\(\int \frac {a+b \text {arccosh}(c x)}{x^4 (d-c^2 d x^2)^3} \, dx\) [54]

Optimal result

Integrand size = 25, antiderivative size = 310 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=-\frac {b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {29 b c^3}{24 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \text {arccosh}(c x))}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arccosh}(c x))}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arccosh}(c x))}{8 d^3 \left (1-c^2 x^2\right )}+\frac {19 b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d^3}+\frac {35 c^3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{4 d^3}+\frac {35 b c^3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{8 d^3}-\frac {35 b c^3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{8 d^3} \]

[Out]

-1/12*b*c^3/d^3/(c*x-1)^(3/2)/(c*x+1)^(3/2)+1/6*b*c/d^3/x^2/(c*x-1)^(3/2)/(c*x+1)^(3/2)+1/3*(-a-b*arccosh(c*x)
)/d^3/x^3/(-c^2*x^2+1)^2-7/3*c^2*(a+b*arccosh(c*x))/d^3/x/(-c^2*x^2+1)^2+35/12*c^4*x*(a+b*arccosh(c*x))/d^3/(-
c^2*x^2+1)^2+35/8*c^4*x*(a+b*arccosh(c*x))/d^3/(-c^2*x^2+1)+19/6*b*c^3*arctan((c*x-1)^(1/2)*(c*x+1)^(1/2))/d^3
+35/4*c^3*(a+b*arccosh(c*x))*arctanh(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d^3+35/8*b*c^3*polylog(2,-c*x-(c*x-1)^(1
/2)*(c*x+1)^(1/2))/d^3-35/8*b*c^3*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d^3-29/24*b*c^3/d^3/(c*x-1)^(1/2)
/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {5932, 5901, 5903, 4267, 2317, 2438, 75, 106, 21, 94, 211, 105, 12} \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\frac {35 c^3 \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{4 d^3}-\frac {7 c^2 (a+b \text {arccosh}(c x))}{3 d^3 x \left (1-c^2 x^2\right )^2}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arccosh}(c x))}{8 d^3 \left (1-c^2 x^2\right )}+\frac {35 c^4 x (a+b \text {arccosh}(c x))}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 b c^3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{8 d^3}-\frac {35 b c^3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{8 d^3}+\frac {19 b c^3 \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )}{6 d^3}-\frac {29 b c^3}{24 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^3}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac {b c}{6 d^3 x^2 (c x-1)^{3/2} (c x+1)^{3/2}} \]

[In]

Int[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^3),x]

[Out]

-1/12*(b*c^3)/(d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + (b*c)/(6*d^3*x^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - (2
9*b*c^3)/(24*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (a + b*ArcCosh[c*x])/(3*d^3*x^3*(1 - c^2*x^2)^2) - (7*c^2*(a
+ b*ArcCosh[c*x]))/(3*d^3*x*(1 - c^2*x^2)^2) + (35*c^4*x*(a + b*ArcCosh[c*x]))/(12*d^3*(1 - c^2*x^2)^2) + (35*
c^4*x*(a + b*ArcCosh[c*x]))/(8*d^3*(1 - c^2*x^2)) + (19*b*c^3*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(6*d^3) +
(35*c^3*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(4*d^3) + (35*b*c^3*PolyLog[2, -E^ArcCosh[c*x]])/(8*d^3)
- (35*b*c^3*PolyLog[2, E^ArcCosh[c*x]])/(8*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 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 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 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 106

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] && LtQ[m, -1] &&
IntegersQ[2*m, 2*n, 2*p]

