Integrand size = 29, antiderivative size = 328 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\frac {b^2 c^2 (1-c x) (1+c x)}{3 x \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {2 c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x}-\frac {4 b c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}} \]
1/3*b^2*c^2*(-c*x+1)*(c*x+1)/x/(-c^2*d*x^2+d)^(1/2)+1/3*b*c*(a+b*arccosh(c *x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/x^2/(-c^2*d*x^2+d)^(1/2)-2/3*c^3*(a+b*arc cosh(c*x))^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)-4/3*b*c^3*(a +b*arccosh(c*x))*ln(1+1/(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*(c*x-1)^(1/2) *(c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+2/3*b^2*c^3*polylog(2,-1/(c*x+(c*x-1)^ (1/2)*(c*x+1)^(1/2))^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)-1 /3*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/d/x^3-2/3*c^2*(a+b*arccosh(c* x))^2*(-c^2*d*x^2+d)^(1/2)/d/x
Time = 0.99 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\frac {-a^2-a^2 c^2 x^2+b^2 c^2 x^2+2 a^2 c^4 x^4-b^2 c^4 x^4+a b c x \sqrt {-1+c x} \sqrt {1+c x}-b^2 (1+c x) \left (1-c x+2 c^2 x^2+2 c^3 x^3 \left (-1+\sqrt {\frac {-1+c x}{1+c x}}\right )\right ) \text {arccosh}(c x)^2+b (1+c x) \text {arccosh}(c x) \left (b c x \sqrt {\frac {-1+c x}{1+c x}}+2 a \left (-1+c x-2 c^2 x^2+2 c^3 x^3\right )-4 b c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-4 a b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \log \left (-1+\sqrt {1+c x}\right )-4 a b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \log \left (1+\sqrt {1+c x}\right )+2 b^2 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{3 x^3 \sqrt {d-c^2 d x^2}} \]
(-a^2 - a^2*c^2*x^2 + b^2*c^2*x^2 + 2*a^2*c^4*x^4 - b^2*c^4*x^4 + a*b*c*x* Sqrt[-1 + c*x]*Sqrt[1 + c*x] - b^2*(1 + c*x)*(1 - c*x + 2*c^2*x^2 + 2*c^3* x^3*(-1 + Sqrt[(-1 + c*x)/(1 + c*x)]))*ArcCosh[c*x]^2 + b*(1 + c*x)*ArcCos h[c*x]*(b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)] + 2*a*(-1 + c*x - 2*c^2*x^2 + 2*c ^3*x^3) - 4*b*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Log[1 + E^(-2*ArcCosh[c*x ])]) - 4*a*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[-1 + Sqrt[1 + c*x]] - 4*a*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 + Sqrt[1 + c*x]] + 2*b^ 2*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, -E^(-2*ArcCosh[c *x])])/(3*x^3*Sqrt[d - c^2*d*x^2])
Result contains complex when optimal does not.
Time = 1.45 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.80, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {6347, 6298, 106, 6332, 6297, 25, 3042, 26, 4201, 2620, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx\) |
| \(\Big \downarrow \) 6347 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \sqrt {d-c^2 d x^2}}dx-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x^3}dx}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \sqrt {d-c^2 d x^2}}dx-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} b c \int \frac {1}{x^2 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {a+b \text {arccosh}(c x)}{2 x^2}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 106 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \sqrt {d-c^2 d x^2}}dx-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}-\frac {a+b \text {arccosh}(c x)}{2 x^2}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 6332 |
| \(\displaystyle \frac {2}{3} c^2 \left (-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{x}dx}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}-\frac {a+b \text {arccosh}(c x)}{2 x^2}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 6297 |
| \(\displaystyle \frac {2}{3} c^2 \left (-\frac {2 c \sqrt {c x-1} \sqrt {c x+1} \int -\left ((a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )d(a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}-\frac {a+b \text {arccosh}(c x)}{2 x^2}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{3} c^2 \left (\frac {2 c \sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )d(a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}-\frac {a+b \text {arccosh}(c x)}{2 x^2}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} c^2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}+\frac {2 c \sqrt {c x-1} \sqrt {c x+1} \int -i (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}-\frac {a+b \text {arccosh}(c x)}{2 x^2}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {2}{3} c^2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}-\frac {2 i c \sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}-\frac {a+b \text {arccosh}(c x)}{2 x^2}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {2}{3} c^2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}-\frac {2 i c \sqrt {c x-1} \sqrt {c x+1} \left (2 i \int \frac {e^{-2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1+e^{-2 \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{\sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}-\frac {a+b \text {arccosh}(c