3.5.86 \(\int \frac {(d+e x^2)^3 (a+b \cosh ^{-1}(c x))}{x^3} \, dx\) [486]

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

[Out]

-1/2*d^3*(a+b*arccosh(c*x))/x^2+3/2*d*e^2*x^2*(a+b*arccosh(c*x))+1/4*e^3*x^4*(a+b*arccosh(c*x))+3*d^2*e*(a+b*a
rccosh(c*x))*ln(x)-1/2*b*c*d^3*(-c^2*x^2+1)/x/(c*x-1)^(1/2)/(c*x+1)^(1/2)+3/32*b*e^2*(8*c^2*d+e)*x*(-c^2*x^2+1
)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/16*b*e^3*x^3*(-c^2*x^2+1)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/2*I*b*d^2*e*arcs
in(c*x)^2*(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+3*b*d^2*e*arcsin(c*x)*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))
^2)*(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3*b*d^2*e*arcsin(c*x)*ln(x)*(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/2
)/(c*x+1)^(1/2)-3/2*I*b*d^2*e*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/2)/(c*x+1)
^(1/2)-3/32*b*e^2*(8*c^2*d+e)*arctanh(c*x/(c^2*x^2-1)^(1/2))*(c^2*x^2-1)^(1/2)/c^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.17, antiderivative size = 476, normalized size of antiderivative = 1.00, 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 = {272, 45, 5958, 12, 6874, 1624, 1821, 1598, 470, 327, 223, 212, 2365, 2363, 4721, 3798, 2221, 2317, 2438} \begin {gather*} -\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+3 d^2 e \log (x) \left (a+b \cosh ^{-1}(c x)\right )+\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} \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 i b d^2 e \sqrt {1-c^2 x^2} \text {ArcSin}(c x)^2}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b d^2 e \sqrt {1-c^2 x^2} \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(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) \text {ArcSin}(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 {b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \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 {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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(b*c*d^3*(1 - c^2*x^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*
ArcCosh[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]*
ArcSin[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 12

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

Rule 45

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 212

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

Rule 223

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 272

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 327

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 470

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 1598

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 1624

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 1821

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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 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 2363

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-e, 2]*(x/Sqr
t[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 2365

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :>
Dist[Sqrt[1 + e1*(e2/(d1*d2))*x^2]/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), Int[(a + b*Log[c*x^n])/Sqrt[1 + e1*(e2/(
d1*d2))*x^2], x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 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 3798

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 4721

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

Rule 5958

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 6874

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

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.49, size = 278, normalized size = 0.58 \begin {gather*} \frac {1}{4} \left (-\frac {2 a d^3}{x^2}+6 a d e^2 x^2+a e^3 x^4+\frac {2 b d^3 \left (c x \sqrt {-1+c x} \sqrt {1+c x}-\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)-\frac {3 b d e^2 \left (c x \sqrt {-1+c x} \sqrt {1+c x}+2 \tanh ^{-1}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{c^2}-\frac {b e^3 \left (c x \sqrt {\frac {-1+c x}{1+c x}} \left (3+3 c x+2 c^2 x^2+2 c^3 x^3\right )+6 \tanh ^{-1}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{8 c^4}+6 b d^2 e \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+2 \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )\right )+12 a d^2 e \log (x)-6 b d^2 e \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )\right ) \end {gather*}

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)/(1 + c*x)]*(3 + 3*c*x + 2*c^2*x^2 + 2*c^3*x
^3) + 6*ArcTanh[Sqrt[(-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 = 2.49, size = 296, normalized size = 0.62 \[\frac {a \,e^{3} x^{4}}{4}+\frac {3 a d \,e^{2} x^{2}}{2}-\frac {a \,d^{3}}{2 x^{2}}+3 a \,d^{2} e \ln \left (c x \right )-\frac {3 b \,\mathrm {arccosh}\left (c x \right ) e^{3}}{32 c^{4}}+\frac {c b \,d^{3} \sqrt {c x +1}\, \sqrt {c x -1}}{2 x}+\frac {3 b \,\mathrm {arccosh}\left (c x \right ) x^{2} d \,e^{2}}{2}+\frac {3 b \,d^{2} e \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}-\frac {3 b \sqrt {c x -1}\, \sqrt {c x +1}\, x d \,e^{2}}{4 c}+\frac {b \,\mathrm {arccosh}\left (c x \right ) e^{3} x^{4}}{4}+3 b \,d^{2} e \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {3 b \,d^{2} e \mathrm {arccosh}\left (c x \right )^{2}}{2}-\frac {b \,\mathrm {arccosh}\left (c x \right ) d^{3}}{2 x^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3} e^{3}}{16 c}-\frac {3 b \sqrt {c x -1}\, \sqrt {c x +1}\, x \,e^{3}}{32 c^{3}}-\frac {3 b \,\mathrm {arccosh}\left (c x \right ) d \,e^{2}}{4 c^{2}}-\frac {c^{2} d^{3} b}{2}\]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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*x^4*e^3 + 1/2*b*d^3*(sqrt(c^2*x^2 - 1)*c/x - arccosh(c*x)/x^2) + 3/2*a*d*x^2*e^2 + 3*a*d^2*e*log(x) - 1/
2*a*d^3/x^2 + integrate(b*x^3*e^3*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + 3*b*d*x*e^2*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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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*x^6*e^3 + 3*a*d*x^4*e^2 + 3*a*d^2*x^2*e + a*d^3 + (b*x^6*e^3 + 3*b*d*x^4*e^2 + 3*b*d^2*x^2*e + b*d
^3)*arccosh(c*x))/x^3, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}}{x^{3}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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