3.45 \(\int \frac{a+b \cosh ^{-1}(c x)}{x^4 (d-c^2 d x^2)^2} \, dx\)

Optimal. Leaf size=248 \[ \frac{5 b c^3 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{2 d^2}-\frac{5 b c^3 \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{2 d^2}+\frac{5 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}-\frac{a+b \cosh ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac{5 c^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2}-\frac{b c^3}{3 d^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{13 b c^3 \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{6 d^2}-\frac{b c}{6 d^2 x^2 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

-(b*c^3)/(3*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c)/(6*d^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (a + b*ArcCos
h[c*x])/(3*d^2*x^3*(1 - c^2*x^2)) - (5*c^2*(a + b*ArcCosh[c*x]))/(3*d^2*x*(1 - c^2*x^2)) + (5*c^4*x*(a + b*Arc
Cosh[c*x]))/(2*d^2*(1 - c^2*x^2)) + (13*b*c^3*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(6*d^2) + (5*c^3*(a + b*Ar
cCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/d^2 + (5*b*c^3*PolyLog[2, -E^ArcCosh[c*x]])/(2*d^2) - (5*b*c^3*PolyLog[2,
E^ArcCosh[c*x]])/(2*d^2)

________________________________________________________________________________________

Rubi [A]  time = 0.290807, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {5746, 103, 12, 104, 21, 92, 205, 5689, 74, 5694, 4182, 2279, 2391} \[ \frac{5 b c^3 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{2 d^2}-\frac{5 b c^3 \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{2 d^2}+\frac{5 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}-\frac{a+b \cosh ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac{5 c^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2}-\frac{b c^3}{3 d^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{13 b c^3 \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{6 d^2}-\frac{b c}{6 d^2 x^2 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*c^3)/(3*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c)/(6*d^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (a + b*ArcCos
h[c*x])/(3*d^2*x^3*(1 - c^2*x^2)) - (5*c^2*(a + b*ArcCosh[c*x]))/(3*d^2*x*(1 - c^2*x^2)) + (5*c^4*x*(a + b*Arc
Cosh[c*x]))/(2*d^2*(1 - c^2*x^2)) + (13*b*c^3*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(6*d^2) + (5*c^3*(a + b*Ar
cCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/d^2 + (5*b*c^3*PolyLog[2, -E^ArcCosh[c*x]])/(2*d^2) - (5*b*c^3*PolyLog[2,
E^ArcCosh[c*x]])/(2*d^2)

Rule 5746

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[(b*c*n*(-d)^p)/(f*(m + 1)
), Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], 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]) /; FreeQ[{a, b,
c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m] && IntegerQ[p]

Rule 103

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] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 12

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

Rule 104

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 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 92

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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 5689

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[(b*c*n*(-d)^p)/(2*(p + 1)), Int[x*(1 + c*x)^(p + 1/2)
*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] + Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p
+ 1)*(a + b*ArcCosh[c*x])^n, x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p,
-1] && IntegerQ[p]

Rule 74

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 5694

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 4182

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 2279

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 2391

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]

Rubi steps

\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{x^4 \left (d-c^2 d x^2\right )^2} \, dx &=-\frac{a+b \cosh ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac{1}{3} \left (5 c^2\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x^2 \left (d-c^2 d x^2\right )^2} \, dx-\frac{(b c) \int \frac{1}{x^3 (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2}\\ &=-\frac{b c}{6 d^2 x^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a+b \cosh ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}+\left (5 c^4\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx-\frac{(b c) \int \frac{3 c^2}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{6 d^2}-\frac{\left (5 b c^3\right ) \int \frac{1}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2}\\ &=\frac{5 b c^3}{3 d^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c}{6 d^2 x^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a+b \cosh ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}+\frac{5 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac{\left (5 b c^2\right ) \int \frac{c+c^2 x}{x \sqrt{-1+c x} (1+c x)^{3/2}} \, dx}{3 d^2}-\frac{\left (b c^3\right ) \int \frac{1}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2}+\frac{\left (5 b c^5\right ) \int \frac{x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2}+\frac{\left (5 c^4\right ) \int \frac{a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx}{2 d}\\ &=-\frac{b c^3}{3 d^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c}{6 d^2 x^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a+b \cosh ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}+\frac{5 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac{\left (b c^2\right ) \int \frac{c+c^2 x}{x \sqrt{-1+c x} (1+c x)^{3/2}} \, dx}{2 d^2}-\frac{\left (5 c^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2}+\frac{\left (5 b c^3\right ) \int \frac{1}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 d^2}\\ &=-\frac{b c^3}{3 d^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c}{6 d^2 x^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a+b \cosh ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}+\frac{5 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac{5 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2}+\frac{\left (b c^3\right ) \int \frac{1}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 d^2}+\frac{\left (5 b c^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2}-\frac{\left (5 b c^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2}+\frac{\left (5 b c^4\right ) \operatorname{Subst}\left (\int \frac{1}{c+c x^2} \, dx,x,\sqrt{-1+c x} \sqrt{1+c x}\right )}{3 d^2}\\ &=-\frac{b c^3}{3 d^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c}{6 d^2 x^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a+b \cosh ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}+\frac{5 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac{5 b c^3 \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{3 d^2}+\frac{5 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2}+\frac{\left (5 b c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d^2}-\frac{\left (5 b c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d^2}+\frac{\left (b c^4\right ) \operatorname{Subst}\left (\int \frac{1}{c+c x^2} \, dx,x,\sqrt{-1+c x} \sqrt{1+c x}\right )}{2 d^2}\\ &=-\frac{b c^3}{3 d^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c}{6 d^2 x^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a+b \cosh ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}+\frac{5 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac{13 b c^3 \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{6 d^2}+\frac{5 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2}+\frac{5 b c^3 \text{Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{2 d^2}-\frac{5 b c^3 \text{Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{2 d^2}\\ \end{align*}

