*a^2*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 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x^3 \cosh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x^3 \cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x \cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{3 a^2 \sqrt{1-a^2 x^2}}-\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int x^2 \cosh ^{-1}(a x)^2 \, dx}{a \sqrt{1-a^2 x^2}}\\ &=-\frac{x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{3 a \sqrt{1-a^2 x^2}}-\frac{2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^4 \sqrt{1-a^2 x^2}}-\frac{x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x^3 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{3 \sqrt{1-a^2 x^2}}-\frac{\left (2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \cosh ^{-1}(a x)^2 \, dx}{a^3 \sqrt{1-a^2 x^2}}\\ &=-\frac{2 x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)}{9 a^2 \sqrt{1-a^2 x^2}}-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{a^3 \sqrt{1-a^2 x^2}}-\frac{x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{3 a \sqrt{1-a^2 x^2}}-\frac{2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^4 \sqrt{1-a^2 x^2}}-\frac{x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (4 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{9 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (4 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{a^2 \sqrt{1-a^2 x^2}}-\frac{\left (2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int x^2 \, dx}{9 a \sqrt{1-a^2 x^2}}\\ &=-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{27 a \sqrt{1-a^2 x^2}}-\frac{40 (1-a x) (1+a x) \cosh ^{-1}(a x)}{9 a^4 \sqrt{1-a^2 x^2}}-\frac{2 x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)}{9 a^2 \sqrt{1-a^2 x^2}}-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{a^3 \sqrt{1-a^2 x^2}}-\frac{x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{3 a \sqrt{1-a^2 x^2}}-\frac{2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^4 \sqrt{1-a^2 x^2}}-\frac{x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^2 \sqrt{1-a^2 x^2}}-\frac{\left (4 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int 1 \, dx}{9 a^3 \sqrt{1-a^2 x^2}}-\frac{\left (4 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int 1 \, dx}{a^3 \sqrt{1-a^2 x^2}}\\ &=-\frac{40 x \sqrt{-1+a x} \sqrt{1+a x}}{9 a^3 \sqrt{1-a^2 x^2}}-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{27 a \sqrt{1-a^2 x^2}}-\frac{40 (1-a x) (1+a x) \cosh ^{-1}(a x)}{9 a^4 \sqrt{1-a^2 x^2}}-\frac{2 x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)}{9 a^2 \sqrt{1-a^2 x^2}}-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{a^3 \sqrt{1-a^2 x^2}}-\frac{x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{3 a \sqrt{1-a^2 x^2}}-\frac{2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^4 \sqrt{1-a^2 x^2}}-\frac{x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^2 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.155989, size = 140, normalized size = 0.58 \[ \frac{\sqrt{1-a^2 x^2} \left (2 a x \left (a^2 x^2+60\right )-9 \sqrt{a x-1} \sqrt{a x+1} \left (a^2 x^2+2\right ) \cosh ^{-1}(a x)^3+9 a x \left (a^2 x^2+6\right ) \cosh ^{-1}(a x)^2-6 \sqrt{a x-1} \sqrt{a x+1} \left (a^2 x^2+20\right ) \cosh ^{-1}(a x)\right )}{27 a^4 \sqrt{a x-1} \sqrt{a x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*ArcCosh[a*x]^3)/Sqrt[1 - a^2*x^2],x]

[Out]

(Sqrt[1 - a^2*x^2]*(2*a*x*(60 + a^2*x^2) - 6*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(20 + a^2*x^2)*ArcCosh[a*x] + 9*a*x*
(6 + a^2*x^2)*ArcCosh[a*x]^2 - 9*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(2 + a^2*x^2)*ArcCosh[a*x]^3))/(27*a^4*Sqrt[-1 +
a*x]*Sqrt[1 + a*x])

________________________________________________________________________________________

Maple [A]  time = 0.198, size = 375, normalized size = 1.5 \begin{align*} -{\frac{9\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}-9\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+6\,{\rm arccosh} \left (ax\right )-2}{216\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 4\,{x}^{4}{a}^{4}-5\,{a}^{2}{x}^{2}+4\,{a}^{3}{x}^{3}\sqrt{ax-1}\sqrt{ax+1}-3\,\sqrt{ax+1}\sqrt{ax-1}ax+1 \right ) }-{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}-9\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+18\,{\rm arccosh} \left (ax\right )-18}{8\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( \sqrt{ax+1}\sqrt{ax-1}ax+{a}^{2}{x}^{2}-1 \right ) }-{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}+9\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+18\,{\rm arccosh} \left (ax\right )+18}{8\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ({a}^{2}{x}^{2}-\sqrt{ax+1}\sqrt{ax-1}ax-1 \right ) }-{\frac{9\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}+9\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+6\,{\rm arccosh} \left (ax\right )+2}{216\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 4\,{x}^{4}{a}^{4}-5\,{a}^{2}{x}^{2}-4\,{a}^{3}{x}^{3}\sqrt{ax-1}\sqrt{ax+1}+3\,\sqrt{ax+1}\sqrt{ax-1}ax+1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/216*(-a^2*x^2+1)^(1/2)*(4*x^4*a^4-5*a^2*x^2+4*a^3*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)-3*(a*x+1)^(1/2)*(a*x-1)^(
1/2)*a*x+1)*(9*arccosh(a*x)^3-9*arccosh(a*x)^2+6*arccosh(a*x)-2)/a^4/(a^2*x^2-1)-3/8*(-a^2*x^2+1)^(1/2)*((a*x+
1)^(1/2)*(a*x-1)^(1/2)*a*x+a^2*x^2-1)*(arccosh(a*x)^3-3*arccosh(a*x)^2+6*arccosh(a*x)-6)/a^4/(a^2*x^2-1)-3/8*(
-a^2*x^2+1)^(1/2)*(a^2*x^2-(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x-1)*(arccosh(a*x)^3+3*arccosh(a*x)^2+6*arccosh(a*x)+
6)/a^4/(a^2*x^2-1)-1/216*(-a^2*x^2+1)^(1/2)*(4*x^4*a^4-5*a^2*x^2-4*a^3*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)+3*(a*x+
1)^(1/2)*(a*x-1)^(1/2)*a*x+1)*(9*arccosh(a*x)^3+9*arccosh(a*x)^2+6*arccosh(a*x)+2)/a^4/(a^2*x^2-1)

