\(\int \frac {x^4 (a+b \text {arccosh}(c x))^2}{(d-c^2 d x^2)^{3/2}} \, dx\) [205]

Optimal result

Integrand size = 29, antiderivative size = 440 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {b^2 x (1-c x) (1+c x)}{4 c^4 d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{4 c^5 d \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^4 d^2}-\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^3}{2 b c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}} \]

[Out]

1/4*b^2*x*(-c*x+1)*(c*x+1)/c^4/d/(-c^2*d*x^2+d)^(1/2)+x^3*(a+b*arccosh(c*x))^2/c^2/d/(-c^2*d*x^2+d)^(1/2)-1/4*
b^2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5/d/(-c^2*d*x^2+d)^(1/2)+1/2*b*x^2*(a+b*arccosh(c*x))*(c*x-1)^(
1/2)*(c*x+1)^(1/2)/c^3/d/(-c^2*d*x^2+d)^(1/2)+(a+b*arccosh(c*x))^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5/d/(-c^2*d*x
^2+d)^(1/2)-1/2*(a+b*arccosh(c*x))^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c^5/d/(-c^2*d*x^2+d)^(1/2)-2*b*(a+b*arccosh
(c*x))*ln(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^5/d/(-c^2*d*x^2+d)^(1/2)-b^2*po
lylog(2,(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5/d/(-c^2*d*x^2+d)^(1/2)+3/2*x*(a+b
*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^4/d^2

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {5934, 5938, 5892, 5883, 92, 54, 5912, 5913, 3797, 2221, 2317, 2438} \[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^3}{2 b c^5 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^4 d^2}+\frac {b x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c^3 d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{4 c^5 d \sqrt {d-c^2 d x^2}}+\frac {b^2 x (1-c x) (c x+1)}{4 c^4 d \sqrt {d-c^2 d x^2}} \]

[In]

Int[(x^4*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]

[Out]

(b^2*x*(1 - c*x)*(1 + c*x))/(4*c^4*d*Sqrt[d - c^2*d*x^2]) - (b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(4
*c^5*d*Sqrt[d - c^2*d*x^2]) + (b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2*c^3*d*Sqrt[d - c^2*
d*x^2]) + (x^3*(a + b*ArcCosh[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcC
osh[c*x])^2)/(c^5*d*Sqrt[d - c^2*d*x^2]) + (3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*c^4*d^2) - (Sqr
t[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3)/(2*b*c^5*d*Sqrt[d - c^2*d*x^2]) - (2*b*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]*(a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(c^5*d*Sqrt[d - c^2*d*x^2]) - (b^2*Sqrt[-1 + c*x]*Sq
rt[1 + c*x]*PolyLog[2, E^(2*ArcCosh[c*x])])/(c^5*d*Sqrt[d - c^2*d*x^2])

Rule 54

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[b*(x/a)]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 92

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a + b*x
)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 2221

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]
- Dist[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]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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]

Rule 2438

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]

Rule 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 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 5892

