\(\int \frac {x^5 (a+b \text {arccosh}(c x))^2}{(d-c^2 d x^2)^{5/2}} \, dx\) [214]

Optimal result

Integrand size = 29, antiderivative size = 568 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^3}-\frac {22 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {11 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}} \]

[Out]

1/3*x^4*(a+b*arccosh(c*x))^2/c^2/d/(-c^2*d*x^2+d)^(3/2)-1/3*b^2*x^2/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-7/3*b^2*(-c*x
+1)*(c*x+1)/c^6/d^2/(-c^2*d*x^2+d)^(1/2)-4/3*x^2*(a+b*arccosh(c*x))^2/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-16/3*a*b*x*
(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5/d^2/(-c^2*d*x^2+d)^(1/2)-16/3*b^2*x*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c
^5/d^2/(-c^2*d*x^2+d)^(1/2)+11/3*b*x*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5/d^2/(-c^2*d*x^2+d)^(1/
2)+1/3*b*x^3*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3/d^2/(-c^2*x^2+1)/(-c^2*d*x^2+d)^(1/2)-22/3*b*(
a+b*arccosh(c*x))*arctanh(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^6/d^2/(-c^2*d*x^2+d)^
(1/2)-11/3*b^2*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^6/d^2/(-c^2*d*x^2+d)^
(1/2)+11/3*b^2*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^6/d^2/(-c^2*d*x^2+d)^(
1/2)-8/3*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^6/d^3

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {5934, 5914, 5879, 75, 5912, 5938, 5903, 4267, 2317, 2438, 100, 21} \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {22 b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^3}+\frac {11 b x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {c x-1} \sqrt {c x+1}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {11 b^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 (1-c x) (c x+1)}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}} \]

[In]

Int[(x^5*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]

[Out]

-1/3*(b^2*x^2)/(c^4*d^2*Sqrt[d - c^2*d*x^2]) - (16*a*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^5*d^2*Sqrt[d - c^2
*d*x^2]) - (7*b^2*(1 - c*x)*(1 + c*x))/(3*c^6*d^2*Sqrt[d - c^2*d*x^2]) - (16*b^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x
]*ArcCosh[c*x])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (11*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(
3*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c^3*d^2*(1 - c^2
*x^2)*Sqrt[d - c^2*d*x^2]) + (x^4*(a + b*ArcCosh[c*x])^2)/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (4*x^2*(a + b*ArcC
osh[c*x])^2)/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (8*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(3*c^6*d^3) - (2
2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(3*c^6*d^2*Sqrt[d - c^2*d*x^2])
- (11*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(3*c^6*d^2*Sqrt[d - c^2*d*x^2]) + (11*b^2
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/(3*c^6*d^2*Sqrt[d - c^2*d*x^2])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 75

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

Rule 100

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

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 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 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 5903

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

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 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 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^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{3 c^2 d}+\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^4 (a+b \text {arccosh}(c x))}{(-1+c x)^2 (1+c x)^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{3 c^4 d^2}+\frac {\left (8 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2 (a+b \text {arccosh}(c x))}{(-1+c x) (1+c x)} \, dx}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (-1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^3}-\frac {\left (16 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int (a+b \text {arccosh}(c x)) \, dx}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2 (a+b \text {arccosh}(c x))}{-1+c^2 x^2} \, dx}{c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2 (a+b \text {arccosh}(c x))}{-1+c^2 x^2} \, dx}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 c^2 d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^3}+\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{-1+c^2 x^2} \, dx}{c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{-1+c^2 x^2} \, dx}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \text {arccosh}(c x) \, dx}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x (-2-2 c x)}{\sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c^4 d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^3}+\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arccosh}(c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arccosh}(c x))}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c^4 d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^3}-\frac {22 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^3}-\frac {22 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \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^{\text {arccosh}(c x)}\right )}{c^6 d^2 \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^{\text {arccosh}(c x)}\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^3}-\frac {22 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {11 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 4.39 (sec) , antiderivative size = 490, normalized size of antiderivative = 0.86 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {8 a^2 \left (8-12 c^2 x^2+3 c^4 x^4\right )+2 a b \left (25 \text {arccosh}(c x)-36 \text {arccosh}(c x) \cosh (2 \text {arccosh}(c x))+3 \text {arccosh}(c x) \cosh (4 \text {arccosh}(c x))+33 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-33 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )+4 \sinh (2 \text {arccosh}(c x))-11 \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right ) \sinh (3 \text {arccosh}(c x))+11 \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right ) \sinh (3 \text {arccosh}(c x))-3 \sinh (4 \text {arccosh}(c x))\right )+b^2 \left (22+25 \text {arccosh}(c x)^2-4 \left (7+9 \text {arccosh}(c x)^2\right ) \cosh (2 \text {arccosh}(c x))+3 \left (2+\text {arccosh}(c x)^2\right ) \cosh (4 \text {arccosh}(c x))-66 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1-e^{-\text {arccosh}(c x)}\right )+66 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1+e^{-\text {arccosh}(c x)}\right )+88 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(c x)}\right )-88 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \operatorname {PolyLog}\left (2,e^{-\text {arccosh}(c x)}\right )+8 \text {arccosh}(c x) \sinh (2 \text {arccosh}(c x))+22 \text {arccosh}(c x) \log \left (1-e^{-\text {arccosh}(c x)}\right ) \sinh (3 \text {arccosh}(c x))-22 \text {arccosh}(c x) \log \left (1+e^{-\text {arccosh}(c x)}\right ) \sinh (3 \text {arccosh}(c x))-6 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )}{24 c^6 d \left (d-c^2 d x^2\right )^{3/2}} \]