________________________________________________________________________________________

Maxima [C]  time = 1.80281, size = 177, normalized size = 0.73 \begin{align*} -\frac{1}{3} \,{\left (\frac{\sqrt{-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname{arcosh}\left (a x\right )^{3} + \frac{2}{27} \, a{\left (\frac{3 \,{\left (-i \, \sqrt{a^{2} x^{2} - 1} x^{2} - \frac{20 i \, \sqrt{a^{2} x^{2} - 1}}{a^{2}}\right )} \operatorname{arcosh}\left (a x\right )}{a^{3}} + \frac{i \, a^{2} x^{3} + 60 i \, x}{a^{4}}\right )} + \frac{{\left (i \, a^{2} x^{3} + 6 i \, x\right )} \operatorname{arcosh}\left (a x\right )^{2}}{3 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(sqrt(-a^2*x^2 + 1)*x^2/a^2 + 2*sqrt(-a^2*x^2 + 1)/a^4)*arccosh(a*x)^3 + 2/27*a*(3*(-I*sqrt(a^2*x^2 - 1)*
x^2 - 20*I*sqrt(a^2*x^2 - 1)/a^2)*arccosh(a*x)/a^3 + (I*a^2*x^3 + 60*I*x)/a^4) + 1/3*(I*a^2*x^3 + 6*I*x)*arcco
sh(a*x)^2/a^3

________________________________________________________________________________________

Fricas [A]  time = 2.20353, size = 447, normalized size = 1.84 \begin{align*} -\frac{9 \,{\left (a^{4} x^{4} + a^{2} x^{2} - 2\right )} \sqrt{-a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} - 9 \,{\left (a^{3} x^{3} + 6 \, a x\right )} \sqrt{a^{2} x^{2} - 1} \sqrt{-a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} + 6 \,{\left (a^{4} x^{4} + 19 \, a^{2} x^{2} - 20\right )} \sqrt{-a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - 2 \,{\left (a^{3} x^{3} + 60 \, a x\right )} \sqrt{a^{2} x^{2} - 1} \sqrt{-a^{2} x^{2} + 1}}{27 \,{\left (a^{6} x^{2} - a^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(9*(a^4*x^4 + a^2*x^2 - 2)*sqrt(-a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 - 1))^3 - 9*(a^3*x^3 + 6*a*x)*sqrt(
a^2*x^2 - 1)*sqrt(-a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 - 1))^2 + 6*(a^4*x^4 + 19*a^2*x^2 - 20)*sqrt(-a^2*x^2 +
1)*log(a*x + sqrt(a^2*x^2 - 1)) - 2*(a^3*x^3 + 60*a*x)*sqrt(a^2*x^2 - 1)*sqrt(-a^2*x^2 + 1))/(a^6*x^2 - a^4)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \operatorname{acosh}^{3}{\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**3*acosh(a*x)**3/(-a**2*x**2+1)**(1/2),x)

[Out]

Integral(x**3*acosh(a*x)**3/sqrt(-(a*x - 1)*(a*x + 1)), x)

________________________________________________________________________________________

Giac [C]  time = 1.33952, size = 200, normalized size = 0.82 \begin{align*} \frac{{\left ({\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{-a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3}}{3 \, a^{4}} + \frac{-2 i \, a^{2} x^{3} + 9 \,{\left (-i \, a^{2} x^{3} - 6 i \, x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} - 120 i \, x - \frac{3 \,{\left (-2 i \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} - 42 i \, \sqrt{a^{2} x^{2} - 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{a}}{27 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 - 1))^3/a^4 + 1/27*(-2*I*a^2*x^3 + 9*
(-I*a^2*x^3 - 6*I*x)*log(a*x + sqrt(a^2*x^2 - 1))^2 - 120*I*x - 3*(-2*I*(a^2*x^2 - 1)^(3/2) - 42*I*sqrt(a^2*x^
2 - 1))*log(a*x + sqrt(a^2*x^2 - 1))/a)/a^3