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

Rule 5912

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

Rule 5913

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

Rule 5934

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

Rubi steps \begin{align*} \text {integral}& = \frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{c^2 d}-\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3 (a+b \text {arccosh}(c x))}{(-1+c x) (1+c x)} \, dx}{c d \sqrt {d-c^2 d x^2}} \\ & = \frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^4 d^2}-\frac {3 \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{2 c^4 d}+\frac {\left (3 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x (a+b \text {arccosh}(c x)) \, dx}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3 (a+b \text {arccosh}(c x))}{-1+c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}} \\ & = \frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^4 d^2}-\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^3}{2 b c^5 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x (a+b \text {arccosh}(c x))}{-1+c^2 x^2} \, dx}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (3 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 c^2 d \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2 x (1-c x) (1+c x)}{4 c^4 d \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^4 d^2}-\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^3}{2 b c^5 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}(\int (a+b x) \coth (x) \, dx,x,\text {arccosh}(c x))}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (3 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 c^4 d \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2 x (1-c x) (1+c x)}{4 c^4 d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{4 c^5 d \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^4 d^2}-\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^3}{2 b c^5 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\text {arccosh}(c x)\right )}{c^5 d \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2 x (1-c x) (1+c x)}{4 c^4 d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{4 c^5 d \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^4 d^2}-\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^3}{2 b c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{c^5 d \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2 x (1-c x) (1+c x)}{4 c^4 d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{4 c^5 d \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^4 d^2}-\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^3}{2 b c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arccosh}(c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2 x (1-c x) (1+c x)}{4 c^4 d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{4 c^5 d \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^4 d^2}-\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^3}{2 b c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 1.69 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.78 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-4 a^2 c \sqrt {d} x \left (-3+c^2 x^2\right )+12 a^2 \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+2 a b \sqrt {d} \left (8 c x \text {arccosh}(c x)-\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (6 \text {arccosh}(c x)^2-\cosh (2 \text {arccosh}(c x))+8 \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )+2 \text {arccosh}(c x) \sinh (2 \text {arccosh}(c x))\right )\right )+b^2 \sqrt {d} \left (8 c x \text {arccosh}(c x)^2+8 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,e^{-2 \text {arccosh}(c x)}\right )-\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (4 \text {arccosh}(c x)^3-2 \text {arccosh}(c x) \left (\cosh (2 \text {arccosh}(c x))-8 \log \left (1-e^{-2 \text {arccosh}(c x)}\right )\right )+\sinh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x)^2 (4+\sinh (2 \text {arccosh}(c x)))\right )\right )}{8 c^5 d^{3/2} \sqrt {d-c^2 d x^2}} \]

[In]

Integrate[(x^4*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]

[Out]

(-4*a^2*c*Sqrt[d]*x*(-3 + c^2*x^2) + 12*a^2*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1
+ c^2*x^2))] + 2*a*b*Sqrt[d]*(8*c*x*ArcCosh[c*x] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(6*ArcCosh[c*x]^2 - Co
sh[2*ArcCosh[c*x]] + 8*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)] + 2*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]])) + b^2
*Sqrt[d]*(8*c*x*ArcCosh[c*x]^2 + 8*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, E^(-2*ArcCosh[c*x])] - Sqrt
[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(4*ArcCosh[c*x]^3 - 2*ArcCosh[c*x]*(Cosh[2*ArcCosh[c*x]] - 8*Log[1 - E^(-2*Ar
cCosh[c*x])]) + Sinh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]^2*(4 + Sinh[2*ArcCosh[c*x]]))))/(8*c^5*d^(3/2)*Sqrt[d -
c^2*d*x^2])

Maple [A] (verified)

Time = 1.46 (sec) , antiderivative size = 756, normalized size of antiderivative = 1.72

method	result	size
default	\(-\frac {a^{2} x^{3}}{2 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 a^{2} x}{2 c^{4} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {3 a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{4} d \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-2 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+2 \operatorname {arccosh}\left (c x \right )^{3} x^{2} c^{2}-6 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x c -4 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}+8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+3 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+8 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+8 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\, c x -2 \operatorname {arccosh}\left (c x \right )^{3}+4 \operatorname {arccosh}\left (c x \right )^{2}-8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right )-8 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-8 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{4 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{5}}+\frac {a b \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{3} x^{3}-2 c^{4} x^{4}+6 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-12 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x -8 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{2} c^{2}+3 c^{2} x^{2}-6 \operatorname {arccosh}\left (c x \right )^{2}+8 \,\operatorname {arccosh}\left (c x \right )-8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )-1\right )}{4 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{5}}\)	\(756\)
parts	\(-\frac {a^{2} x^{3}}{2 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 a^{2} x}{2 c^{4} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {3 a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{4} d \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-2 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+2 \operatorname {arccosh}\left (c x \right )^{3} x^{2} c^{2}-6 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x c -4 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}+8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+3 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+8 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+8 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\, c x -2 \operatorname {arccosh}\left (c x \right )^{3}+4 \operatorname {arccosh}\left (c x \right )^{2}-8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-8 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right )-8 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-8 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{4 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{5}}+\frac {a b \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{3} x^{3}-2 c^{4} x^{4}+6 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-12 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x -8 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{2} c^{2}+3 c^{2} x^{2}-6 \operatorname {arccosh}\left (c x \right )^{2}+8 \,\operatorname {arccosh}\left (c x \right )-8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )-1\right )}{4 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{5}}\)	\(756\)

