\(\int \frac {x^3 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx\) [254]

Optimal result

Integrand size = 24, antiderivative size = 243 \[ \int \frac {x^3 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {40 x \sqrt {-1+a x}}{9 a^3 \sqrt {1-a x}}-\frac {2 x^3 \sqrt {-1+a x}}{27 a \sqrt {1-a x}}-\frac {40 \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{9 a^4}-\frac {2 x^2 \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{9 a^2}-\frac {2 x \sqrt {-1+a x} \text {arccosh}(a x)^2}{a^3 \sqrt {1-a x}}-\frac {x^3 \sqrt {-1+a x} \text {arccosh}(a x)^2}{3 a \sqrt {1-a x}}-\frac {2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{3 a^2} \]

[Out]

-40/9*x*(a*x-1)^(1/2)/a^3/(-a*x+1)^(1/2)-2/27*x^3*(a*x-1)^(1/2)/a/(-a*x+1)^(1/2)-2*x*arccosh(a*x)^2*(a*x-1)^(1
/2)/a^3/(-a*x+1)^(1/2)-1/3*x^3*arccosh(a*x)^2*(a*x-1)^(1/2)/a/(-a*x+1)^(1/2)-40/9*arccosh(a*x)*(-a*x+1)^(1/2)*
(a*x+1)^(1/2)/a^4-2/9*x^2*arccosh(a*x)*(-a*x+1)^(1/2)*(a*x+1)^(1/2)/a^2-2/3*arccosh(a*x)^3*(-a^2*x^2+1)^(1/2)/
a^4-1/3*x^2*arccosh(a*x)^3*(-a^2*x^2+1)^(1/2)/a^2

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5938, 5914, 5879, 5915, 8, 5883, 5939, 30} \[ \int \frac {x^3 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {40 \sqrt {1-a x} \sqrt {a x+1} \text {arccosh}(a x)}{9 a^4}-\frac {2 x \sqrt {a x-1} \text {arccosh}(a x)^2}{a^3 \sqrt {1-a x}}-\frac {40 x \sqrt {a x-1}}{9 a^3 \sqrt {1-a x}}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{3 a^2}-\frac {2 x^2 \sqrt {1-a x} \sqrt {a x+1} \text {arccosh}(a x)}{9 a^2}-\frac {2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{3 a^4}-\frac {x^3 \sqrt {a x-1} \text {arccosh}(a x)^2}{3 a \sqrt {1-a x}}-\frac {2 x^3 \sqrt {a x-1}}{27 a \sqrt {1-a x}} \]

[In]

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

Rule 8

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

Rule 30

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

Rule 5879

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 5883

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)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5914

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

Rule 5915

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sy
mbol] :> 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/(2*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Int[(1 + c*x)^(p + 1/2)*(-
1 + c*x)^(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] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]

Rule 5938

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] + (Dist[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] - Dist[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*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && I
GtQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rule 5939

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

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.58 \[ \int \frac {x^3 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (2 a x \left (60+a^2 x^2\right )-6 \sqrt {-1+a x} \sqrt {1+a x} \left (20+a^2 x^2\right ) \text {arccosh}(a x)+9 a x \left (6+a^2 x^2\right ) \text {arccosh}(a x)^2-9 \sqrt {-1+a x} \sqrt {1+a x} \left (2+a^2 x^2\right ) \text {arccosh}(a x)^3\right )}{27 a^4 \sqrt {-1+a x} \sqrt {1+a x}} \]

[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] (verified)

Time = 1.38 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.54

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (4 a^{4} x^{4}-5 a^{2} x^{2}+4 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}-3 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +1\right ) \left (9 \operatorname {arccosh}\left (a x \right )^{3}-9 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )-2\right )}{216 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \left (\sqrt {a x -1}\, \sqrt {a x +1}\, a x +a^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (a x \right )^{3}-3 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )-6\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x -1}\, \sqrt {a x +1}\, a x -1\right ) \left (\operatorname {arccosh}\left (a x \right )^{3}+3 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )+6\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (4 a^{4} x^{4}-5 a^{2} x^{2}-4 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}+3 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +1\right ) \left (9 \operatorname {arccosh}\left (a x \right )^{3}+9 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )+2\right )}{216 a^{4} \left (a^{2} x^{2}-1\right )}\)	\(375\)

[In]

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

[Out]

-1/216*(-a^2*x^2+1)^(1/2)*(4*a^4*x^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*a^4*x^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)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.84 \[ \int \frac {x^3 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {9 \, {\left (a^{4} x^{4} + a^{2} x^{2} - 2\right )} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} - 9 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 6 \, {\left (a^{4} x^{4} + 19 \, a^{2} x^{2} - 20\right )} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 2 \, {\left (a^{3} x^{3} + 60 \, a x\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1}}{27 \, {\left (a^{6} x^{2} - a^{4}\right )}} \]

[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]

\[ \int \frac {x^3 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^{3} \operatorname {acosh}^{3}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]

[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)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.54 \[ \int \frac {x^3 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {1}{3} \, {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arcosh}\left (a x\right )^{3} + \frac {2}{27} \, a {\left (\frac {3 \, {\left (-i \, \sqrt {a^{2} x^{2} - 1} x^{2} - \frac {20 i \, \sqrt {a^{2} x^{2} - 1}}{a^{2}}\right )} \operatorname {arcosh}\left (a x\right )}{a^{3}} + \frac {i \, a^{2} x^{3} + 60 i \, x}{a^{4}}\right )} + \frac {{\left (i \, a^{2} x^{3} + 6 i \, x\right )} \operatorname {arcosh}\left (a x\right )^{2}}{3 \, a^{3}} \]

[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

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^3\,{\mathrm {acosh}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \]

[In]

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

[Out]

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