[In]

Integrate[(x^5*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]

[Out]

-1/24*(8*a^2*(8 - 12*c^2*x^2 + 3*c^4*x^4) + 2*a*b*(25*ArcCosh[c*x] - 36*ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] + 3*
ArcCosh[c*x]*Cosh[4*ArcCosh[c*x]] + 33*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Cosh[ArcCosh[c*x]/2]] - 33*Sqr
t[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Sinh[ArcCosh[c*x]/2]] + 4*Sinh[2*ArcCosh[c*x]] - 11*Log[Cosh[ArcCosh[c*x
]/2]]*Sinh[3*ArcCosh[c*x]] + 11*Log[Sinh[ArcCosh[c*x]/2]]*Sinh[3*ArcCosh[c*x]] - 3*Sinh[4*ArcCosh[c*x]]) + b^2
*(22 + 25*ArcCosh[c*x]^2 - 4*(7 + 9*ArcCosh[c*x]^2)*Cosh[2*ArcCosh[c*x]] + 3*(2 + ArcCosh[c*x]^2)*Cosh[4*ArcCo
sh[c*x]] - 66*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])] + 66*Sqrt[(-1 + c*x
)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] + 88*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*P
olyLog[2, -E^(-ArcCosh[c*x])] - 88*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*PolyLog[2, E^(-ArcCosh[c*x])] + 8*
ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]] + 22*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])]*Sinh[3*ArcCosh[c*x]] - 22*ArcCo
sh[c*x]*Log[1 + E^(-ArcCosh[c*x])]*Sinh[3*ArcCosh[c*x]] - 6*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]))/(c^6*d*(d - c^
2*d*x^2)^(3/2))

Maple [A] (verified)

Time = 1.46 (sec) , antiderivative size = 922, normalized size of antiderivative = 1.62

method	result	size
default	\(a^{2} \left (-\frac {x^{4}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {\frac {4 x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {8}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}}{c^{2}}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}-2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}+2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}+\sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x +c^{2} x^{2}-5 \operatorname {arccosh}\left (c x \right )^{2}-1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{6}}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}\right )-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (6 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-6 c^{5} x^{5}-11 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{4} x^{4}+11 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) c^{4} x^{4}-24 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+11 c^{3} x^{3}+22 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-22 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}+16 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-5 c x -11 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+11 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )\right )}{3 c^{6} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) d^{3}}\)	\(922\)
parts	\(a^{2} \left (-\frac {x^{4}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {\frac {4 x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {8}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}}{c^{2}}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}-2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}+2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}+\sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x +c^{2} x^{2}-5 \operatorname {arccosh}\left (c x \right )^{2}-1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{6}}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}\right )-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (6 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-6 c^{5} x^{5}-11 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{4} x^{4}+11 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) c^{4} x^{4}-24 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+11 c^{3} x^{3}+22 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-22 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}+16 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-5 c x -11 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+11 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )\right )}{3 c^{6} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) d^{3}}\)	\(922\)

