3.164 \(\int (c-a^2 c x^2)^3 \cosh ^{-1}(a x)^2 \, dx\)

Optimal. Leaf size=266 \[ -\frac{2}{343} a^6 c^3 x^7+\frac{234 a^4 c^3 x^5}{6125}-\frac{1514 a^2 c^3 x^3}{11025}+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^2+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac{16}{35} c^3 x \cosh ^{-1}(a x)^2+\frac{2 c^3 (a x-1)^{7/2} (a x+1)^{7/2} \cosh ^{-1}(a x)}{49 a}-\frac{12 c^3 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)}{175 a}+\frac{16 c^3 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{105 a}-\frac{32 c^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{35 a}+\frac{4322 c^3 x}{3675} \]

[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

________________________________________________________________________________________

Rubi [A]  time = 0.676279, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5681, 5718, 194, 5654, 8} \[ -\frac{2}{343} a^6 c^3 x^7+\frac{234 a^4 c^3 x^5}{6125}-\frac{1514 a^2 c^3 x^3}{11025}+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^2+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac{16}{35} c^3 x \cosh ^{-1}(a x)^2+\frac{2 c^3 (a x-1)^{7/2} (a x+1)^{7/2} \cosh ^{-1}(a x)}{49 a}-\frac{12 c^3 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)}{175 a}+\frac{16 c^3 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{105 a}-\frac{32 c^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{35 a}+\frac{4322 c^3 x}{3675} \]

Antiderivative was successfully verified.

[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 5681

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[(b*c*n*(-d)^p)/(2*p + 1), Int[x*(-1 + c*x)^(p - 1/2)*(1 + c*x)^
(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] + Dist[(2*d*p)/(2*p + 1), Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos
h[c*x])^n, x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && IntegerQ[p]

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 194

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 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]

Rubi steps

\begin{align*} \int \left (c-a^2 c x^2\right )^3 \cosh ^{-1}(a x)^2 \, dx &=\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^2+\frac{1}{7} (6 c) \int \left (c-a^2 c x^2\right )^2 \cosh ^{-1}(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} \cosh ^{-1}(a x) \, dx\\ &=\frac{2 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)}{49 a}+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^2+\frac{1}{35} \left (24 c^2\right ) \int \left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)^2 \, dx-\frac{1}{49} \left (2 c^3\right ) \int \left (-1+a^2 x^2\right )^3 \, dx-\frac{1}{35} \left (12 a c^3\right ) \int x (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x) \, dx\\ &=-\frac{12 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{175 a}+\frac{2 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)}{49 a}+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(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}{35} \left (16 c^3\right ) \int \cosh ^{-1}(a x)^2 \, dx+\frac{1}{35} \left (16 a c^3\right ) \int x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-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{16 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{105 a}-\frac{12 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{175 a}+\frac{2 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)}{49 a}+\frac{16}{35} c^3 x \cosh ^{-1}(a x)^2+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(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 a c^3\right ) \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{962 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} \cosh ^{-1}(a x)}{35 a}+\frac{16 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{105 a}-\frac{12 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{175 a}+\frac{2 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)}{49 a}+\frac{16}{35} c^3 x \cosh ^{-1}(a x)^2+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^2+\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} \cosh ^{-1}(a x)}{35 a}+\frac{16 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{105 a}-\frac{12 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{175 a}+\frac{2 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)}{49 a}+\frac{16}{35} c^3 x \cosh ^{-1}(a x)^2+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^2\\ \end{align*}

Mathematica [A]  time = 0.280084, size = 125, normalized size = 0.47 \[ \frac{c^3 \left (-2250 a^7 x^7+14742 a^5 x^5-52990 a^3 x^3-11025 a x \left (5 a^6 x^6-21 a^4 x^4+35 a^2 x^2-35\right ) \cosh ^{-1}(a x)^2+210 \sqrt{a x-1} \sqrt{a x+1} \left (75 a^6 x^6-351 a^4 x^4+757 a^2 x^2-2161\right ) \cosh ^{-1}(a x)+453810 a x\right )}{385875 a} \]

Antiderivative was successfully verified.

