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

Optimal. Leaf size=476 \[ -\frac{3 i b d^2 e \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt{c x-1} \sqrt{c x+1}}+3 d^2 e \log (x) \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3 i b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b d^2 e \sqrt{1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b e^2 x \left (1-c^2 x^2\right ) \left (8 c^2 d+e\right )}{32 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b e^2 \sqrt{c^2 x^2-1} \left (8 c^2 d+e\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{32 c^4 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

-(b*c*d^3*(1 - c^2*x^2))/(2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*e^2*(8*c^2*d + e)*x*(1 - c^2*x^2))/(32*c^3*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^3*x^3*(1 - c^2*x^2))/(16*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^3*(a + b*Ar
cCosh[c*x]))/(2*x^2) + (3*d*e^2*x^2*(a + b*ArcCosh[c*x]))/2 + (e^3*x^4*(a + b*ArcCosh[c*x]))/4 - (((3*I)/2)*b*
d^2*e*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*b*e^2*(8*c^2*d + e)*Sqrt[-1 + c^2*x
^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(32*c^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*d^2*e*Sqrt[1 - c^2*x^2]*Ar
cSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + 3*d^2*e*(a + b*ArcCosh[c*x])*Log[x]
- (3*b*d^2*e*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (((3*I)/2)*b*d^2*e*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 = 1.76026, antiderivative size = 476, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 19, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.905, Rules used = {266, 43, 5790, 12, 6742, 1610, 1807, 1584, 459, 321, 217, 206, 2328, 2326, 4625, 3717, 2190, 2279, 2391} \[ -\frac{3 i b d^2 e \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt{c x-1} \sqrt{c x+1}}+3 d^2 e \log (x) \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3 i b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b d^2 e \sqrt{1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b e^2 x \left (1-c^2 x^2\right ) \left (8 c^2 d+e\right )}{32 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b e^2 \sqrt{c^2 x^2-1} \left (8 c^2 d+e\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{32 c^4 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*c*d^3*(1 - c^2*x^2))/(2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*e^2*(8*c^2*d + e)*x*(1 - c^2*x^2))/(32*c^3*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^3*x^3*(1 - c^2*x^2))/(16*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^3*(a + b*Ar
cCosh[c*x]))/(2*x^2) + (3*d*e^2*x^2*(a + b*ArcCosh[c*x]))/2 + (e^3*x^4*(a + b*ArcCosh[c*x]))/4 - (((3*I)/2)*b*
d^2*e*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*b*e^2*(8*c^2*d + e)*Sqrt[-1 + c^2*x
^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(32*c^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*d^2*e*Sqrt[1 - c^2*x^2]*Ar
cSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + 3*d^2*e*(a + b*ArcCosh[c*x])*Log[x]
- (3*b*d^2*e*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (((3*I)/2)*b*d^2*e*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 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 1610

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[((
a + b*x)^FracPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1807

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[(R*(c*x)^(m + 1)*(a + b*x^2)^(p + 1))/(a*c*(m + 1)), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 459

