3.484 \(\int \frac{(d+e x^2)^3 (a+b \cosh ^{-1}(c x))}{x} \, dx\)

Optimal. Leaf size=509 \[ -\frac{i b d^3 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3}{2} d^2 e x^2 \left (a+b \cosh ^{-1}(c x)\right )+d^3 \log (x) \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3 b d^2 e \cosh ^{-1}(c x)}{4 c^2}-\frac{i b d^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b d^3 \sqrt{1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{9 b d e^2 x \sqrt{c x-1} \sqrt{c x+1}}{32 c^3}-\frac{9 b d e^2 \cosh ^{-1}(c x)}{32 c^4}-\frac{5 b e^3 x^3 \sqrt{c x-1} \sqrt{c x+1}}{144 c^3}-\frac{5 b e^3 x \sqrt{c x-1} \sqrt{c x+1}}{96 c^5}-\frac{5 b e^3 \cosh ^{-1}(c x)}{96 c^6}-\frac{3 b d^2 e x \sqrt{c x-1} \sqrt{c x+1}}{4 c}-\frac{3 b d e^2 x^3 \sqrt{c x-1} \sqrt{c x+1}}{16 c}-\frac{b e^3 x^5 \sqrt{c x-1} \sqrt{c x+1}}{36 c} \]

[Out]

(-3*b*d^2*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c) - (9*b*d*e^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(32*c^3) - (5*b
*e^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(96*c^5) - (3*b*d*e^2*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*c) - (5*b*e^3
*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(144*c^3) - (b*e^3*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(36*c) - (3*b*d^2*e*Ar
cCosh[c*x])/(4*c^2) - (9*b*d*e^2*ArcCosh[c*x])/(32*c^4) - (5*b*e^3*ArcCosh[c*x])/(96*c^6) + (3*d^2*e*x^2*(a +
b*ArcCosh[c*x]))/2 + (3*d*e^2*x^4*(a + b*ArcCosh[c*x]))/4 + (e^3*x^6*(a + b*ArcCosh[c*x]))/6 - ((I/2)*b*d^3*Sq
rt[1 - c^2*x^2]*ArcSin[c*x]^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^3*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^3*(a + b*ArcCosh[c*x])*Log[x] - (b*d^3*Sqrt[1 - c^2*
x^2]*ArcSin[c*x]*Log[x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((I/2)*b*d^3*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*A
rcSin[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])

________________________________________________________________________________________

Rubi [A]  time = 1.09202, antiderivative size = 509, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 15, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {266, 43, 5790, 12, 6742, 90, 52, 100, 2328, 2326, 4625, 3717, 2190, 2279, 2391} \[ -\frac{i b d^3 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3}{2} d^2 e x^2 \left (a+b \cosh ^{-1}(c x)\right )+d^3 \log (x) \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3 b d^2 e \cosh ^{-1}(c x)}{4 c^2}-\frac{i b d^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b d^3 \sqrt{1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{9 b d e^2 x \sqrt{c x-1} \sqrt{c x+1}}{32 c^3}-\frac{9 b d e^2 \cosh ^{-1}(c x)}{32 c^4}-\frac{5 b e^3 x^3 \sqrt{c x-1} \sqrt{c x+1}}{144 c^3}-\frac{5 b e^3 x \sqrt{c x-1} \sqrt{c x+1}}{96 c^5}-\frac{5 b e^3 \cosh ^{-1}(c x)}{96 c^6}-\frac{3 b d^2 e x \sqrt{c x-1} \sqrt{c x+1}}{4 c}-\frac{3 b d e^2 x^3 \sqrt{c x-1} \sqrt{c x+1}}{16 c}-\frac{b e^3 x^5 \sqrt{c x-1} \sqrt{c x+1}}{36 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*d^2*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c) - (9*b*d*e^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(32*c^3) - (5*b
*e^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(96*c^5) - (3*b*d*e^2*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*c) - (5*b*e^3
*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(144*c^3) - (b*e^3*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(36*c) - (3*b*d^2*e*Ar
cCosh[c*x])/(4*c^2) - (9*b*d*e^2*ArcCosh[c*x])/(32*c^4) - (5*b*e^3*ArcCosh[c*x])/(96*c^6) + (3*d^2*e*x^2*(a +
b*ArcCosh[c*x]))/2 + (3*d*e^2*x^4*(a + b*ArcCosh[c*x]))/4 + (e^3*x^6*(a + b*ArcCosh[c*x]))/6 - ((I/2)*b*d^3*Sq
rt[1 - c^2*x^2]*ArcSin[c*x]^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^3*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^3*(a + b*ArcCosh[c*x])*Log[x] - (b*d^3*Sqrt[1 - c^2*
x^2]*ArcSin[c*x]*Log[x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((I/2)*b*d^3*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*A
rcSin[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 12