[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]  time = 0.148, size = 188, normalized size = 0.7 \begin{align*} -{\frac{{c}^{3}}{385875\,a} \left ( 55125\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}{a}^{7}{x}^{7}-15750\,{\rm arccosh} \left (ax\right )\sqrt{ax-1}\sqrt{ax+1}{a}^{6}{x}^{6}-231525\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}{a}^{5}{x}^{5}+73710\,{\rm arccosh} \left (ax\right ){a}^{4}{x}^{4}\sqrt{ax-1}\sqrt{ax+1}+2250\,{a}^{7}{x}^{7}+385875\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}{a}^{3}{x}^{3}-158970\,{\rm arccosh} \left (ax\right )\sqrt{ax-1}\sqrt{ax+1}{a}^{2}{x}^{2}-14742\,{x}^{5}{a}^{5}-385875\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}ax+453810\,{\rm arccosh} \left (ax\right )\sqrt{ax-1}\sqrt{ax+1}+52990\,{x}^{3}{a}^{3}-453810\,ax \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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*ar
ccosh(a*x)^2*a^5*x^5+73710*arccosh(a*x)*a^4*x^4*(a*x-1)^(1/2)*(a*x+1)^(1/2)+2250*a^7*x^7+385875*arccosh(a*x)^2
*a^3*x^3-158970*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^2*x^2-14742*x^5*a^5-385875*arccosh(a*x)^2*a*x+45381
0*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+52990*x^3*a^3-453810*a*x)

________________________________________________________________________________________

Maxima [A]  time = 1.20855, size = 240, normalized size = 0.9 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [A]  time = 2.03693, size = 416, normalized size = 1.56 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [A]  time = 15.0503, size = 243, normalized size = 0.91 \begin{align*} \begin{cases} - \frac{a^{6} c^{3} x^{7} \operatorname{acosh}^{2}{\left (a x \right )}}{7} - \frac{2 a^{6} c^{3} x^{7}}{343} + \frac{2 a^{5} c^{3} x^{6} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{49} + \frac{3 a^{4} c^{3} x^{5} \operatorname{acosh}^{2}{\left (a x \right )}}{5} + \frac{234 a^{4} c^{3} x^{5}}{6125} - \frac{234 a^{3} c^{3} x^{4} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{1225} - a^{2} c^{3} x^{3} \operatorname{acosh}^{2}{\left (a x \right )} - \frac{1514 a^{2} c^{3} x^{3}}{11025} + \frac{1514 a c^{3} x^{2} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{3675} + c^{3} x \operatorname{acosh}^{2}{\left (a x \right )} + \frac{4322 c^{3} x}{3675} - \frac{4322 c^{3} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{3675 a} & \text{for}\: a \neq 0 \\- \frac{\pi ^{2} c^{3} x}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**6*c**3*x**7*acosh(a*x)**2/7 - 2*a**6*c**3*x**7/343 + 2*a**5*c**3*x**6*sqrt(a**2*x**2 - 1)*acosh
(a*x)/49 + 3*a**4*c**3*x**5*acosh(a*x)**2/5 + 234*a**4*c**3*x**5/6125 - 234*a**3*c**3*x**4*sqrt(a**2*x**2 - 1)
*acosh(a*x)/1225 - a**2*c**3*x**3*acosh(a*x)**2 - 1514*a**2*c**3*x**3/11025 + 1514*a*c**3*x**2*sqrt(a**2*x**2
- 1)*acosh(a*x)/3675 + c**3*x*acosh(a*x)**2 + 4322*c**3*x/3675 - 4322*c**3*sqrt(a**2*x**2 - 1)*acosh(a*x)/(367
5*a), Ne(a, 0)), (-pi**2*c**3*x/4, True))

________________________________________________________________________________________

Giac [A]  time = 1.18543, size = 227, normalized size = 0.85 \begin{align*} -\frac{2}{385875} \,{\left (1125 \, a^{6} x^{7} - 7371 \, a^{4} x^{5} + 26495 \, a^{2} x^{3} - 226905 \, x - \frac{105 \,{\left (75 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{7}{2}} - 126 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 280 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} - 1680 \, \sqrt{a^{2} x^{2} - 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{a}\right )} c^{3} - \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 )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/385875*(1125*a^6*x^7 - 7371*a^4*x^5 + 26495*a^2*x^3 - 226905*x - 105*(75*(a^2*x^2 - 1)^(7/2) - 126*(a^2*x^2
- 1)^(5/2) + 280*(a^2*x^2 - 1)^(3/2) - 1680*sqrt(a^2*x^2 - 1))*log(a*x + sqrt(a^2*x^2 - 1))/a)*c^3 - 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)*log(a*x + sqrt(a^2*x^2 - 1))^2