\(\int (c-a^2 c x^2)^3 \text {arccosh}(a x)^2 \, dx\) [164]

Optimal result

Integrand size = 20, antiderivative size = 266 \[ \int \left (c-a^2 c x^2\right )^3 \text {arccosh}(a x)^2 \, dx=\frac {4322 c^3 x}{3675}-\frac {1514 a^2 c^3 x^3}{11025}+\frac {234 a^4 c^3 x^5}{6125}-\frac {2}{343} a^6 c^3 x^7-\frac {32 c^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{35 a}+\frac {16 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)}{105 a}-\frac {12 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)}{175 a}+\frac {2 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \text {arccosh}(a x)}{49 a}+\frac {16}{35} c^3 x \text {arccosh}(a x)^2+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^2+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2 \]

[Out]

4322/3675*c^3*x-1514/11025*a^2*c^3*x^3+234/6125*a^4*c^3*x^5-2/343*a^6*c^3*x^7+16/105*c^3*(a*x-1)^(3/2)*(a*x+1)
^(3/2)*arccosh(a*x)/a-12/175*c^3*(a*x-1)^(5/2)*(a*x+1)^(5/2)*arccosh(a*x)/a+2/49*c^3*(a*x-1)^(7/2)*(a*x+1)^(7/
2)*arccosh(a*x)/a+16/35*c^3*x*arccosh(a*x)^2+8/35*c^3*x*(-a^2*x^2+1)*arccosh(a*x)^2+6/35*c^3*x*(-a^2*x^2+1)^2*
arccosh(a*x)^2+1/7*c^3*x*(-a^2*x^2+1)^3*arccosh(a*x)^2-32/35*c^3*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5897, 5879, 5915, 8, 41, 200} \[ \int \left (c-a^2 c x^2\right )^3 \text {arccosh}(a x)^2 \, dx=-\frac {2}{343} a^6 c^3 x^7+\frac {234 a^4 c^3 x^5}{6125}+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^2-\frac {1514 a^2 c^3 x^3}{11025}+\frac {16}{35} c^3 x \text {arccosh}(a x)^2+\frac {2 c^3 (a x-1)^{7/2} (a x+1)^{7/2} \text {arccosh}(a x)}{49 a}-\frac {12 c^3 (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)}{175 a}+\frac {16 c^3 (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{105 a}-\frac {32 c^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{35 a}+\frac {4322 c^3 x}{3675} \]

[In]

Int[(c - a^2*c*x^2)^3*ArcCosh[a*x]^2,x]

[Out]

(4322*c^3*x)/3675 - (1514*a^2*c^3*x^3)/11025 + (234*a^4*c^3*x^5)/6125 - (2*a^6*c^3*x^7)/343 - (32*c^3*Sqrt[-1
+ a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(35*a) + (16*c^3*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x])/(105*a) - (
12*c^3*(-1 + a*x)^(5/2)*(1 + a*x)^(5/2)*ArcCosh[a*x])/(175*a) + (2*c^3*(-1 + a*x)^(7/2)*(1 + a*x)^(7/2)*ArcCos
h[a*x])/(49*a) + (16*c^3*x*ArcCosh[a*x]^2)/35 + (8*c^3*x*(1 - a^2*x^2)*ArcCosh[a*x]^2)/35 + (6*c^3*x*(1 - a^2*
x^2)^2*ArcCosh[a*x]^2)/35 + (c^3*x*(1 - a^2*x^2)^3*ArcCosh[a*x]^2)/7

