\(\int \frac {(a+b \text {arccosh}(c x))^2}{(d-c^2 d x^2)^{5/2}} \, dx\) [219]

Optimal result

Integrand size = 26, antiderivative size = 331 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}} \]

[Out]

1/3*x*(a+b*arccosh(c*x))^2/d/(-c^2*d*x^2+d)^(3/2)-1/3*b^2*x/d^2/(-c^2*d*x^2+d)^(1/2)+2/3*x*(a+b*arccosh(c*x))^
2/d^2/(-c^2*d*x^2+d)^(1/2)+1/3*b*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/d^2/(-c^2*x^2+1)/(-c^2*d*x^2
+d)^(1/2)+2/3*(a+b*arccosh(c*x))^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/d^2/(-c^2*d*x^2+d)^(1/2)-4/3*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/d^2/(-c^2*d*x^2+d)^(1/2)-2/3*b^2*
polylog(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/d^2/(-c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5901, 5899, 5913, 3797, 2221, 2317, 2438, 5912, 5914, 39} \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 x (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 x}{3 d^2 \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

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

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 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 5899

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

Rule 5901

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(
p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 1))), x] + (Dist[(2*p + 3)/(2*d*(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] && LtQ[p, -1] && NeQ[p, -3/2]

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 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]

Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{3 d}+\frac {\left (2 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x (a+b \text {arccosh}(c x))}{(-1+c x)^2 (1+c x)^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x (a+b \text {arccosh}(c x))}{\left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))^2}{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 {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}(\int (a+b x) \coth (x) \, dx,x,\text {arccosh}(c x))}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 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 )}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 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 )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 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 )}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 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 )}{3 c d^2 \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 \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arccosh}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 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 )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 1.64 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {\frac {a^2 c x \left (-3+2 c^2 x^2\right )}{-1+c^2 x^2}+a b \left (2 c x \left (2+\frac {1}{1-c^2 x^2}\right ) \text {arccosh}(c x)+\frac {\sqrt {\frac {-1+c x}{1+c x}} \left (-1+\left (4-4 c^2 x^2\right ) \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )\right )}{-1+c x}\right )+b^2 \left (-\frac {\text {arccosh}(c x) \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)+c x \text {arccosh}(c x)\right )}{-1+c^2 x^2}+c x \left (-1+2 \text {arccosh}(c x)^2\right )-2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1-e^{-2 \text {arccosh}(c x)}\right )\right )+2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,e^{-2 \text {arccosh}(c x)}\right )\right )}{3 c d^2 \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

((a^2*c*x*(-3 + 2*c^2*x^2))/(-1 + c^2*x^2) + a*b*(2*c*x*(2 + (1 - c^2*x^2)^(-1))*ArcCosh[c*x] + (Sqrt[(-1 + c*
x)/(1 + c*x)]*(-1 + (4 - 4*c^2*x^2)*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)]))/(-1 + c*x)) + b^2*(-((ArcCosh[
c*x]*(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) + c*x*ArcCosh[c*x]))/(-1 + c^2*x^2)) + c*x*(-1 + 2*ArcCosh[c*x]^2)
- 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 - E^(-2*ArcCosh[c*x])]) + 2*Sqrt
[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, E^(-2*ArcCosh[c*x])]))/(3*c*d^2*Sqrt[d - c^2*d*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2434\) vs. \(2(313)=626\).

Time = 1.16 (sec) , antiderivative size = 2435, normalized size of antiderivative = 7.36

method	result	size
default	\(\text {Expression too large to display}\)	\(2435\)
parts	\(\text {Expression too large to display}\)	\(2435\)

[In]

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

[Out]

2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*(c*x-1)*(c*x+1)*x-4/3*b^2*(-d*(c^2*x^2-
1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*c^6*arccosh(c*x)*x^7-2*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6
-10*c^4*x^4+11*c^2*x^2-4)/d^3*c^4*arccosh(c*x)^2*x^5+14/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*
c^2*x^2-4)/d^3*c^4*arccosh(c*x)*x^5+17/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*c^
2*arccosh(c*x)^2*x^3-16/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*c^2*arccosh(c*x)*
x^3-4/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3/c*(c*x-1)^(1/2)*(c*x+1)^(1/2)+4/3*b
^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c/(c^2*x^2-1)*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(
1/2))+2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*c^4*(c*x-1)*(c*x+1)*x^5-4/3*b^2*(
-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*c^2*(c*x-1)*(c*x+1)*x^3+2*b^2*(-d*(c^2*x^2-1))^(
1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*(c*x-1)*(c*x+1)*arccosh(c*x)*x-4/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*
x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c/(c^2*x^2-1)*arccosh(c*x)^2-4*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+1
1*c^2*x^2-4)/d^3*arccosh(c*x)^2*x+2*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*arccosh
(c*x)*x-2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*c^6*x^7+3*b^2*(-d*(c^2*x^2-1))^
(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*c^4*x^5-13/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11
*c^2*x^2-4)/d^3*c^2*x^3+2*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*x+4/3*b^2*(-d*(c^
2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c/(c^2*x^2-1)*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-b^2*(
-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*c^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^4+7/3*b^2*(-d*
(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*c*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2-4/3*b^2*(-d*(c^2*
x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3/c*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)+8/3*b^2*(-d*(
c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3/c*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)^2+a^2*(1/
3/d*x/(-c^2*d*x^2+d)^(3/2)+2/3/d^2*x/(-c^2*d*x^2+d)^(1/2))-1/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3-3*c*x+2*(
c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-2*(c*x-1)^(1/2)*(c*x+1)^(1/2))*(8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln((c*x+(c*x-
1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^5*c^5-8*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^6*c^6-20*(c*x-1)^(1/2)*(c*x
+1)^(1/2)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^3*c^3+24*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^4*c
^4+2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3-2*c^4*x^4+6*c^2*x^2*arccosh(c*x)+12*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln((c
*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x*c-24*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^2*c^2-3*(c*x-1)^(1/2)*
(c*x+1)^(1/2)*c*x+4*c^2*x^2-8*arccosh(c*x)+8*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)-2)/(3*c^6*x^6-10*c^4*x^
4+11*c^2*x^2-4)/d^3/c+2*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*c^3*(c*x-1)^(1/2)*(
c*x+1)^(1/2)*arccosh(c*x)^2*x^4-10/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*c^2*(c
*x-1)*(c*x+1)*arccosh(c*x)*x^3+4/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*c^4*(c*x
-1)*(c*x+1)*arccosh(c*x)*x^5+b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*c*(c*x-1)^(1/2
)*(c*x+1)^(1/2)*arccosh(c*x)*x^2-14/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/d^3*c*(c*
x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)^2*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^3/c/(
c^2*x^2-1)*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+4/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*
x+1)^(1/2)/d^3/c/(c^2*x^2-1)*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))

Fricas [F]

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

[In]

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

[Out]

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

\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

Integral((a + b*acosh(c*x))**2/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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