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 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 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 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*} \text {integral}& = -\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {1}{3} \left (7 c^2\right ) \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^3} \, dx+\frac {(b c) \int \frac {1}{x^3 (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{3 d^3} \\ & = \frac {b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \text {arccosh}(c x))}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {1}{3} \left (35 c^4\right ) \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx+\frac {(b c) \int \frac {5 c^2}{x (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{6 d^3}+\frac {\left (7 b c^3\right ) \int \frac {1}{x (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{3 d^3} \\ & = -\frac {7 b c^3}{9 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \text {arccosh}(c x))}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arccosh}(c x))}{12 d^3 \left (1-c^2 x^2\right )^2}-\frac {\left (7 b c^2\right ) \int \frac {3 c+3 c^2 x}{x (-1+c x)^{3/2} (1+c x)^{5/2}} \, dx}{9 d^3}+\frac {\left (5 b c^3\right ) \int \frac {1}{x (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{6 d^3}-\frac {\left (35 b c^5\right ) \int \frac {x}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{12 d^3}+\frac {\left (35 c^4\right ) \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx}{4 d} \\ & = -\frac {b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \text {arccosh}(c x))}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arccosh}(c x))}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arccosh}(c x))}{8 d^3 \left (1-c^2 x^2\right )}-\frac {\left (5 b c^2\right ) \int \frac {3 c+3 c^2 x}{x (-1+c x)^{3/2} (1+c x)^{5/2}} \, dx}{18 d^3}-\frac {\left (7 b c^3\right ) \int \frac {1}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^3}+\frac {\left (35 b c^5\right ) \int \frac {x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{8 d^3}+\frac {\left (35 c^4\right ) \int \frac {a+b \text {arccosh}(c x)}{d-c^2 d x^2} \, dx}{8 d^2} \\ & = -\frac {b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {49 b c^3}{24 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \text {arccosh}(c x))}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arccosh}(c x))}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arccosh}(c x))}{8 d^3 \left (1-c^2 x^2\right )}+\frac {\left (7 b c^2\right ) \int \frac {c+c^2 x}{x \sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{3 d^3}-\frac {\left (35 c^3\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arccosh}(c x))}{8 d^3}-\frac {\left (5 b c^3\right ) \int \frac {1}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{6 d^3} \\ & = -\frac {b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {29 b c^3}{24 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \text {arccosh}(c x))}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arccosh}(c x))}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arccosh}(c x))}{8 d^3 \left (1-c^2 x^2\right )}+\frac {35 c^3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{4 d^3}+\frac {\left (5 b c^2\right ) \int \frac {c+c^2 x}{x \sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{6 d^3}+\frac {\left (7 b c^3\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 d^3}+\frac {\left (35 b c^3\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{8 d^3}-\frac {\left (35 b c^3\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{8 d^3} \\ & = -\frac {b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {29 b c^3}{24 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \text {arccosh}(c x))}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arccosh}(c x))}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arccosh}(c x))}{8 d^3 \left (1-c^2 x^2\right )}+\frac {35 c^3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{4 d^3}+\frac {\left (5 b c^3\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{6 d^3}+\frac {\left (35 b c^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{8 d^3}-\frac {\left (35 b c^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{8 d^3}+\frac {\left (7 b c^4\right ) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{3 d^3} \\ & = -\frac {b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {29 b c^3}{24 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \text {arccosh}(c x))}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arccosh}(c x))}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arccosh}(c x))}{8 d^3 \left (1-c^2 x^2\right )}+\frac {7 b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{3 d^3}+\frac {35 c^3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{4 d^3}+\frac {35 b c^3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{8 d^3}-\frac {35 b c^3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{8 d^3}+\frac {\left (5 b c^4\right ) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d^3} \\ & = -\frac {b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {29 b c^3}{24 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \text {arccosh}(c x))}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arccosh}(c x))}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arccosh}(c x))}{8 d^3 \left (1-c^2 x^2\right )}+\frac {19 b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d^3}+\frac {35 c^3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{4 d^3}+\frac {35 b c^3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{8 d^3}-\frac {35 b c^3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{8 d^3} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 1.33 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.52 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\frac {-\frac {16 a}{x^3}-\frac {144 a c^2}{x}+\frac {12 a c^4 x}{\left (-1+c^2 x^2\right )^2}-\frac {66 a c^4 x}{-1+c^2 x^2}-\frac {b c^3 \left ((-2+c x) \sqrt {-1+c x} \sqrt {1+c x}-3 \text {arccosh}(c x)\right )}{(-1+c x)^2}+\frac {b c^3 \left (\sqrt {-1+c x} \sqrt {1+c x} (2+c x)-3 \text {arccosh}(c x)\right )}{(1+c x)^2}+33 b c^3 \left (-\frac {1}{\sqrt {\frac {-1+c x}{1+c x}}}+\frac {\text {arccosh}(c x)}{1-c x}\right )+33 b c^3 \left (\sqrt {\frac {-1+c x}{1+c x}}-\frac {\text {arccosh}(c x)}{1+c x}\right )+144 b c^2 \left (-\frac {\text {arccosh}(c x)}{x}+\frac {c \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\right )+\frac {8 b \left (-2 \text {arccosh}(c x)+\frac {c x \left (-1+c^2 x^2+c^2 x^2 \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\right )}{x^3}-105 a c^3 \log (1-c x)+105 a c^3 \log (1+c x)-\frac {105}{2} b c^3 \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1+e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )\right )+\frac {105}{2} b c^3 \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1-e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{48 d^3} \]

[In]

Integrate[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^3),x]

[Out]