Rule 8

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

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

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 5897

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

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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2+\frac {1}{7} (6 c) \int \left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^2 \, dx+\frac {1}{7} \left (2 a c^3\right ) \int x (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x) \, dx \\ & = \frac {2 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \text {arccosh}(a x)}{49 a}+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2+\frac {1}{35} \left (24 c^2\right ) \int \left (c-a^2 c x^2\right ) \text {arccosh}(a x)^2 \, dx-\frac {1}{49} \left (2 c^3\right ) \int (-1+a x)^3 (1+a x)^3 \, dx-\frac {1}{35} \left (12 a c^3\right ) \int x (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x) \, dx \\ & = -\frac {12 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)}{175 a}+\frac {2 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \text {arccosh}(a x)}{49 a}+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^2+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2-\frac {1}{49} \left (2 c^3\right ) \int \left (-1+a^2 x^2\right )^3 \, dx+\frac {1}{175} \left (12 c^3\right ) \int (-1+a x)^2 (1+a x)^2 \, dx+\frac {1}{35} \left (16 c^3\right ) \int \text {arccosh}(a x)^2 \, dx+\frac {1}{35} \left (16 a c^3\right ) \int x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \, dx \\ & = \frac {16 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)}{105 a}-\frac {12 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)}{175 a}+\frac {2 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \text {arccosh}(a x)}{49 a}+\frac {16}{35} c^3 x \text {arccosh}(a x)^2+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^2+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2-\frac {1}{49} \left (2 c^3\right ) \int \left (-1+3 a^2 x^2-3 a^4 x^4+a^6 x^6\right ) \, dx+\frac {1}{175} \left (12 c^3\right ) \int \left (-1+a^2 x^2\right )^2 \, dx-\frac {1}{105} \left (16 c^3\right ) \int (-1+a x) (1+a x) \, dx-\frac {1}{35} \left (32 a c^3\right ) \int \frac {x \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {2 c^3 x}{49}-\frac {2}{49} a^2 c^3 x^3+\frac {6}{245} a^4 c^3 x^5-\frac {2}{343} a^6 c^3 x^7-\frac {32 c^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{35 a}+\frac {16 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)}{105 a}-\frac {12 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)}{175 a}+\frac {2 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \text {arccosh}(a x)}{49 a}+\frac {16}{35} c^3 x \text {arccosh}(a x)^2+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^2+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2+\frac {1}{175} \left (12 c^3\right ) \int \left (1-2 a^2 x^2+a^4 x^4\right ) \, dx-\frac {1}{105} \left (16 c^3\right ) \int \left (-1+a^2 x^2\right ) \, dx+\frac {1}{35} \left (32 c^3\right ) \int 1 \, dx \\ & = \frac {4322 c^3 x}{3675}-\frac {1514 a^2 c^3 x^3}{11025}+\frac {234 a^4 c^3 x^5}{6125}-\frac {2}{343} a^6 c^3 x^7-\frac {32 c^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{35 a}+\frac {16 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)}{105 a}-\frac {12 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)}{175 a}+\frac {2 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \text {arccosh}(a x)}{49 a}+\frac {16}{35} c^3 x \text {arccosh}(a x)^2+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^2+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.47 \[ \int \left (c-a^2 c x^2\right )^3 \text {arccosh}(a x)^2 \, dx=\frac {c^3 \left (453810 a x-52990 a^3 x^3+14742 a^5 x^5-2250 a^7 x^7+210 \sqrt {-1+a x} \sqrt {1+a x} \left (-2161+757 a^2 x^2-351 a^4 x^4+75 a^6 x^6\right ) \text {arccosh}(a x)-11025 a x \left (-35+35 a^2 x^2-21 a^4 x^4+5 a^6 x^6\right ) \text {arccosh}(a x)^2\right )}{385875 a} \]

[In]

Integrate[(c - a^2*c*x^2)^3*ArcCosh[a*x]^2,x]

[Out]

(c^3*(453810*a*x - 52990*a^3*x^3 + 14742*a^5*x^5 - 2250*a^7*x^7 + 210*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-2161 + 75
7*a^2*x^2 - 351*a^4*x^4 + 75*a^6*x^6)*ArcCosh[a*x] - 11025*a*x*(-35 + 35*a^2*x^2 - 21*a^4*x^4 + 5*a^6*x^6)*Arc
Cosh[a*x]^2))/(385875*a)

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.71

method	result	size
derivativedivides	\(-\frac {c^{3} \left (55125 \operatorname {arccosh}\left (a x \right )^{2} a^{7} x^{7}-15750 \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{6} x^{6}-231525 a^{5} x^{5} \operatorname {arccosh}\left (a x \right )^{2}+73710 a^{4} x^{4} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+2250 a^{7} x^{7}+385875 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{2}-158970 a^{2} x^{2} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}-14742 a^{5} x^{5}-385875 a x \operatorname {arccosh}\left (a x \right )^{2}+453810 \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )+52990 a^{3} x^{3}-453810 a x \right )}{385875 a}\)	\(188\)
default	\(-\frac {c^{3} \left (55125 \operatorname {arccosh}\left (a x \right )^{2} a^{7} x^{7}-15750 \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{6} x^{6}-231525 a^{5} x^{5} \operatorname {arccosh}\left (a x \right )^{2}+73710 a^{4} x^{4} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+2250 a^{7} x^{7}+385875 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{2}-158970 a^{2} x^{2} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}-14742 a^{5} x^{5}-385875 a x \operatorname {arccosh}\left (a x \right )^{2}+453810 \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )+52990 a^{3} x^{3}-453810 a x \right )}{385875 a}\)	\(188\)

[In]

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

[Out]

