3.3.11 \(\int \frac {(a+b \cosh ^{-1}(c x))^2}{x^2 (d-c^2 d x^2)^{3/2}} \, dx\) [211]

Optimal. Leaf size=341 \[ -\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}} \]

[Out]

-(a+b*arccosh(c*x))^2/d/x/(-c^2*d*x^2+d)^(1/2)+2*c^2*x*(a+b*arccosh(c*x))^2/d/(-c^2*d*x^2+d)^(1/2)+2*c*(a+b*ar
ccosh(c*x))^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)-4*b*c*(a+b*arccosh(c*x))*arctanh((c*x+(c*x-1)
^(1/2)*(c*x+1)^(1/2))^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)-4*b*c*(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)/d/(-c^2*d*x^2+d)^(1/2)-b^2*c*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)/d/(-c^2*d*x^2+d)^(1/2)-b^2*c*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)/d/(-c^2*d*x^2+d)^(1/2)

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Rubi [A]
time = 0.42, antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {5932, 5899, 5913, 3797, 2221, 2317, 2438, 5912, 5916, 5569, 4267} \begin {gather*} \frac {2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 c \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d x \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {c x-1} \sqrt {c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 5569

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

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 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 5916

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

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]

Rubi steps

\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x^2 (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d x \sqrt {d-c^2 d x^2}}+\frac {\left (2 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x \left (-1+c^2 x^2\right )} \, dx}{d \sqrt {d-c^2 d x^2}}-\frac {\left (2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}-\frac {\left (4 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x) \text {csch}(2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (8 b c \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,\cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {b^2 c \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}\\ \end {align*}

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Mathematica [A]
time = 1.14, size = 315, normalized size = 0.92 \begin {gather*} \frac {a^2 \left (-1+2 c^2 x^2\right )+2 a b \left (c^2 x^2 \cosh ^{-1}(c x)+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)-c x \left (\log (c x)+\log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )\right )\right )\right )+b^2 \left (\cosh ^{-1}(c x) \left (c^2 x^2 \cosh ^{-1}(c x)+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)-2 c x \left (\cosh ^{-1}(c x)+\log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )+\log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )\right )\right )\right )+c x \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+c x \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {PolyLog}\left (2,e^{-2 \cosh ^{-1}(c x)}\right )\right )}{d x \sqrt {d-c^2 d x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(824\) vs. \(2(359)=718\).
time = 2.19, size = 825, normalized size = 2.42

method	result	size
default	\(a^{2} \left (-\frac {1}{d x \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 c^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}\right )-\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right )^{2} c}{\left (c^{2} x^{2}-1\right ) d^{2}}-\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} x \,c^{2}}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2}}{x \left (c^{2} x^{2}-1\right ) d^{2}}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) c}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) c}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c}{\left (c^{2} x^{2}-1\right ) d^{2}}-\frac {4 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) c}{\left (c^{2} x^{2}-1\right ) d^{2}}-\frac {4 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x \,c^{2}}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )}{x \left (c^{2} x^{2}-1\right ) d^{2}}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) c}{\left (c^{2} x^{2}-1\right ) d^{2}}\)	\(825\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*(-1/d/x/(-c^2*d*x^2+d)^(1/2)+2*c^2/d*x/(-c^2*d*x^2+d)^(1/2))-2*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c
*x+1)^(1/2)/(c^2*x^2-1)/d^2*arccosh(c*x)^2*c-2*b^2*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)^2*x/(c^2*x^2-1)/d^2*c^2
+b^2*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)^2/x/(c^2*x^2-1)/d^2+2*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1
)^(1/2)/(c^2*x^2-1)/d^2*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*c+2*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x
-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*c+2*b^2*(-d*(c^2*x^2-1))^(
1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*c+b^2*
(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)
)^2)*c+2*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*arccosh(c*x)*ln(1-c*x-(c*x-1)^
(1/2)*(c*x+1)^(1/2))*c+2*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*polylog(2,c*x+
(c*x-1)^(1/2)*(c*x+1)^(1/2))*c-4*a*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*arccos
h(c*x)*c-4*a*b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)*x/(c^2*x^2-1)/d^2*c^2+2*a*b*(-d*(c^2*x^2-1))^(1/2)*arccosh(
c*x)/x/(c^2*x^2-1)/d^2+2*a*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*ln((c*x+(c*x-1
)^(1/2)*(c*x+1)^(1/2))^4-1)*c

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^2/x^2/(-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^6 - 2*c^2*d^2*x^4 + d
^2*x^2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x^{2} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{x^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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