[In]

int(x^4*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*a^2*x^3/c^2/d/(-c^2*d*x^2+d)^(1/2)+3/2*a^2/c^4*x/d/(-c^2*d*x^2+d)^(1/2)-3/2*a^2/c^4/d/(c^2*d)^(1/2)*arcta
n((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+1/4*b^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*(2*arccosh(
c*x)^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-2*c^4*x^4*arccosh(c*x)+(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+2*arccos
h(c*x)^3*x^2*c^2-6*arccosh(c*x)^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-4*arccosh(c*x)^2*x^2*c^2+8*arccosh(c*x)*ln(1
+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^2*c^2+8*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^2*c^2+3*c^2*x
^2*arccosh(c*x)+8*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^2*c^2+8*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1
/2))*x^2*c^2-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-2*arccosh(c*x)^3+4*arccosh(c*x)^2-8*arccosh(c*x)*ln(1+c*x+(c*x-1)
^(1/2)*(c*x+1)^(1/2))-8*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-arccosh(c*x)-8*polylog(2,-c*x-(c*x-
1)^(1/2)*(c*x+1)^(1/2))-8*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))/(c^2*x^2-1)^2/d^2/c^5+1/4*a*b*(c*x-1)^(1
/2)*(c*x+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*(4*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c^3*x^3-2*c^4*x^4+6*arcco
sh(c*x)^2*x^2*c^2-12*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c*x-8*c^2*x^2*arccosh(c*x)+8*ln((c*x+(c*x-1)^(1/
2)*(c*x+1)^(1/2))^2-1)*x^2*c^2+3*c^2*x^2-6*arccosh(c*x)^2+8*arccosh(c*x)-8*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)
)^2-1)-1)/(c^2*x^2-1)^2/d^2/c^5

Fricas [F]

\[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(x^4*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

integral((b^2*x^4*arccosh(c*x)^2 + 2*a*b*x^4*arccosh(c*x) + a^2*x^4)*sqrt(-c^2*d*x^2 + d)/(c^4*d^2*x^4 - 2*c^2
*d^2*x^2 + d^2), x)

Sympy [F]

\[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(x**4*(a+b*acosh(c*x))**2/(-c**2*d*x**2+d)**(3/2),x)

[Out]

Integral(x**4*(a + b*acosh(c*x))**2/(-d*(c*x - 1)*(c*x + 1))**(3/2), x)

Maxima [F]

\[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(x^4*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxima")

[Out]

-1/2*a^2*(x^3/(sqrt(-c^2*d*x^2 + d)*c^2*d) - 3*x/(sqrt(-c^2*d*x^2 + d)*c^4*d) + 3*arcsin(c*x)/(c^5*d^(3/2))) +
integrate(b^2*x^4*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/(-c^2*d*x^2 + d)^(3/2) + 2*a*b*x^4*log(c*x + sqrt(
c*x + 1)*sqrt(c*x - 1))/(-c^2*d*x^2 + d)^(3/2), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^4*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(3/2),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. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]

[In]

int((x^4*(a + b*acosh(c*x))^2)/(d - c^2*d*x^2)^(3/2),x)

[Out]

int((x^4*(a + b*acosh(c*x))^2)/(d - c^2*d*x^2)^(3/2), x)