-1/385875/a*c^3*(55125*arccosh(a*x)^2*a^7*x^7-15750*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^6*x^6-231525*a^
5*x^5*arccosh(a*x)^2+73710*a^4*x^4*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+2250*a^7*x^7+385875*a^3*x^3*arccos
h(a*x)^2-158970*a^2*x^2*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)-14742*a^5*x^5-385875*a*x*arccosh(a*x)^2+45381
0*(a*x-1)^(1/2)*(a*x+1)^(1/2)*arccosh(a*x)+52990*a^3*x^3-453810*a*x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.66 \[ \int \left (c-a^2 c x^2\right )^3 \text {arccosh}(a x)^2 \, dx=-\frac {2250 \, a^{7} c^{3} x^{7} - 14742 \, a^{5} c^{3} x^{5} + 52990 \, a^{3} c^{3} x^{3} - 453810 \, a c^{3} x + 11025 \, {\left (5 \, a^{7} c^{3} x^{7} - 21 \, a^{5} c^{3} x^{5} + 35 \, a^{3} c^{3} x^{3} - 35 \, a c^{3} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 210 \, {\left (75 \, a^{6} c^{3} x^{6} - 351 \, a^{4} c^{3} x^{4} + 757 \, a^{2} c^{3} x^{2} - 2161 \, c^{3}\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{385875 \, a} \]

[In]

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

[Out]

-1/385875*(2250*a^7*c^3*x^7 - 14742*a^5*c^3*x^5 + 52990*a^3*c^3*x^3 - 453810*a*c^3*x + 11025*(5*a^7*c^3*x^7 -
21*a^5*c^3*x^5 + 35*a^3*c^3*x^3 - 35*a*c^3*x)*log(a*x + sqrt(a^2*x^2 - 1))^2 - 210*(75*a^6*c^3*x^6 - 351*a^4*c
^3*x^4 + 757*a^2*c^3*x^2 - 2161*c^3)*sqrt(a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1)))/a

Sympy [F]

\[ \int \left (c-a^2 c x^2\right )^3 \text {arccosh}(a x)^2 \, dx=- c^{3} \left (\int 3 a^{2} x^{2} \operatorname {acosh}^{2}{\left (a x \right )}\, dx + \int \left (- 3 a^{4} x^{4} \operatorname {acosh}^{2}{\left (a x \right )}\right )\, dx + \int a^{6} x^{6} \operatorname {acosh}^{2}{\left (a x \right )}\, dx + \int \left (- \operatorname {acosh}^{2}{\left (a x \right )}\right )\, dx\right ) \]

[In]

integrate((-a**2*c*x**2+c)**3*acosh(a*x)**2,x)

[Out]

-c**3*(Integral(3*a**2*x**2*acosh(a*x)**2, x) + Integral(-3*a**4*x**4*acosh(a*x)**2, x) + Integral(a**6*x**6*a
cosh(a*x)**2, x) + Integral(-acosh(a*x)**2, x))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.67 \[ \int \left (c-a^2 c x^2\right )^3 \text {arccosh}(a x)^2 \, dx=-\frac {2}{343} \, a^{6} c^{3} x^{7} + \frac {234}{6125} \, a^{4} c^{3} x^{5} - \frac {1514}{11025} \, a^{2} c^{3} x^{3} + \frac {4322}{3675} \, c^{3} x + \frac {2}{3675} \, {\left (75 \, \sqrt {a^{2} x^{2} - 1} a^{4} c^{3} x^{6} - 351 \, \sqrt {a^{2} x^{2} - 1} a^{2} c^{3} x^{4} + 757 \, \sqrt {a^{2} x^{2} - 1} c^{3} x^{2} - \frac {2161 \, \sqrt {a^{2} x^{2} - 1} c^{3}}{a^{2}}\right )} a \operatorname {arcosh}\left (a x\right ) - \frac {1}{35} \, {\left (5 \, a^{6} c^{3} x^{7} - 21 \, a^{4} c^{3} x^{5} + 35 \, a^{2} c^{3} x^{3} - 35 \, c^{3} x\right )} \operatorname {arcosh}\left (a x\right )^{2} \]

[In]

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

[Out]

-2/343*a^6*c^3*x^7 + 234/6125*a^4*c^3*x^5 - 1514/11025*a^2*c^3*x^3 + 4322/3675*c^3*x + 2/3675*(75*sqrt(a^2*x^2
- 1)*a^4*c^3*x^6 - 351*sqrt(a^2*x^2 - 1)*a^2*c^3*x^4 + 757*sqrt(a^2*x^2 - 1)*c^3*x^2 - 2161*sqrt(a^2*x^2 - 1)
*c^3/a^2)*a*arccosh(a*x) - 1/35*(5*a^6*c^3*x^7 - 21*a^4*c^3*x^5 + 35*a^2*c^3*x^3 - 35*c^3*x)*arccosh(a*x)^2

Giac [F(-2)]

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

[In]

integrate((-a^2*c*x^2+c)^3*arccosh(a*x)^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. \[ \int \left (c-a^2 c x^2\right )^3 \text {arccosh}(a x)^2 \, dx=\int {\mathrm {acosh}\left (a\,x\right )}^2\,{\left (c-a^2\,c\,x^2\right )}^3 \,d x \]

[In]

int(acosh(a*x)^2*(c - a^2*c*x^2)^3,x)

[Out]

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