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

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 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 )^3 \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx &=\frac{3}{2} d^2 e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \cosh ^{-1}(c x)\right )+d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-(b c) \int \frac{18 d^2 e x^2+9 d e^2 x^4+2 e^3 x^6+12 d^3 \log (x)}{12 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{3}{2} d^2 e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \cosh ^{-1}(c x)\right )+d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{1}{12} (b c) \int \frac{18 d^2 e x^2+9 d e^2 x^4+2 e^3 x^6+12 d^3 \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{3}{2} d^2 e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \cosh ^{-1}(c x)\right )+d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{1}{12} (b c) \int \left (\frac{18 d^2 e x^2}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{9 d e^2 x^4}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 e^3 x^6}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{12 d^3 \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}\right ) \, dx\\ &=\frac{3}{2} d^2 e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \cosh ^{-1}(c x)\right )+d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\left (b c d^3\right ) \int \frac{\log (x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\frac{1}{2} \left (3 b c d^2 e\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\frac{1}{4} \left (3 b c d e^2\right ) \int \frac{x^4}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\frac{1}{6} \left (b c e^3\right ) \int \frac{x^6}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{3 b d^2 e x \sqrt{-1+c x} \sqrt{1+c x}}{4 c}-\frac{3 b d e^2 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{16 c}-\frac{b e^3 x^5 \sqrt{-1+c x} \sqrt{1+c x}}{36 c}+\frac{3}{2} d^2 e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \cosh ^{-1}(c x)\right )+d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{\left (3 b d^2 e\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{4 c}-\frac{\left (3 b d e^2\right ) \int \frac{3 x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{16 c}-\frac{\left (b e^3\right ) \int \frac{5 x^4}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{36 c}-\frac{\left (b c d^3 \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{3 b d^2 e x \sqrt{-1+c x} \sqrt{1+c x}}{4 c}-\frac{3 b d e^2 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{16 c}-\frac{b e^3 x^5 \sqrt{-1+c x} \sqrt{1+c x}}{36 c}-\frac{3 b d^2 e \cosh ^{-1}(c x)}{4 c^2}+\frac{3}{2} d^2 e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \cosh ^{-1}(c x)\right )+d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{b d^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (9 b d e^2\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{16 c}-\frac{\left (5 b e^3\right ) \int \frac{x^4}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{36 c}+\frac{\left (b d^3 \sqrt{1-c^2 x^2}\right ) \int \frac{\sin ^{-1}(c x)}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{3 b d^2 e x \sqrt{-1+c x} \sqrt{1+c x}}{4 c}-\frac{9 b d e^2 x \sqrt{-1+c x} \sqrt{1+c x}}{32 c^3}-\frac{3 b d e^2 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{16 c}-\frac{5 b e^3 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{144 c^3}-\frac{b e^3 x^5 \sqrt{-1+c x} \sqrt{1+c x}}{36 c}-\frac{3 b d^2 e \cosh ^{-1}(c x)}{4 c^2}+\frac{3}{2} d^2 e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \cosh ^{-1}(c x)\right )+d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{b d^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (9 b d e^2\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{32 c^3}-\frac{\left (5 b e^3\right ) \int \frac{3 x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{144 c^3}+\frac{\left (b d^3 \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{3 b d^2 e x \sqrt{-1+c x} \sqrt{1+c x}}{4 c}-\frac{9 b d e^2 x \sqrt{-1+c x} \sqrt{1+c x}}{32 c^3}-\frac{3 b d e^2 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{16 c}-\frac{5 b e^3 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{144 c^3}-\frac{b e^3 x^5 \sqrt{-1+c x} \sqrt{1+c x}}{36 c}-\frac{3 b d^2 e \cosh ^{-1}(c x)}{4 c^2}-\frac{9 b d e^2 \cosh ^{-1}(c x)}{32 c^4}+\frac{3}{2} d^2 e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac{i b