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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^3} \, dx &=-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-(b c) \int \frac{-2 d^3+6 d e^2 x^4+e^3 x^6+12 d^2 e x^2 \log (x)}{4 x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{1}{4} (b c) \int \frac{-2 d^3+6 d e^2 x^4+e^3 x^6+12 d^2 e x^2 \log (x)}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{1}{4} (b c) \int \left (\frac{-2 d^3+6 d e^2 x^4+e^3 x^6}{x^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{12 d^2 e \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}\right ) \, dx\\ &=-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{1}{4} (b c) \int \frac{-2 d^3+6 d e^2 x^4+e^3 x^6}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx-\left (3 b c d^2 e\right ) \int \frac{\log (x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{\left (3 b c d^2 e \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{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{-2 d^3+6 d e^2 x^4+e^3 x^6}{x^2 \sqrt{-1+c^2 x^2}} \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 b d^2 e \sqrt{1-c^2 x^2}\right ) \int \frac{\sin ^{-1}(c x)}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{6 d e^2 x^3+e^3 x^5}{x \sqrt{-1+c^2 x^2}} \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 b d^2 e \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{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{x^2 \left (6 d e^2+e^3 x^2\right )}{\sqrt{-1+c^2 x^2}} \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3 i b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{-1+c x} \sqrt{1+c x}}+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (6 i b d^2 e \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{\left (b \left (-24 c^2 d e^2-3 e^3\right ) \sqrt{-1+c^2 x^2}\right ) \int \frac{x^2}{\sqrt{-1+c^2 x^2}} \, dx}{16 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 b e^2 \left (8 c^2 d+e\right ) x \left (1-c^2 x^2\right )}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3 i b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 b d^2 e \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}}+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 b d^2 e \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{\left (b \left (-24 c^2 d e^2-3 e^3\right ) \sqrt{-1+c^2 x^2}\right ) \int \frac{1}{\sqrt{-1+c^2 x^2}} \, dx}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 b e^2 \left (8 c^2 d+e\right ) x \left (1-c^2 x^2\right )}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3 i b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 b d^2 e \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}}+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 i b d^2 e \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{\left (b \left (-24 c^2 d e^2-3 e^3\right ) \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,\frac{x}{\sqrt{-1+c^2 x^2}}\right )}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 b e^2 \left (8 c^2 d+e\right ) x \left (1-c^2 x^2\right )}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3 i b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{3 b e^2 \left (8 c^2 d+e\right ) \sqrt{-1+c^2 x^2} \tanh ^{-1}\left (\frac{c x}{\sqrt{-1+c^2 x^2}}\right )}{32 c^4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 b d^2 e \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}}+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{3 i b d^2 e \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 = 0.657731, size = 267, normalized size = 0.56 \[ \frac{1}{4} \left (-6 b d^2 e \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+12 a d^2 e \log (x)-\frac{2 a d^3}{x^2}+6 a d e^2 x^2+a e^3 x^4-\frac{3 b d e^2 \left (c x \sqrt{c x-1} \sqrt{c x+1}+2 \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )\right )}{c^2}-\frac{b e^3 \left (c x \sqrt{c x-1} \sqrt{c x+1} \left (2 c^2 x^2+3\right )+6 \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )\right )}{8 c^4}+6 b d^2 e \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )+\frac{2 b d^3 \left (c x \sqrt{c x-1} \sqrt{c x+1}-\cosh ^{-1}(c x)\right )}{x^2}+6 b d e^2 x^2 \cosh ^{-1}(c x)+b e^3 x^4 \cosh ^{-1}(c x)\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*a*d^3)/x^2 + 6*a*d*e^2*x^2 + a*e^3*x^4 + (2*b*d^3*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - ArcCosh[c*x]))/x^2
+ 6*b*d*e^2*x^2*ArcCosh[c*x] + b*e^3*x^4*ArcCosh[c*x] - (3*b*d*e^2*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*ArcTa
nh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/c^2 - (b*e^3*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3 + 2*c^2*x^2) + 6*ArcTanh[Sq
rt[(-1 + c*x)/(1 + c*x)]]))/(8*c^4) + 6*b*d^2*e*ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])]) +
12*a*d^2*e*Log[x] - 6*b*d^2*e*PolyLog[2, -E^(-2*ArcCosh[c*x])])/4

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Maple [A]  time = 0.165, size = 296, normalized size = 0.6 \begin{align*}{\frac{a{e}^{3}{x}^{4}}{4}}+{\frac{3\,a{x}^{2}d{e}^{2}}{2}}+3\,a{d}^{2}e\ln \left ( cx \right ) -{\frac{{d}^{3}a}{2\,{x}^{2}}}+{\frac{b{\rm arccosh} \left (cx\right ){e}^{3}{x}^{4}}{4}}-{\frac{b{d}^{3}{\rm arccosh} \left (cx\right )}{2\,{x}^{2}}}+{\frac{3\,b{\rm arccosh} \left (cx\right ){x}^{2}d{e}^{2}}{2}}+{\frac{3\,b{d}^{2}e}{2}{\it polylog} \left ( 2,- \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2} \right ) }-{\frac{b{e}^{3}{x}^{3}}{16\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,b{e}^{3}x}{32\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{{d}^{3}b{c}^{2}}{2}}+{\frac{c{d}^{3}b}{2\,x}\sqrt{cx-1}\sqrt{cx+1}}+3\,b{d}^{2}e{\rm arccosh} \left (cx\right )\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}+1 \right ) -{\frac{3\,bd{\rm arccosh} \left (cx\right ){e}^{2}}{4\,{c}^{2}}}-{\frac{3\,bxd{e}^{2}}{4\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,b{d}^{2}e \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{2}}-{\frac{3\,b{\rm arccosh} \left (cx\right ){e}^{3}}{32\,{c}^{4}}} \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^3,x)

[Out]

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

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

[Out]

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

[Out]

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

[Out]

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