\(\int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \sqrt {d-c^2 d x^2}} \, dx\) [202]

Optimal result

Integrand size = 29, antiderivative size = 430 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \sqrt {d-c^2 d x^2}} \, dx=\frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{x \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 d x^2}+\frac {c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2 \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{\sqrt {d-c^2 d x^2}}-\frac {i b c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {i b c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {i b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {i b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {5932, 5946, 4265, 2611, 2320, 6724, 5883, 94, 211} \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \sqrt {d-c^2 d x^2}} \, dx=\frac {c^2 \sqrt {c x-1} \sqrt {c x+1} \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}-\frac {i b c^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}}+\frac {i b c^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{x \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 d x^2}+\frac {i b^2 c^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {i b^2 c^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {c x-1} \sqrt {c x+1} \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )}{\sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(x*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x^2]*(a + b*
ArcCosh[c*x])^2)/(2*d*x^2) + (c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/
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]])/Sqrt[d - c^2
*d*x^2] - (I*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]])/Sqrt[d -
c^2*d*x^2] + (I*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[c*x]])/Sqrt[d
- c^2*d*x^2] + (I*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, (-I)*E^ArcCosh[c*x]])/Sqrt[d - c^2*d*x^2] -
(I*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, I*E^ArcCosh[c*x]])/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 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 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 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 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 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 {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 d x^2}+\frac {1}{2} c^2 \int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {d-c^2 d x^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} \, dx}{\sqrt {d-c^2 d x^2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{x \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 d x^2}+\frac {\left (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 \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}{\sqrt {d-c^2 d x^2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{x \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 d x^2}+\frac {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 )}{\sqrt {d-c^2 d x^2}}-\frac {\left (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 )}{\sqrt {d-c^2 d x^2}}+\frac {\left (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 )}{\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 )}{\sqrt {d-c^2 d x^2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{x \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 d x^2}+\frac {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 )}{\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 )}{\sqrt {d-c^2 d x^2}}-\frac {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 )}{\sqrt {d-c^2 d x^2}}+\frac {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 )}{\sqrt {d-c^2 d x^2}}+\frac {\left (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 )}{\sqrt {d-c^2 d x^2}}-\frac {\left (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 )}{\sqrt {d-c^2 d x^2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{x \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 d x^2}+\frac {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 )}{\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 )}{\sqrt {d-c^2 d x^2}}-\frac {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 )}{\sqrt {d-c^2 d x^2}}+\frac {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 )}{\sqrt {d-c^2 d x^2}}+\frac {\left (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 )}{\sqrt {d-c^2 d x^2}}-\frac {\left (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 )}{\sqrt {d-c^2 d x^2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{x \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 d x^2}+\frac {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 )}{\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 )}{\sqrt {d-c^2 d x^2}}-\frac {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 )}{\sqrt {d-c^2 d x^2}}+\frac {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 )}{\sqrt {d-c^2 d x^2}}+\frac {i b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {i b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{\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. \(5036\) vs. \(2(430)=860\).

Time = 63.66 (sec) , antiderivative size = 5036, normalized size of antiderivative = 11.71 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^3 \sqrt {d-c^2 d x^2}} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((a+b*arccosh(c*x))^2/x^3/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((a+b*arccosh(c*x))^2/x^3/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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