d^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{-1+c x} \sqrt{1+c x}}+d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{b d^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 b e^3\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{48 c^3}-\frac{\left (2 i b d^3 \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{3 b d^2 e x \sqrt{-1+c x} \sqrt{1+c x}}{4 c}-\frac{9 b d e^2 x \sqrt{-1+c x} \sqrt{1+c x}}{32 c^3}-\frac{5 b e^3 x \sqrt{-1+c x} \sqrt{1+c x}}{96 c^5}-\frac{3 b d e^2 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{16 c}-\frac{5 b e^3 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{144 c^3}-\frac{b e^3 x^5 \sqrt{-1+c x} \sqrt{1+c x}}{36 c}-\frac{3 b d^2 e \cosh ^{-1}(c x)}{4 c^2}-\frac{9 b d e^2 \cosh ^{-1}(c x)}{32 c^4}+\frac{3}{2} d^2 e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac{i b d^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^3 \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^3 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{b d^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 b e^3\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{96 c^5}-\frac{\left (b d^3 \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{3 b d^2 e x \sqrt{-1+c x} \sqrt{1+c x}}{4 c}-\frac{9 b d e^2 x \sqrt{-1+c x} \sqrt{1+c x}}{32 c^3}-\frac{5 b e^3 x \sqrt{-1+c x} \sqrt{1+c x}}{96 c^5}-\frac{3 b d e^2 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{16 c}-\frac{5 b e^3 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{144 c^3}-\frac{b e^3 x^5 \sqrt{-1+c x} \sqrt{1+c x}}{36 c}-\frac{3 b d^2 e \cosh ^{-1}(c x)}{4 c^2}-\frac{9 b d e^2 \cosh ^{-1}(c x)}{32 c^4}-\frac{5 b e^3 \cosh ^{-1}(c x)}{96 c^6}+\frac{3}{2} d^2 e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac{i b d^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^3 \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^3 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{b d^3 \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^3 \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{3 b d^2 e x \sqrt{-1+c x} \sqrt{1+c x}}{4 c}-\frac{9 b d e^2 x \sqrt{-1+c x} \sqrt{1+c x}}{32 c^3}-\frac{5 b e^3 x \sqrt{-1+c x} \sqrt{1+c x}}{96 c^5}-\frac{3 b d e^2 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{16 c}-\frac{5 b e^3 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{144 c^3}-\frac{b e^3 x^5 \sqrt{-1+c x} \sqrt{1+c x}}{36 c}-\frac{3 b d^2 e \cosh ^{-1}(c x)}{4 c^2}-\frac{9 b d e^2 \cosh ^{-1}(c x)}{32 c^4}-\frac{5 b e^3 \cosh ^{-1}(c x)}{96 c^6}+\frac{3}{2} d^2 e x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac{i b d^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^3 \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^3 \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{b d^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{i b d^3 \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 = 0.743615, size = 314, normalized size = 0.62 \[ \frac{1}{2} b d^3 \left (\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )-\text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )\right )+\frac{3}{2} a d^2 e x^2+a d^3 \log (x)+\frac{3}{4} a d e^2 x^4+\frac{1}{6} a e^3 x^6-\frac{3 b d^2 e \left (-2 c^2 x^2 \cosh ^{-1}(c x)+c x \sqrt{c x-1} \sqrt{c x+1}+2 \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )\right )}{4 c^2}-\frac{3 b d e^2 \left (c x \sqrt{c x-1} \sqrt{c x+1} \left (2 c^2 x^2+3\right )-8 c^4 x^4 \cosh ^{-1}(c x)+6 \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )\right )}{32 c^4}-\frac{b e^3 \left (c x \sqrt{c x-1} \sqrt{c x+1} \left (8 c^4 x^4+10 c^2 x^2+15\right )-48 c^6 x^6 \cosh ^{-1}(c x)+30 \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )\right )}{288 c^6} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(3*a*d^2*e*x^2)/2 + (3*a*d*e^2*x^4)/4 + (a*e^3*x^6)/6 - (3*b*d^2*e*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 2*c^2*x
^2*ArcCosh[c*x] + 2*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(4*c^2) - (3*b*d*e^2*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*
x]*(3 + 2*c^2*x^2) - 8*c^4*x^4*ArcCosh[c*x] + 6*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(32*c^4) - (b*e^3*(c*x*S
qrt[-1 + c*x]*Sqrt[1 + c*x]*(15 + 10*c^2*x^2 + 8*c^4*x^4) - 48*c^6*x^6*ArcCosh[c*x] + 30*ArcTanh[Sqrt[(-1 + c*
x)/(1 + c*x)]]))/(288*c^6) + a*d^3*Log[x] + (b*d^3*(ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])
]) - PolyLog[2, -E^(-2*ArcCosh[c*x])]))/2