[In]

int(x^5*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

a^2*(-x^4/c^2/d/(-c^2*d*x^2+d)^(3/2)+4/c^2*(x^2/c^2/d/(-c^2*d*x^2+d)^(3/2)-2/3/d/c^4/(-c^2*d*x^2+d)^(3/2)))+b^
2*(-1/2*(-d*(c^2*x^2-1))^(1/2)*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(arccosh(c*x)^2-2*arccosh(c*x)+2)/d
^3/c^6/(c^2*x^2-1)-1/2*(-d*(c^2*x^2-1))^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(arccosh(c*x)^2+2*a
rccosh(c*x)+2)/d^3/c^6/(c^2*x^2-1)+1/3*(-d*(c^2*x^2-1))^(1/2)*(6*arccosh(c*x)^2*x^2*c^2+(c*x+1)^(1/2)*arccosh(
c*x)*(c*x-1)^(1/2)*c*x+c^2*x^2-5*arccosh(c*x)^2-1)/(c^2*x^2-1)^2/d^3/c^6+11/3*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(
1/2)*(c*x+1)^(1/2)/d^3/c^6/(c^2*x^2-1)*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+11/3*(-d*(c^2*x^2-1)
)^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c^6/(c^2*x^2-1)*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-11/3*(-d*(
c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c^6/(c^2*x^2-1)*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^
(1/2))-11/3*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c^6/(c^2*x^2-1)*polylog(2,c*x+(c*x-1)^(1/2)
*(c*x+1)^(1/2)))-1/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(6*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arc
cosh(c*x)*c^4*x^4-6*c^5*x^5-11*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*c^4*x^4+11*ln((c*x-1)^(1/2)*(c*x+1)^(1/2)
+c*x-1)*c^4*x^4-24*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c^2*x^2+11*c^3*x^3+22*ln(1+c*x+(c*x-1)^(1/2)*(c*x+
1)^(1/2))*x^2*c^2-22*ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)*x^2*c^2+16*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)
-5*c*x-11*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+11*ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1))/c^6/(c^6*x^6-3*c^4*x
^4+3*c^2*x^2-1)/d^3

Fricas [F]

\[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^5*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(-(b^2*x^5*arccosh(c*x)^2 + 2*a*b*x^5*arccosh(c*x) + a^2*x^5)*sqrt(-c^2*d*x^2 + d)/(c^6*d^3*x^6 - 3*c^
4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(x**5*(a+b*acosh(c*x))**2/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^5*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

-1/3*a^2*(3*x^4/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 12*x^2/((-c^2*d*x^2 + d)^(3/2)*c^4*d) + 8/((-c^2*d*x^2 + d)^(
3/2)*c^6*d)) - 1/3*(3*b^2*c^4*sqrt(d)*x^4 - 12*b^2*c^2*sqrt(d)*x^2 + 8*b^2*sqrt(d))*sqrt(c*x + 1)*sqrt(-c*x +
1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/(c^10*d^3*x^4 - 2*c^8*d^3*x^2 + c^6*d^3) - integrate(2/3*((12*b^2*
c^3*x^3 + 3*(a*b*c^5 - b^2*c^5)*x^5 - 8*b^2*c*x)*(c*x + 1)*sqrt(c*x - 1) + (15*b^2*c^4*x^4 + 3*(a*b*c^6 - b^2*
c^6)*x^6 - 20*b^2*c^2*x^2 + 8*b^2)*sqrt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c^12*
d^(5/2)*x^7 - 3*c^10*d^(5/2)*x^5 + 3*c^8*d^(5/2)*x^3 - c^6*d^(5/2)*x + (c^11*d^(5/2)*x^6 - 3*c^9*d^(5/2)*x^4 +
3*c^7*d^(5/2)*x^2 - c^5*d^(5/2))*sqrt(c*x + 1)*sqrt(c*x - 1)), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^5*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),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. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^5\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]

[In]

int((x^5*(a + b*acosh(c*x))^2)/(d - c^2*d*x^2)^(5/2),x)

[Out]

int((x^5*(a + b*acosh(c*x))^2)/(d - c^2*d*x^2)^(5/2), x)