3.3.20 \(\int \frac {(a+b \cosh ^{-1}(c x))^2}{x (d-c^2 d x^2)^{5/2}} \, dx\) [220]

Optimal. Leaf size=597 \[ -\frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \text {ArcTan}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {14 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {7 b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}} \]

[Out]

1/3*(a+b*arccosh(c*x))^2/d/(-c^2*d*x^2+d)^(3/2)-1/3*b^2/d^2/(-c^2*d*x^2+d)^(1/2)+(a+b*arccosh(c*x))^2/d^2/(-c^
2*d*x^2+d)^(1/2)+1/3*b*c*x*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(-c^2*x^2+1)/(-c^2*d*x^2+d)^(1/2
)+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^2/(-c^2*d*x^2+d
)^(1/2)+14/3*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)/d^2/(-c
^2*d*x^2+d)^(1/2)+7/3*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)/d^2/(-c^2*d*
x^2+d)^(1/2)-2*I*b*(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^2/(-c^2*d*x^2+d)^(1/2)+2*I*b*(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^2/(-c^2*d*x^2+d)^(1/2)-7/3*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)/d^2/(-c^2*d*x^2+d)^(1/2)+2*I*b^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)/d^2/(-c^2*d*x^2+d)^(1/2)-2*I*b^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)/d^2/(-c^2*d*x^2+d)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.65, antiderivative size = 597, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 13, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {5936, 5946, 4265, 2611, 2320, 6724, 5889, 5903, 4267, 2317, 2438, 5901, 75} \begin {gather*} \frac {2 \sqrt {c x-1} \sqrt {c x+1} \text {ArcTan}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {b c x \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {14 b \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {7 b^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*b^2/(d^2*Sqrt[d - c^2*d*x^2]) + (b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*d^2*(1 - c^2
*x^2)*Sqrt[d - c^2*d*x^2]) + (a + b*ArcCosh[c*x])^2/(3*d*(d - c^2*d*x^2)^(3/2)) + (a + b*ArcCosh[c*x])^2/(d^2*
Sqrt[d - c^2*d*x^2]) + (2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/(d^2*Sqr
t[d - c^2*d*x^2]) + (14*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(3*d^2*Sq
rt[d - c^2*d*x^2]) + (7*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(3*d^2*Sqrt[d - c^2*d*x^
2]) - ((2*I)*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(d^2*Sqrt[d
- c^2*d*x^2]) + ((2*I)*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[c*x]])/(d^2*
Sqrt[d - c^2*d*x^2]) - (7*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/(3*d^2*Sqrt[d - c^2*d*x
^2]) + ((2*I)*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, (-I)*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) - ((
2*I)*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, I*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2])