((-16*a)/x^3 - (144*a*c^2)/x + (12*a*c^4*x)/(-1 + c^2*x^2)^2 - (66*a*c^4*x)/(-1 + c^2*x^2) - (b*c^3*((-2 + c*x
)*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 3*ArcCosh[c*x]))/(-1 + c*x)^2 + (b*c^3*(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2 + c*x
) - 3*ArcCosh[c*x]))/(1 + c*x)^2 + 33*b*c^3*(-(1/Sqrt[(-1 + c*x)/(1 + c*x)]) + ArcCosh[c*x]/(1 - c*x)) + 33*b*
c^3*(Sqrt[(-1 + c*x)/(1 + c*x)] - ArcCosh[c*x]/(1 + c*x)) + 144*b*c^2*(-(ArcCosh[c*x]/x) + (c*Sqrt[-1 + c^2*x^
2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (8*b*(-2*ArcCosh[c*x] + (c*x*(-1 + c^2*x^2 +
c^2*x^2*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/x^3 - 105*a*c^3*Log[1
- c*x] + 105*a*c^3*Log[1 + c*x] - (105*b*c^3*(ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 + E^ArcCosh[c*x]]) - 4*Pol
yLog[2, -E^ArcCosh[c*x]]))/2 + (105*b*c^3*(ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 - E^ArcCosh[c*x]]) - 4*PolyLog
[2, E^ArcCosh[c*x]]))/2)/(48*d^3)

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.01

method	result	size
derivativedivides	\(c^{3} \left (-\frac {a \left (\frac {1}{3 c^{3} x^{3}}+\frac {3}{c x}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {11}{16 \left (c x +1\right )}-\frac {35 \ln \left (c x +1\right )}{16}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {11}{16 \left (c x -1\right )}+\frac {35 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b \left (\frac {105 \,\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}+29 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}-175 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )-27 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+56 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +8 \,\operatorname {arccosh}\left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}-\frac {19 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3}-\frac {35 \operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {35 \operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {35 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}\right )\)	\(314\)
default	\(c^{3} \left (-\frac {a \left (\frac {1}{3 c^{3} x^{3}}+\frac {3}{c x}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {11}{16 \left (c x +1\right )}-\frac {35 \ln \left (c x +1\right )}{16}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {11}{16 \left (c x -1\right )}+\frac {35 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b \left (\frac {105 \,\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}+29 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}-175 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )-27 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+56 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +8 \,\operatorname {arccosh}\left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}-\frac {19 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3}-\frac {35 \operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {35 \operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {35 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}\right )\)	\(314\)
parts	\(-\frac {a \left (\frac {c^{3}}{16 \left (c x +1\right )^{2}}+\frac {11 c^{3}}{16 \left (c x +1\right )}-\frac {35 c^{3} \ln \left (c x +1\right )}{16}+\frac {1}{3 x^{3}}+\frac {3 c^{2}}{x}-\frac {c^{3}}{16 \left (c x -1\right )^{2}}+\frac {11 c^{3}}{16 \left (c x -1\right )}+\frac {35 c^{3} \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b \,c^{3} \left (\frac {105 \,\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}+29 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}-175 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )-27 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+56 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +8 \,\operatorname {arccosh}\left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}-\frac {19 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3}-\frac {35 \operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {35 \operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {35 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}\)	\(328\)

[In]

int((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)

[Out]

c^3*(-a/d^3*(1/3/c^3/x^3+3/c/x+1/16/(c*x+1)^2+11/16/(c*x+1)-35/16*ln(c*x+1)-1/16/(c*x-1)^2+11/16/(c*x-1)+35/16
*ln(c*x-1))-b/d^3*(1/24*(105*arccosh(c*x)*c^6*x^6+29*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^5-175*c^4*x^4*arccosh(c
*x)-27*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+56*c^2*x^2*arccosh(c*x)-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+8*arccosh
(c*x))/(c^4*x^4-2*c^2*x^2+1)/c^3/x^3-19/3*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-35/8*dilog(c*x+(c*x-1)^(1/2)
*(c*x+1)^(1/2))-35/8*dilog(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-35/8*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)
^(1/2))))

Fricas [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{4}} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^3,x, algorithm="fricas")

[Out]

integral(-(b*arccosh(c*x) + a)/(c^6*d^3*x^10 - 3*c^4*d^3*x^8 + 3*c^2*d^3*x^6 - d^3*x^4), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((a+b*acosh(c*x))/x**4/(-c**2*d*x**2+d)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{4}} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^3,x, algorithm="maxima")

[Out]

