\(\int \frac {(a+b \text {arccosh}(c x))^2}{x^3 (d-c^2 d x^2)^{3/2}} \, dx\) [212]

Optimal result

Integrand size = 29, antiderivative size = 650 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \text {arccosh}(c x))^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2 \arctan \left (e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {4 b c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}} \]

[Out]

3/2*c^2*(a+b*arccosh(c*x))^2/d/(-c^2*d*x^2+d)^(1/2)-1/2*(a+b*arccosh(c*x))^2/d/x^2/(-c^2*d*x^2+d)^(1/2)+b*c*(a
+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/x/(-c^2*d*x^2+d)^(1/2)+3*c^2*(a+b*arccosh(c*x))^2*arctan(c*x+(c
*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)-b^2*c^2*arctan((c*x-1)^(1/2)*(c*
x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)+4*b*c^2*(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)/d/(-c^2*d*x^2+d)^(1/2)+2*b^2*c^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)/d/(-c^2*d*x^2+d)^(1/2)-3*I*b*c^2*(a+b*arccosh(c*x))*polylog(2,-I*(
c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)+3*I*b*c^2*(a+b*arccosh(c*
x))*polylog(2,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)-2*b^2*c^
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)/d/(-c^2*d*x^2+d)^(1/2)+3*I*b^2*c^2*po
lylog(3,-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)-3*I*b^2*c^2*p
olylog(3,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 650, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {5932, 5936, 5946, 4265, 2611, 2320, 6724, 5889, 5903, 4267, 2317, 2438, 5912, 94, 211} \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {3 c^2 \sqrt {c x-1} \sqrt {c x+1} \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {4 b c^2 \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \text {arccosh}(c x))^2}{2 d \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{d x \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b^2 c^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b^2 c^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {c x-1} \sqrt {c x+1} \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )}{d \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

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

Rule 94

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

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 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 4265

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

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 5889

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

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 5932

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] + (Dist[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] + Dist[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*ArcCos
h[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]

Rule 5936

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/(2*d*f*(p + 1))), x] + (Dist[(m + 2*p + 3)/(2*d*(
p + 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*c*(n/(2*f*(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*ArcCo
sh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &
&  !GtQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])

Rule 5946

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

Rule 6724

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

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(5362\) vs. \(2(650)=1300\).

Time = 63.67 (sec) , antiderivative size = 5362, normalized size of antiderivative = 8.25 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

integrate((a+b*arccosh(c*x))^2/x^3/(-c^2*d*x^2+d)^(3/2),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^4*d^2*x^7 - 2*c^2*d^2*x^5 + d
^2*x^3), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate((b*arccosh(c*x) + a)^2/((-c^2*d*x^2 + d)^(3/2)*x^3), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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