\(\int \frac {a+b \text {arccosh}(c x)}{x^3 (d-c^2 d x^2)^3} \, dx\) [53]

Optimal result

Integrand size = 25, antiderivative size = 250 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\frac {b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {5 b c^3 x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d^3 \left (1-c^2 x^2\right )}+\frac {6 c^2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^3}+\frac {3 b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {3 b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 d^3} \]

[Out]

1/2*b*c/d^3/x/(c*x-1)^(3/2)/(c*x+1)^(3/2)-5/12*b*c^3*x/d^3/(c*x-1)^(3/2)/(c*x+1)^(3/2)+3/4*c^2*(a+b*arccosh(c*
x))/d^3/(-c^2*x^2+1)^2+1/2*(-a-b*arccosh(c*x))/d^3/x^2/(-c^2*x^2+1)^2+3/2*c^2*(a+b*arccosh(c*x))/d^3/(-c^2*x^2
+1)+6*c^2*(a+b*arccosh(c*x))*arctanh((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d^3+3/2*b*c^2*polylog(2,-(c*x+(c*x-1
)^(1/2)*(c*x+1)^(1/2))^2)/d^3-3/2*b*c^2*polylog(2,(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d^3-2/3*b*c^3*x/d^3/(c*
x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5932, 5936, 5916, 5569, 4267, 2317, 2438, 39, 40, 105, 12} \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\frac {6 c^2 \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d^3}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d^3 \left (1-c^2 x^2\right )}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {3 b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {2 b c^3 x}{3 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 b c^3 x}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac {b c}{2 d^3 x (c x-1)^{3/2} (c x+1)^{3/2}} \]

[In]

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

[Out]

(b*c)/(2*d^3*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - (5*b*c^3*x)/(12*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - (2*
b*c^3*x)/(3*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*c^2*(a + b*ArcCosh[c*x]))/(4*d^3*(1 - c^2*x^2)^2) - (a + b*
ArcCosh[c*x])/(2*d^3*x^2*(1 - c^2*x^2)^2) + (3*c^2*(a + b*ArcCosh[c*x]))/(2*d^3*(1 - c^2*x^2)) + (6*c^2*(a + b
*ArcCosh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])])/d^3 + (3*b*c^2*PolyLog[2, -E^(2*ArcCosh[c*x])])/(2*d^3) - (3*b*c^2
*PolyLog[2, E^(2*ArcCosh[c*x])])/(2*d^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 40

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-x)*(a + b*x)^(m + 1)*((c + d*x)^(m
+ 1)/(2*a*c*(m + 1))), x] + Dist[(2*m + 3)/(2*a*c*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /;
FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]

Rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 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 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 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]

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])

Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\left (3 c^2\right ) \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx+\frac {(b c) \int \frac {1}{x^2 (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{2 d^3} \\ & = \frac {b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {(b c) \int \frac {4 c^2}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{2 d^3}-\frac {\left (3 b c^3\right ) \int \frac {1}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{4 d^3}+\frac {\left (3 c^2\right ) \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx}{d} \\ & = \frac {b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b c^3 x}{4 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d^3 \left (1-c^2 x^2\right )}+\frac {\left (b c^3\right ) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^3}+\frac {\left (3 b c^3\right ) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^3}+\frac {\left (2 b c^3\right ) \int \frac {1}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^3}+\frac {\left (3 c^2\right ) \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )} \, dx}{d^2} \\ & = \frac {b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {5 b c^3 x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c^3 x}{d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d^3 \left (1-c^2 x^2\right )}-\frac {\left (3 c^2\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\text {arccosh}(c x))}{d^3}-\frac {\left (4 b c^3\right ) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^3} \\ & = \frac {b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {5 b c^3 x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d^3 \left (1-c^2 x^2\right )}-\frac {\left (6 c^2\right ) \text {Subst}(\int (a+b x) \text {csch}(2 x) \, dx,x,\text {arccosh}(c x))}{d^3} \\ & = \frac {b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {5 b c^3 x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d^3 \left (1-c^2 x^2\right )}+\frac {6 c^2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^3}+\frac {\left (3 b c^2\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{d^3}-\frac {\left (3 b c^2\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{d^3} \\ & = \frac {b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {5 b c^3 x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d^3 \left (1-c^2 x^2\right )}+\frac {6 c^2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^3}+\frac {\left (3 b c^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {\left (3 b c^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arccosh}(c x)}\right )}{2 d^3} \\ & = \frac {b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {5 b c^3 x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c^3 x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d^3 \left (1-c^2 x^2\right )}+\frac {6 c^2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^3}+\frac {3 b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {3 b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 d^3} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 1.08 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.61 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx=\frac {-\frac {2 a}{x^2}+\frac {a c^2}{\left (-1+c^2 x^2\right )^2}-\frac {4 a c^2}{-1+c^2 x^2}-\frac {b c^2 \left ((-2+c x) \sqrt {-1+c x} \sqrt {1+c x}-3 \text {arccosh}(c x)\right )}{12 (-1+c x)^2}-\frac {b c^2 \left (\sqrt {-1+c x} \sqrt {1+c x} (2+c x)-3 \text {arccosh}(c x)\right )}{12 (1+c x)^2}+\frac {2 b \left (c x \sqrt {-1+c x} \sqrt {1+c x}-\text {arccosh}(c x)\right )}{x^2}+\frac {9}{4} b c^2 \left (-\frac {1}{\sqrt {\frac {-1+c x}{1+c x}}}+\frac {\text {arccosh}(c x)}{1-c x}\right )-\frac {9}{4} b c^2 \left (\sqrt {\frac {-1+c x}{1+c x}}-\frac {\text {arccosh}(c x)}{1+c x}\right )+12 a c^2 \log (x)-6 a c^2 \log \left (1-c^2 x^2\right )+6 b c^2 \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right )+3 b c^2 \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1+e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )\right )+3 b c^2 \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1-e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{4 d^3} \]