Mathematica [A]  time = 1.66823, size = 377, normalized size = 1.52 \[ -\frac{-30 b c^3 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )+30 b c^3 \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )+\frac{6 a c^4 x}{c^2 x^2-1}+\frac{24 a c^2}{x}+15 a c^3 \log (1-c x)-15 a c^3 \log (c x+1)+\frac{4 a}{x^3}-\frac{26 b c^3 \sqrt{c^2 x^2-1} \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b c^4 x \sqrt{\frac{c x-1}{c x+1}}}{c x-1}-3 b c^3 \sqrt{\frac{c x-1}{c x+1}}+\frac{3 b c^3 \sqrt{\frac{c x-1}{c x+1}}}{c x-1}-\frac{2 b c^3}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b c^3 \cosh ^{-1}(c x)}{c x-1}+\frac{3 b c^3 \cosh ^{-1}(c x)}{c x+1}+\frac{24 b c^2 \cosh ^{-1}(c x)}{x}+30 b c^3 \cosh ^{-1}(c x) \log \left (1-e^{\cosh ^{-1}(c x)}\right )-30 b c^3 \cosh ^{-1}(c x) \log \left (e^{\cosh ^{-1}(c x)}+1\right )+\frac{2 b c}{x^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{4 b \cosh ^{-1}(c x)}{x^3}}{12 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((4*a)/x^3 + (24*a*c^2)/x - 3*b*c^3*Sqrt[(-1 + c*x)/(1 + c*x)] + (3*b*c^3*Sqrt[(-1 + c*x)/(1 + c*x)])/(-1 + c
*x) + (3*b*c^4*x*Sqrt[(-1 + c*x)/(1 + c*x)])/(-1 + c*x) - (2*b*c^3)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c)/(
x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (6*a*c^4*x)/(-1 + c^2*x^2) + (4*b*ArcCosh[c*x])/x^3 + (24*b*c^2*ArcCosh[c*
x])/x + (3*b*c^3*ArcCosh[c*x])/(-1 + c*x) + (3*b*c^3*ArcCosh[c*x])/(1 + c*x) - (26*b*c^3*Sqrt[-1 + c^2*x^2]*Ar
cTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + 30*b*c^3*ArcCosh[c*x]*Log[1 - E^ArcCosh[c*x]] - 30*
b*c^3*ArcCosh[c*x]*Log[1 + E^ArcCosh[c*x]] + 15*a*c^3*Log[1 - c*x] - 15*a*c^3*Log[1 + c*x] - 30*b*c^3*PolyLog[
2, -E^ArcCosh[c*x]] + 30*b*c^3*PolyLog[2, E^ArcCosh[c*x]])/(12*d^2)

