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

Optimal. Leaf size=342 \[ -\frac{i b d^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt{c x-1} \sqrt{c x+1}}+d^2 \log (x) \left (a+b \cosh ^{-1}(c x)\right )+d e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{i b d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b d^2 \sqrt{1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b e x \sqrt{c x-1} \sqrt{c x+1} \left (16 c^2 d+3 e\right )}{32 c^3}-\frac{b e \left (16 c^2 d+3 e\right ) \cosh ^{-1}(c x)}{32 c^4}-\frac{b e^2 x^3 \sqrt{c x-1} \sqrt{c x+1}}{16 c} \]

[Out]

-(b*e*(16*c^2*d + 3*e)*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(32*c^3) - (b*e^2*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16
*c) - (b*e*(16*c^2*d + 3*e)*ArcCosh[c*x])/(32*c^4) + d*e*x^2*(a + b*ArcCosh[c*x]) + (e^2*x^4*(a + b*ArcCosh[c*
x]))/4 - ((I/2)*b*d^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*Sqrt[1 - c^2*x^
2]*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d^2*(a + b*ArcCosh[c*x])*Log[x
] - (b*d^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((I/2)*b*d^2*Sqrt[1 - c^2*x^
2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])

________________________________________________________________________________________

Rubi [A]  time = 0.786414, antiderivative size = 369, normalized size of antiderivative = 1.08, number of steps used = 16, number of rules used = 15, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {266, 43, 5790, 6742, 90, 52, 100, 12, 2328, 2326, 4625, 3717, 2190, 2279, 2391} \[ -\frac{i b d^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt{c x-1} \sqrt{c x+1}}+d^2 \log (x) \left (a+b \cosh ^{-1}(c x)\right )+d e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{i b d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b d^2 \sqrt{1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b d e \cosh ^{-1}(c x)}{2 c^2}-\frac{3 b e^2 x \sqrt{c x-1} \sqrt{c x+1}}{32 c^3}-\frac{3 b e^2 \cosh ^{-1}(c x)}{32 c^4}-\frac{b d e x \sqrt{c x-1} \sqrt{c x+1}}{2 c}-\frac{b e^2 x^3 \sqrt{c x-1} \sqrt{c x+1}}{16 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c) - (3*b*e^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(32*c^3) - (b*e^2*x^3
*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*c) - (b*d*e*ArcCosh[c*x])/(2*c^2) - (3*b*e^2*ArcCosh[c*x])/(32*c^4) + d*e*x
^2*(a + b*ArcCosh[c*x]) + (e^2*x^4*(a + b*ArcCosh[c*x]))/4 - ((I/2)*b*d^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(Sq
rt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])])/(Sqrt[-1 +
c*x]*Sqrt[1 + c*x]) + d^2*(a + b*ArcCosh[c*x])*Log[x] - (b*d^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[x])/(Sqrt[-1
+ c*x]*Sqrt[1 + c*x]) - ((I/2)*b*d^2*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(Sqrt[-1 + c*x]*Sqrt
[1 + c*x])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 5790

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[
1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] &
& (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 100

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))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 12

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

Rule 2328

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :>
Dist[Sqrt[1 + (e1*e2*x^2)/(d1*d2)]/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), Int[(a + b*Log[c*x^n])/Sqrt[1 + (e1*e2*x
^2)/(d1*d2)], x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0]

Rule 2326

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(ArcSin[(Rt[-e, 2]*x)/S
qrt[d]]*(a + b*Log[c*x^n]))/Rt[-e, 2], x] - Dist[(b*n)/Rt[-e, 2], Int[ArcSin[(Rt[-e, 2]*x)/Sqrt[d]]/x, x], x]
/; FreeQ[{a, b, c, d, e, n}, x] && GtQ[d, 0] && NegQ[e]