x)}{2 x^2}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {2}{3} c^2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}-\frac {2 i c \sqrt {c x-1} \sqrt {c x+1} \left (2 i \left (\frac {1}{2} b \int \log \left (1+e^{-2 \text {arccosh}(c x)}\right )d(a+b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{\sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}-\frac {a+b \text {arccosh}(c x)}{2 x^2}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2}{3} c^2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}-\frac {2 i c \sqrt {c x-1} \sqrt {c x+1} \left (2 i \left (-\frac {1}{4} b^2 \int e^{2 \text {arccosh}(c x)} \log \left (1+e^{-2 \text {arccosh}(c x)}\right )de^{-2 \text {arccosh}(c x)}-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{\sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}-\frac {a+b \text {arccosh}(c x)}{2 x^2}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {2}{3} c^2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}-\frac {2 i c \sqrt {c x-1} \sqrt {c x+1} \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{\sqrt {d-c^2 d x^2}}\right )-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x}-\frac {a+b \text {arccosh}(c x)}{2 x^2}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 d x^3}\) |
-1/3*(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(d*x^3) - (2*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*x) - (a + b*A rcCosh[c*x])/(2*x^2)))/(3*Sqrt[d - c^2*d*x^2]) + (2*c^2*(-((Sqrt[d - c^2*d *x^2]*(a + b*ArcCosh[c*x])^2)/(d*x)) - ((2*I)*c*Sqrt[-1 + c*x]*Sqrt[1 + c* x]*((-1/2*I)*(a + b*ArcCosh[c*x])^2 + (2*I)*(-1/2*(b*(a + b*ArcCosh[c*x])* Log[1 + E^(-2*ArcCosh[c*x])]) + (b^2*PolyLog[2, -a - b*ArcCosh[c*x]])/4))) /Sqrt[d - c^2*d*x^2]))/3
3.3.3.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0] && NeQ[m, -1]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[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]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + Simp[b*c*(n/(f*(m + 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, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3 , 0] && NeQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1 ))) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] + Simp [b*c*(n/(f*(m + 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] && ILtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1520\) vs. \(2(310)=620\).
Time = 1.21 (sec) , antiderivative size = 1521, normalized size of antiderivative = 4.64
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1521\) |
| parts | \(\text {Expression too large to display}\) | \(1521\) |
2/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*x^2-1)*p olylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*c^3+a^2*(-1/3/d/x^3*(-c^2*d *x^2+d)^(1/2)-2/3*c^2/d/x*(-c^2*d*x^2+d)^(1/2))-1/3*b^2*(-d*(c^2*x^2-1))^( 1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^3+2/3*b^2*(-d *(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^5*c^8+1/3*b^2*(-d*(c^2*x^2 -1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^3*c^6-2/3*b^2*(-d*(c^2*x^2-1))^(1/2 )/d/(3*c^4*x^4-2*c^2*x^2-1)*x*c^4-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4* x^4-2*c^2*x^2-1)/x*c^2+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x ^2-1)/x^3*arccosh(c*x)^2-4/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x +1)^(1/2)/d/(c^2*x^2-1)*arccosh(c*x)^2*c^3-4/3*b^2*(-d*(c^2*x^2-1))^(1/2)/ d/(3*c^4*x^4-2*c^2*x^2-1)*x^3*arccosh(c*x)*(c*x-1)*(c*x+1)*c^6+2*b^2*(-d*( c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^2*arccosh(c*x)^2*(c*x-1)^(1/ 2)*(c*x+1)^(1/2)*c^5-2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2 -1)*x*arccosh(c*x)*(c*x-1)*(c*x+1)*c^4-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(3 *c^4*x^4-2*c^2*x^2-1)/x^2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c+4/3*b ^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*x^2-1)*arccos h(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*c^3+4/3*b^2*(-d*(c^2*x^2- 1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^5*arccosh(c*x)*c^8-2*b^2*(-d*(c^2*x^ 2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^3*arccosh(c*x)^2*c^6-2/3*b^2*(-d*( c^2*x^2-1))^(1/2)/d/(3*c^4*x^4-2*c^2*x^2-1)*x^3*arccosh(c*x)*c^6+1/3*b^...
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d} x^{4}} \,d x } \]
integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)/(c^2*d*x^6 - d*x^4), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x^{4} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d} x^{4}} \,d x } \]
1/3*(4*c^2*sqrt(-d)*log(x)/d - sqrt(-d)/(d*x^2))*a*b*c - 2/3*a*b*(2*sqrt(- c^2*d*x^2 + d)*c^2/(d*x) + sqrt(-c^2*d*x^2 + d)/(d*x^3))*arccosh(c*x) - 1/ 3*a^2*(2*sqrt(-c^2*d*x^2 + d)*c^2/(d*x) + sqrt(-c^2*d*x^2 + d)/(d*x^3)) + b^2*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/(sqrt(-c^2*d*x^2 + d)*x^4), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d} x^{4}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{x^4\,\sqrt {d-c^2\,d\,x^2}} \,d x \]