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Maple [A]  time = 0.188, size = 352, normalized size = 1.4 \begin{align*} -{\frac{{c}^{3}a}{4\,{d}^{2} \left ( cx-1 \right ) }}-{\frac{5\,{c}^{3}a\ln \left ( cx-1 \right ) }{4\,{d}^{2}}}-{\frac{a}{3\,{d}^{2}{x}^{3}}}-2\,{\frac{{c}^{2}a}{{d}^{2}x}}-{\frac{{c}^{3}a}{4\,{d}^{2} \left ( cx+1 \right ) }}+{\frac{5\,{c}^{3}a\ln \left ( cx+1 \right ) }{4\,{d}^{2}}}-{\frac{5\,{c}^{4}b{\rm arccosh} \left (cx\right )x}{2\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b{c}^{3}}{3\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}+{\frac{5\,{c}^{2}b{\rm arccosh} \left (cx\right )}{3\,{d}^{2}x \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{bc}{6\,{d}^{2}{x}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}+{\frac{b{\rm arccosh} \left (cx\right )}{3\,{d}^{2}{x}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{13\,b{c}^{3}}{3\,{d}^{2}}\arctan \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{5\,b{c}^{3}}{2\,{d}^{2}}{\it dilog} \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{5\,b{c}^{3}}{2\,{d}^{2}}{\it dilog} \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{5\,b{c}^{3}{\rm arccosh} \left (cx\right )}{2\,{d}^{2}}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*c^3*a/d^2/(c*x-1)-5/4*c^3*a/d^2*ln(c*x-1)-1/3*a/d^2/x^3-2*c^2*a/d^2/x-1/4*c^3*a/d^2/(c*x+1)+5/4*c^3*a/d^2
*ln(c*x+1)-5/2*c^4*b/d^2/(c^2*x^2-1)*arccosh(c*x)*x-1/3*c^3*b/d^2/(c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)+5/3*
c^2*b/d^2/x/(c^2*x^2-1)*arccosh(c*x)-1/6*c*b/d^2/x^2/(c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)+1/3*b/d^2/x^3/(c^
2*x^2-1)*arccosh(c*x)+13/3*c^3*b/d^2*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+5/2*c^3*b/d^2*dilog(c*x+(c*x-1)^(
1/2)*(c*x+1)^(1/2))+5/2*c^3*b/d^2*dilog(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+5/2*c^3*b/d^2*arccosh(c*x)*ln(1+c*x
+(c*x-1)^(1/2)*(c*x+1)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(15*c^3*log(c*x + 1)/d^2 - 15*c^3*log(c*x - 1)/d^2 - 2*(15*c^4*x^4 - 10*c^2*x^2 - 2)/(c^2*d^2*x^5 - d^2*x
^3))*a + 1/192*(8640*c^7*integrate(1/24*x^5*log(c*x - 1)/(c^4*d^2*x^6 - 2*c^2*d^2*x^4 + d^2*x^2), x) - 120*c^6
*(2*x/(c^4*d^2*x^2 - c^2*d^2) + log(c*x + 1)/(c^3*d^2) - log(c*x - 1)/(c^3*d^2)) - 2880*c^6*integrate(1/24*x^4
*log(c*x - 1)/(c^4*d^2*x^6 - 2*c^2*d^2*x^4 + d^2*x^2), x) + 45*(c*(2/(c^4*d^2*x - c^3*d^2) - log(c*x + 1)/(c^3
*d^2) + log(c*x - 1)/(c^3*d^2)) + 4*log(c*x - 1)/(c^4*d^2*x^2 - c^2*d^2))*c^5 + 80*c^4*(2*x/(c^2*d^2*x^2 - d^2
) - log(c*x + 1)/(c*d^2) + log(c*x - 1)/(c*d^2)) + 2880*c^4*integrate(1/24*x^2*log(c*x - 1)/(c^4*d^2*x^6 - 2*c
^2*d^2*x^4 + d^2*x^2), x) + 16*c^2*(2*(3*c^2*x^2 - 2)/(c^2*d^2*x^3 - d^2*x) - 3*c*log(c*x + 1)/d^2 + 3*c*log(c
*x - 1)/d^2) - 4*(15*(c^5*x^5 - c^3*x^3)*log(c*x + 1)^2 + 30*(c^5*x^5 - c^3*x^3)*log(c*x + 1)*log(c*x - 1) + 4
*(30*c^4*x^4 - 20*c^2*x^2 - 15*(c^5*x^5 - c^3*x^3)*log(c*x + 1) + 15*(c^5*x^5 - c^3*x^3)*log(c*x - 1) - 4)*log
(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*d^2*x^5 - d^2*x^3) + 192*integrate(-1/12*(30*c^5*x^4 - 20*c^3*x^2 -
15*(c^6*x^5 - c^4*x^3)*log(c*x + 1) + 15*(c^6*x^5 - c^4*x^3)*log(c*x - 1) - 4*c)/(c^5*d^2*x^8 - 2*c^3*d^2*x^6
+ c*d^2*x^4 + (c^4*d^2*x^7 - 2*c^2*d^2*x^5 + d^2*x^3)*sqrt(c*x + 1)*sqrt(c*x - 1)), x))*b

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{arcosh}\left (c x\right ) + a}{c^{4} d^{2} x^{8} - 2 \, c^{2} d^{2} x^{6} + d^{2} x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx + \int \frac{b \operatorname{acosh}{\left (c x \right )}}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a/(c**4*x**8 - 2*c**2*x**6 + x**4), x) + Integral(b*acosh(c*x)/(c**4*x**8 - 2*c**2*x**6 + x**4), x))
/d**2

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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