Rule 4625

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Tan[x], x], x, ArcSin[c*
x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 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{\left (d+e x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx &=d e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-(b c) \int \frac{d e x^2+\frac{e^2 x^4}{4}+d^2 \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=d e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-(b c) \int \left (\frac{d e x^2}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{e^2 x^4}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{d^2 \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}\right ) \, dx\\ &=d e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\left (b c d^2\right ) \int \frac{\log (x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-(b c d e) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\frac{1}{4} \left (b c e^2\right ) \int \frac{x^4}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{b d e x \sqrt{-1+c x} \sqrt{1+c x}}{2 c}-\frac{b e^2 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{16 c}+d e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{(b d e) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 c}-\frac{\left (b e^2\right ) \int \frac{3 x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{16 c}-\frac{\left (b c d^2 \sqrt{1-c^2 x^2}\right ) \int \frac{\log (x)}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b d e x \sqrt{-1+c x} \sqrt{1+c x}}{2 c}-\frac{b e^2 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{16 c}-\frac{b d e \cosh ^{-1}(c x)}{2 c^2}+d e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{b d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 b e^2\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{16 c}+\frac{\left (b d^2 \sqrt{1-c^2 x^2}\right ) \int \frac{\sin ^{-1}(c x)}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b d e x \sqrt{-1+c x} \sqrt{1+c x}}{2 c}-\frac{3 b e^2 x \sqrt{-1+c x} \sqrt{1+c x}}{32 c^3}-\frac{b e^2 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{16 c}-\frac{b d e \cosh ^{-1}(c x)}{2 c^2}+d e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{b d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 b e^2\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{32 c^3}+\frac{\left (b d^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b d e x \sqrt{-1+c x} \sqrt{1+c x}}{2 c}-\frac{3 b e^2 x \sqrt{-1+c x} \sqrt{1+c x}}{32 c^3}-\frac{b e^2 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{16 c}-\frac{b d e \cosh ^{-1}(c x)}{2 c^2}-\frac{3 b e^2 \cosh ^{-1}(c x)}{32 c^4}+d e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{i b d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{-1+c x} \sqrt{1+c x}}+d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{b d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 i b d^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b d e x \sqrt{-1+c x} \sqrt{1+c x}}{2 c}-\frac{3 b e^2 x \sqrt{-1+c x} \sqrt{1+c x}}{32 c^3}-\frac{b e^2 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{16 c}-\frac{b d e \cosh ^{-1}(c x)}{2 c^2}-\frac{3 b e^2 \cosh ^{-1}(c x)}{32 c^4}+d e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{i b d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{b d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b d^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b d e x \sqrt{-1+c x} \sqrt{1+c x}}{2 c}-\frac{3 b e^2 x \sqrt{-1+c x} \sqrt{1+c x}}{32 c^3}-\frac{b e^2 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{16 c}-\frac{b d e \cosh ^{-1}(c x)}{2 c^2}-\frac{3 b e^2 \cosh ^{-1}(c x)}{32 c^4}+d e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{i b d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{b d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (i b d^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b d e x \sqrt{-1+c x} \sqrt{1+c x}}{2 c}-\frac{3 b e^2 x \sqrt{-1+c x} \sqrt{1+c x}}{32 c^3}-\frac{b e^2 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{16 c}-\frac{b d e \cosh ^{-1}(c x)}{2 c^2}-\frac{3 b e^2 \cosh ^{-1}(c x)}{32 c^4}+d e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{i b d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{b d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{i b d^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 0.415412, size = 217, normalized size = 0.63 \[ -\frac{1}{2} b d^2 \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+a d^2 \log (x)+a d e x^2+\frac{1}{4} a e^2 x^4-\frac{b d e \left (c x \sqrt{c x-1} \sqrt{c x+1}+2 \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )\right )}{2 c^2}-\frac{b e^2 \left (c x \sqrt{c x-1} \sqrt{c x+1} \left (2 c^2 x^2+3\right )+6 \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )\right )}{32 c^4}+\frac{1}{2} b d^2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )+b d e x^2 \cosh ^{-1}(c x)+\frac{1}{4} b e^2 x^4 \cosh ^{-1}(c x) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

a*d*e*x^2 + (a*e^2*x^4)/4 + b*d*e*x^2*ArcCosh[c*x] + (b*e^2*x^4*ArcCosh[c*x])/4 - (b*d*e*(c*x*Sqrt[-1 + c*x]*S
qrt[1 + c*x] + 2*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(2*c^2) - (b*e^2*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3 +
2*c^2*x^2) + 6*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(32*c^4) + (b*d^2*ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 +
E^(-2*ArcCosh[c*x])]))/2 + a*d^2*Log[x] - (b*d^2*PolyLog[2, -E^(-2*ArcCosh[c*x])])/2

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Maple [A]  time = 0.239, size = 225, normalized size = 0.7 \begin{align*}{\frac{a{e}^{2}{x}^{4}}{4}}+a{x}^{2}de+a{d}^{2}\ln \left ( cx \right ) -{\frac{bdex}{2\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}{d}^{2}}{2}}+{\frac{b{\rm arccosh} \left (cx\right ){e}^{2}{x}^{4}}{4}}+b{\rm arccosh} \left (cx\right ){x}^{2}de+{\frac{b{d}^{2}}{2}{\it polylog} \left ( 2,- \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2} \right ) }-{\frac{b{e}^{2}{x}^{3}}{16\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,b{e}^{2}x}{32\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}+b{d}^{2}{\rm arccosh} \left (cx\right )\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}+1 \right ) -{\frac{bde{\rm arccosh} \left (cx\right )}{2\,{c}^{2}}}-{\frac{3\,b{\rm arccosh} \left (cx\right ){e}^{2}}{32\,{c}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*e^2*x^4+a*x^2*d*e+a*d^2*ln(c*x)-1/2*b*d*e*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-1/2*b*arccosh(c*x)^2*d^2+1/4*b
*arccosh(c*x)*e^2*x^4+b*arccosh(c*x)*x^2*d*e+1/2*b*d^2*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)-1/16*b*
e^2*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-3/32*b*e^2*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3+b*d^2*arccosh(c*x)*ln((c*x+
(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+1)-1/2*b*d*e*arccosh(c*x)/c^2-3/32*b*e^2*arccosh(c*x)/c^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \, a e^{2} x^{4} + a d e x^{2} + a d^{2} \log \left (x\right ) + \int b e^{2} x^{3} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) + 2 \, b d e x \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) + \frac{b d^{2} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*e^2*x^4 + a*d*e*x^2 + a*d^2*log(x) + integrate(b*e^2*x^3*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + 2*b*d*
e*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + b*d^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*acosh(c*x))*(d + e*x**2)**2/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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