[In]

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

[Out]

((-2*a)/x^2 + (a*c^2)/(-1 + c^2*x^2)^2 - (4*a*c^2)/(-1 + c^2*x^2) - (b*c^2*((-2 + c*x)*Sqrt[-1 + c*x]*Sqrt[1 +
c*x] - 3*ArcCosh[c*x]))/(12*(-1 + c*x)^2) - (b*c^2*(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2 + c*x) - 3*ArcCosh[c*x]))
/(12*(1 + c*x)^2) + (2*b*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - ArcCosh[c*x]))/x^2 + (9*b*c^2*(-(1/Sqrt[(-1 + c*x
)/(1 + c*x)]) + ArcCosh[c*x]/(1 - c*x)))/4 - (9*b*c^2*(Sqrt[(-1 + c*x)/(1 + c*x)] - ArcCosh[c*x]/(1 + c*x)))/4
+ 12*a*c^2*Log[x] - 6*a*c^2*Log[1 - c^2*x^2] + 6*b*c^2*(ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[
c*x])]) - PolyLog[2, -E^(-2*ArcCosh[c*x])]) + 3*b*c^2*(ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 + E^ArcCosh[c*x]])
- 4*PolyLog[2, -E^ArcCosh[c*x]]) + 3*b*c^2*(ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 - E^ArcCosh[c*x]]) - 4*PolyL
og[2, E^ArcCosh[c*x]]))/(4*d^3)

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.56

method	result	size
derivativedivides	\(c^{2} \left (-\frac {a \left (\frac {1}{2 c^{2} x^{2}}-3 \ln \left (c x \right )-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {9}{16 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{2}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {9}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{2}\right )}{d^{3}}-\frac {b \left (\frac {8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}-8 c^{6} x^{6}+18 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+16 c^{4} x^{4}-27 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-6 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -8 c^{2} x^{2}+6 \,\operatorname {arccosh}\left (c x \right )}{12 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2} x^{2}}-3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3}}\right )\)	\(391\)
default	\(c^{2} \left (-\frac {a \left (\frac {1}{2 c^{2} x^{2}}-3 \ln \left (c x \right )-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {9}{16 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{2}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {9}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{2}\right )}{d^{3}}-\frac {b \left (\frac {8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}-8 c^{6} x^{6}+18 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+16 c^{4} x^{4}-27 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-6 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -8 c^{2} x^{2}+6 \,\operatorname {arccosh}\left (c x \right )}{12 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2} x^{2}}-3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3}}\right )\)	\(391\)
parts	\(-\frac {a \left (-\frac {c^{2}}{16 \left (c x +1\right )^{2}}-\frac {9 c^{2}}{16 \left (c x +1\right )}+\frac {3 c^{2} \ln \left (c x +1\right )}{2}+\frac {1}{2 x^{2}}-3 c^{2} \ln \left (x \right )-\frac {c^{2}}{16 \left (c x -1\right )^{2}}+\frac {9 c^{2}}{16 \left (c x -1\right )}+\frac {3 c^{2} \ln \left (c x -1\right )}{2}\right )}{d^{3}}-\frac {b \,c^{2} \left (\frac {8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}-8 c^{6} x^{6}+18 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+16 c^{4} x^{4}-27 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-6 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -8 c^{2} x^{2}+6 \,\operatorname {arccosh}\left (c x \right )}{12 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2} x^{2}}-3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3}}\)	\(406\)

[In]

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

[Out]

c^2*(-a/d^3*(1/2/c^2/x^2-3*ln(c*x)-1/16/(c*x+1)^2-9/16/(c*x+1)+3/2*ln(c*x+1)-1/16/(c*x-1)^2+9/16/(c*x-1)+3/2*l
n(c*x-1))-b/d^3*(1/12/(c^4*x^4-2*c^2*x^2+1)/c^2/x^2*(8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^5-8*c^6*x^6+18*c^4*x^
4*arccosh(c*x)-3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+16*c^4*x^4-27*c^2*x^2*arccosh(c*x)-6*(c*x-1)^(1/2)*(c*x+1
)^(1/2)*c*x-8*c^2*x^2+6*arccosh(c*x))-3*arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)-3/2*polylog(2,-
(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)+3*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+3*polylog(2,c*x+(c*x
-1)^(1/2)*(c*x+1)^(1/2))+3*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+3*polylog(2,-c*x-(c*x-1)^(1/2)*(
c*x+1)^(1/2))))

Fricas [F]

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

[In]

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

[Out]

integral(-(b*arccosh(c*x) + a)/(c^6*d^3*x^9 - 3*c^4*d^3*x^7 + 3*c^2*d^3*x^5 - d^3*x^3), x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-1/4*a*((6*c^4*x^4 - 9*c^2*x^2 + 2)/(c^4*d^3*x^6 - 2*c^2*d^3*x^4 + d^3*x^2) + 6*c^2*log(c*x + 1)/d^3 + 6*c^2*l
og(c*x - 1)/d^3 - 12*c^2*log(x)/d^3) - b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c^6*d^3*x^9 - 3*c^4
*d^3*x^7 + 3*c^2*d^3*x^5 - d^3*x^3), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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