1/6144*(1935360*c^9*integrate(1/96*x^7*log(c*x - 1)/(c^6*d^3*x^8 - 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 - d^3*x^2), x
) - 1680*c^8*(2*(5*c^2*x^3 - 3*x)/(c^8*d^3*x^4 - 2*c^6*d^3*x^2 + c^4*d^3) + 3*log(c*x + 1)/(c^5*d^3) - 3*log(c
*x - 1)/(c^5*d^3)) - 645120*c^8*integrate(1/96*x^6*log(c*x - 1)/(c^6*d^3*x^8 - 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 -
d^3*x^2), x) + 630*(c*(2*(5*c^2*x^2 + 3*c*x - 6)/(c^8*d^3*x^3 - c^7*d^3*x^2 - c^6*d^3*x + c^5*d^3) - 5*log(c*
x + 1)/(c^5*d^3) + 5*log(c*x - 1)/(c^5*d^3)) + 16*(2*c^2*x^2 - 1)*log(c*x - 1)/(c^8*d^3*x^4 - 2*c^6*d^3*x^2 +
c^4*d^3))*c^7 + 2800*c^6*(2*(c^2*x^3 + x)/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3) - log(c*x + 1)/(c^3*d^3) + l
og(c*x - 1)/(c^3*d^3)) + 1290240*c^6*integrate(1/96*x^4*log(c*x - 1)/(c^6*d^3*x^8 - 3*c^4*d^3*x^6 + 3*c^2*d^3*
x^4 - d^3*x^2), x) + 315*(c*(2*(3*c^2*x^2 - 3*c*x - 2)/(c^6*d^3*x^3 - c^5*d^3*x^2 - c^4*d^3*x + c^3*d^3) - 3*l
og(c*x + 1)/(c^3*d^3) + 3*log(c*x - 1)/(c^3*d^3)) - 16*log(c*x - 1)/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3))*c
^5 + 896*c^4*(2*(3*c^2*x^3 - 5*x)/(c^4*d^3*x^4 - 2*c^2*d^3*x^2 + d^3) - 3*log(c*x + 1)/(c*d^3) + 3*log(c*x - 1
)/(c*d^3)) - 645120*c^4*integrate(1/96*x^2*log(c*x - 1)/(c^6*d^3*x^8 - 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 - d^3*x^2
), x) + 128*c^2*(2*(15*c^4*x^4 - 25*c^2*x^2 + 8)/(c^4*d^3*x^5 - 2*c^2*d^3*x^3 + d^3*x) - 15*c*log(c*x + 1)/d^3
+ 15*c*log(c*x - 1)/d^3) - 32*(105*(c^7*x^7 - 2*c^5*x^5 + c^3*x^3)*log(c*x + 1)^2 + 210*(c^7*x^7 - 2*c^5*x^5
+ c^3*x^3)*log(c*x + 1)*log(c*x - 1) + 4*(210*c^6*x^6 - 350*c^4*x^4 + 112*c^2*x^2 - 105*(c^7*x^7 - 2*c^5*x^5 +
c^3*x^3)*log(c*x + 1) + 105*(c^7*x^7 - 2*c^5*x^5 + c^3*x^3)*log(c*x - 1) + 16)*log(c*x + sqrt(c*x + 1)*sqrt(c
*x - 1)))/(c^4*d^3*x^7 - 2*c^2*d^3*x^5 + d^3*x^3) + 6144*integrate(-1/48*(210*c^7*x^6 - 350*c^5*x^4 + 112*c^3*
x^2 - 105*(c^8*x^7 - 2*c^6*x^5 + c^4*x^3)*log(c*x + 1) + 105*(c^8*x^7 - 2*c^6*x^5 + c^4*x^3)*log(c*x - 1) + 16
*c)/(c^7*d^3*x^10 - 3*c^5*d^3*x^8 + 3*c^3*d^3*x^6 - c*d^3*x^4 + (c^6*d^3*x^9 - 3*c^4*d^3*x^7 + 3*c^2*d^3*x^5 -
d^3*x^3)*sqrt(c*x + 1)*sqrt(c*x - 1)), x))*b + 1/48*a*(105*c^3*log(c*x + 1)/d^3 - 105*c^3*log(c*x - 1)/d^3 -
2*(105*c^6*x^6 - 175*c^4*x^4 + 56*c^2*x^2 + 8)/(c^4*d^3*x^7 - 2*c^2*d^3*x^5 + d^3*x^3))

Giac [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{4}} \,d x } \]

[In]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]

[In]

int((a + b*acosh(c*x))/(x^4*(d - c^2*d*x^2)^3),x)

[Out]

int((a + b*acosh(c*x))/(x^4*(d - c^2*d*x^2)^3), x)