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Maple [A]  time = 0.139, size = 351, normalized size = 0.7 \begin{align*}{\frac{a{e}^{3}{x}^{6}}{6}}+{\frac{3\,ad{e}^{2}{x}^{4}}{4}}+{\frac{3\,a{d}^{2}e{x}^{2}}{2}}+{d}^{3}a\ln \left ( cx \right ) +{\frac{b{d}^{3}}{2}{\it polylog} \left ( 2,- \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2} \right ) }-{\frac{{d}^{3}b \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{2}}+{\frac{b{\rm arccosh} \left (cx\right ){e}^{3}{x}^{6}}{6}}+{\frac{3\,b{\rm arccosh} \left (cx\right )d{e}^{2}{x}^{4}}{4}}+{\frac{3\,b{\rm arccosh} \left (cx\right ){d}^{2}e{x}^{2}}{2}}-{\frac{3\,bd{e}^{2}{x}^{3}}{16\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{9\,bd{e}^{2}x}{32\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,b{d}^{2}ex}{4\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{e}^{3}{x}^{5}}{36\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{5\,b{e}^{3}{x}^{3}}{144\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{5\,b{e}^{3}x}{96\,{c}^{5}}\sqrt{cx-1}\sqrt{cx+1}}+{d}^{3}b{\rm arccosh} \left (cx\right )\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}+1 \right ) -{\frac{9\,bd{\rm arccosh} \left (cx\right ){e}^{2}}{32\,{c}^{4}}}-{\frac{3\,b{d}^{2}{\rm arccosh} \left (cx\right )e}{4\,{c}^{2}}}-{\frac{5\,b{\rm arccosh} \left (cx\right ){e}^{3}}{96\,{c}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^3*(a+b*arccosh(c*x))/x,x)

[Out]

1/6*a*e^3*x^6+3/4*a*d*e^2*x^4+3/2*a*d^2*e*x^2+d^3*a*ln(c*x)+1/2*d^3*b*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1
/2))^2)-1/2*d^3*b*arccosh(c*x)^2+1/6*b*arccosh(c*x)*e^3*x^6+3/4*b*arccosh(c*x)*d*e^2*x^4+3/2*b*arccosh(c*x)*d^
2*e*x^2-3/16*b*d*e^2*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-9/32*b*d*e^2*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-3/4*b*d^
2*e*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-1/36*b*e^3*x^5*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-5/144*b*e^3*x^3*(c*x-1)^(1/2)
*(c*x+1)^(1/2)/c^3-5/96*b*e^3*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5+d^3*b*arccosh(c*x)*ln((c*x+(c*x-1)^(1/2)*(c*x+
1)^(1/2))^2+1)-9/32*b*d*e^2*arccosh(c*x)/c^4-3/4*b*d^2*e*arccosh(c*x)/c^2-5/96*b*e^3*arccosh(c*x)/c^6

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{6} \, a e^{3} x^{6} + \frac{3}{4} \, a d e^{2} x^{4} + \frac{3}{2} \, a d^{2} e x^{2} + a d^{3} \log \left (x\right ) + \int b e^{3} x^{5} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) + 3 \, b d e^{2} x^{3} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) + 3 \, b d^{2} e x \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) + \frac{b d^{3} \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)^3*(a+b*arccosh(c*x))/x,x, algorithm="maxima")

[Out]

1/6*a*e^3*x^6 + 3/4*a*d*e^2*x^4 + 3/2*a*d^2*e*x^2 + a*d^3*log(x) + integrate(b*e^3*x^5*log(c*x + sqrt(c*x + 1)
*sqrt(c*x - 1)) + 3*b*d*e^2*x^3*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + 3*b*d^2*e*x*log(c*x + sqrt(c*x + 1)*s
qrt(c*x - 1)) + b*d^3*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^{3} x^{6} + 3 \, a d e^{2} x^{4} + 3 \, a d^{2} e x^{2} + a d^{3} +{\left (b e^{3} x^{6} + 3 \, b d e^{2} x^{4} + 3 \, b d^{2} e x^{2} + b d^{3}\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)^3*(a+b*arccosh(c*x))/x,x, algorithm="fricas")

[Out]

integral((a*e^3*x^6 + 3*a*d*e^2*x^4 + 3*a*d^2*e*x^2 + a*d^3 + (b*e^3*x^6 + 3*b*d*e^2*x^4 + 3*b*d^2*e*x^2 + b*d
^3)*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 )^{3}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*acosh(c*x))*(d + e*x**2)**3/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 )}^{3}{\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)^3*(a+b*arccosh(c*x))/x,x, algorithm="giac")

[Out]

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