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 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 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 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 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*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{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 {a+b \cosh ^{-1}(c x)}{\left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b c x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{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 {a+b \cosh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{d^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 {x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {14 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b \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,\cosh ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b \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,\cosh ^{-1}(c x)\right )}{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,\cosh ^{-1}(c x)\right )}{3 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,\cosh ^{-1}(c x)\right )}{3 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 \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{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 \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {14 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{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^{\cosh ^{-1}(c x)}\right )}{3 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^{\cosh ^{-1}(c x)}\right )}{3 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^{\cosh ^{-1}(c x)}\right )}{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^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {14 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {7 b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {14 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {7 b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 9.30, size = 806, normalized size = 1.35 \begin {gather*} \sqrt {-d \left (-1+c^2 x^2\right )} \left (\frac {a^2}{3 d^3 \left (-1+c^2 x^2\right )^2}-\frac {a^2}{d^3 \left (-1+c^2 x^2\right )}\right )+\frac {a^2 \log (c x)}{d^{5/2}}-\frac {a^2 \log \left (d+\sqrt {d} \sqrt {-d \left (-1+c^2 x^2\right )}\right )}{d^{5/2}}+\frac {a b \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (14 \cosh ^{-1}(c x) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )-\text {csch}^2\left (\frac {1}{2} \cosh ^{-1}(c x)\right )-\frac {1}{2} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x) \text {csch}^4\left (\frac {1}{2} \cosh ^{-1}(c x)\right )-24 i \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )+24 i \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )-28 \log \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )-24 i \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )+24 i \text {PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )-\text {sech}^2\left (\frac {1}{2} \cosh ^{-1}(c x)\right )-\frac {8 \cosh ^{-1}(c x) \sinh ^4\left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3}-14 \cosh ^{-1}(c x) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{12 d^2 \sqrt {-d (-1+c x) (1+c x)}}+\frac {b^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (-4 \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )+14 \cosh ^{-1}(c x)^2 \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )-2 \cosh ^{-1}(c x) \text {csch}^2\left (\frac {1}{2} \cosh ^{-1}(c x)\right )-\frac {1}{2} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)^2 \text {csch}^4\left (\frac {1}{2} \cosh ^{-1}(c x)\right )-56 \cosh ^{-1}(c x) \log \left (1-e^{-\cosh ^{-1}(c x)}\right )-24 i \cosh ^{-1}(c x)^2 \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )+24 i \cosh ^{-1}(c x)^2 \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )+56 \cosh ^{-1}(c x) \log \left (1+e^{-\cosh ^{-1}(c x)}\right )-56 \text {PolyLog}\left (2,-e^{-\cosh ^{-1}(c x)}\right )-48 i \cosh ^{-1}(c x) \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )+48 i \cosh ^{-1}(c x) \text {PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )+56 \text {PolyLog}\left (2,e^{-\cosh ^{-1}(c x)}\right )-48 i \text {PolyLog}\left (3,-i e^{-\cosh ^{-1}(c x)}\right )+48 i \text {PolyLog}\left (3,i e^{-\cosh ^{-1}(c x)}\right )-2 \cosh ^{-1}(c x) \text {sech}^2\left (\frac {1}{2} \cosh ^{-1}(c x)\right )-\frac {8 \cosh ^{-1}(c x)^2 \sinh ^4\left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3}+4 \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )-14 \cosh ^{-1}(c x)^2 \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{24 d^2 \sqrt {-d (-1+c x) (1+c x)}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*(a^2/(3*d^3*(-1 + c^2*x^2)^2) - a^2/(d^3*(-1 + c^2*x^2))) + (a^2*Log[c*x])/d^(5/2) -
(a^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/d^(5/2) + (a*b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(14*ArcC
osh[c*x]*Coth[ArcCosh[c*x]/2] - Csch[ArcCosh[c*x]/2]^2 - (Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Cs
ch[ArcCosh[c*x]/2]^4)/2 - (24*I)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] + (24*I)*ArcCosh[c*x]*Log[1 + I/E^ArcC
osh[c*x]] - 28*Log[Tanh[ArcCosh[c*x]/2]] - (24*I)*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (24*I)*PolyLog[2, I/E^ArcC
osh[c*x]] - Sech[ArcCosh[c*x]/2]^2 - (8*ArcCosh[c*x]*Sinh[ArcCosh[c*x]/2]^4)/(((-1 + c*x)/(1 + c*x))^(3/2)*(1
+ c*x)^3) - 14*ArcCosh[c*x]*Tanh[ArcCosh[c*x]/2]))/(12*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]) + (b^2*Sqrt[(-1 +
c*x)/(1 + c*x)]*(1 + c*x)*(-4*Coth[ArcCosh[c*x]/2] + 14*ArcCosh[c*x]^2*Coth[ArcCosh[c*x]/2] - 2*ArcCosh[c*x]*C
sch[ArcCosh[c*x]/2]^2 - (Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2*Csch[ArcCosh[c*x]/2]^4)/2 - 56*Ar
cCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])] - (24*I)*ArcCosh[c*x]^2*Log[1 - I/E^ArcCosh[c*x]] + (24*I)*ArcCosh[c*x]^
2*Log[1 + I/E^ArcCosh[c*x]] + 56*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] - 56*PolyLog[2, -E^(-ArcCosh[c*x])] -
(48*I)*ArcCosh[c*x]*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (48*I)*ArcCosh[c*x]*PolyLog[2, I/E^ArcCosh[c*x]] + 56*P
olyLog[2, E^(-ArcCosh[c*x])] - (48*I)*PolyLog[3, (-I)/E^ArcCosh[c*x]] + (48*I)*PolyLog[3, I/E^ArcCosh[c*x]] -
2*ArcCosh[c*x]*Sech[ArcCosh[c*x]/2]^2 - (8*ArcCosh[c*x]^2*Sinh[ArcCosh[c*x]/2]^4)/(((-1 + c*x)/(1 + c*x))^(3/2
)*(1 + c*x)^3) + 4*Tanh[ArcCosh[c*x]/2] - 14*ArcCosh[c*x]^2*Tanh[ArcCosh[c*x]/2]))/(24*d^2*Sqrt[-(d*(-1 + c*x)
*(1 + c*x))])

________________________________________________________________________________________

Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}{x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

________________________________________________________________________________________

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 \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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/(-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), x)

________________________________________________________________________________________

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\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________