\(\int \frac {(d+e x^2)^3 (a+b \text {arccosh}(c x))}{x} \, dx\) [484]

Optimal result

Integrand size = 21, antiderivative size = 509 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=-\frac {3 b d^2 e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {9 b d e^2 x \sqrt {-1+c x} \sqrt {1+c x}}{32 c^3}-\frac {5 b e^3 x \sqrt {-1+c x} \sqrt {1+c x}}{96 c^5}-\frac {3 b d e^2 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}-\frac {5 b e^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{144 c^3}-\frac {b e^3 x^5 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}-\frac {3 b d^2 e \text {arccosh}(c x)}{4 c^2}-\frac {9 b d e^2 \text {arccosh}(c x)}{32 c^4}-\frac {5 b e^3 \text {arccosh}(c x)}{96 c^6}+\frac {3}{2} d^2 e x^2 (a+b \text {arccosh}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {arccosh}(c x))-\frac {i b d^3 \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^3 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+d^3 (a+b \text {arccosh}(c x)) \log (x)-\frac {b d^3 \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b d^3 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

-3/4*b*d^2*e*arccosh(c*x)/c^2-9/32*b*d*e^2*arccosh(c*x)/c^4-5/96*b*e^3*arccosh(c*x)/c^6+3/2*d^2*e*x^2*(a+b*arc
cosh(c*x))+3/4*d*e^2*x^4*(a+b*arccosh(c*x))+1/6*e^3*x^6*(a+b*arccosh(c*x))+d^3*(a+b*arccosh(c*x))*ln(x)-3/4*b*
d^2*e*x*(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^3-5/96*b*e^3*x*(c*x-1)^(1/2
)*(c*x+1)^(1/2)/c^5-3/16*b*d*e^2*x^3*(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^3-1/36*b*e^3*x^5*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-1/2*I*b*d^3*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)+b*d^3*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)-b*d^3*arcsin(c*x)*ln(x)*(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/2*I*b*d^
3*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {272, 45, 5958, 12, 6874, 92, 54, 102, 2365, 2363, 4721, 3798, 2221, 2317, 2438} \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=d^3 \log (x) (a+b \text {arccosh}(c x))+\frac {3}{2} d^2 e x^2 (a+b \text {arccosh}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {arccosh}(c x))-\frac {5 b e^3 \text {arccosh}(c x)}{96 c^6}-\frac {9 b d e^2 \text {arccosh}(c x)}{32 c^4}-\frac {3 b d^2 e \text {arccosh}(c x)}{4 c^2}-\frac {i b d^3 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {i b d^3 \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^3 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b d^3 \sqrt {1-c^2 x^2} \log (x) \arcsin (c x)}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {5 b e^3 x \sqrt {c x-1} \sqrt {c x+1}}{96 c^5}-\frac {9 b d e^2 x \sqrt {c x-1} \sqrt {c x+1}}{32 c^3}-\frac {5 b e^3 x^3 \sqrt {c x-1} \sqrt {c x+1}}{144 c^3}-\frac {3 b d^2 e x \sqrt {c x-1} \sqrt {c x+1}}{4 c}-\frac {3 b d e^2 x^3 \sqrt {c x-1} \sqrt {c x+1}}{16 c}-\frac {b e^3 x^5 \sqrt {c x-1} \sqrt {c x+1}}{36 c} \]

[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 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 54

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 92

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 102

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 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 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*} \text {integral}& = \frac {3}{2} d^2 e x^2 (a+b \text {arccosh}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {arccosh}(c x))+d^3 (a+b \text {arccosh}(c x)) \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 (a+b \text {arccosh}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {arccosh}(c x))+d^3 (a+b \text {arccosh}(c x)) \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 (a+b \text {arccosh}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {arccosh}(c x))+d^3 (a+b \text {arccosh}(c x)) \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 (a+b \text {arccosh}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {arccosh}(c x))+d^3 (a+b \text {arccosh}(c x)) \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 (a+b \text {arccosh}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {arccosh}(c x))+d^3 (a+b \text {arccosh}(c x)) \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 \text {arccosh}(c x)}{4 c^2}+\frac {3}{2} d^2 e x^2 (a+b \text {arccosh}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {arccosh}(c x))+d^3 (a+b \text {arccosh}(c x)) \log (x)-\frac {b d^3 \sqrt {1-c^2 x^2} \arcsin (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 {\arcsin (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 \text {arccosh}(c x)}{4 c^2}+\frac {3}{2} d^2 e x^2 (a+b \text {arccosh}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {arccosh}(c x))+d^3 (a+b \text {arccosh}(c x)) \log (x)-\frac {b d^3 \sqrt {1-c^2 x^2} \arcsin (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 ) \text {Subst}(\int x \cot (x) \, dx,x,\arcsin (c x))}{\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 \text {arccosh}(c x)}{4 c^2}-\frac {9 b d e^2 \text {arccosh}(c x)}{32 c^4}+\frac {3}{2} d^2 e x^2 (a+b \text {arccosh}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {arccosh}(c x))-\frac {i b d^3 \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+d^3 (a+b \text {arccosh}(c x)) \log (x)-\frac {b d^3 \sqrt {1-c^2 x^2} \arcsin (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 ) \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\arcsin (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 \text {arccosh}(c x)}{4 c^2}-\frac {9 b d e^2 \text {arccosh}(c x)}{32 c^4}+\frac {3}{2} d^2 e x^2 (a+b \text {arccosh}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {arccosh}(c x))-\frac {i b d^3 \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^3 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+d^3 (a+b \text {arccosh}(c x)) \log (x)-\frac {b d^3 \sqrt {1-c^2 x^2} \arcsin (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 ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\arcsin (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 \text {arccosh}(c x)}{4 c^2}-\frac {9 b d e^2 \text {arccosh}(c x)}{32 c^4}-\frac {5 b e^3 \text {arccosh}(c x)}{96 c^6}+\frac {3}{2} d^2 e x^2 (a+b \text {arccosh}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {arccosh}(c x))-\frac {i b d^3 \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^3 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+d^3 (a+b \text {arccosh}(c x)) \log (x)-\frac {b d^3 \sqrt {1-c^2 x^2} \arcsin (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 ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \arcsin (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 \text {arccosh}(c x)}{4 c^2}-\frac {9 b d e^2 \text {arccosh}(c x)}{32 c^4}-\frac {5 b e^3 \text {arccosh}(c x)}{96 c^6}+\frac {3}{2} d^2 e x^2 (a+b \text {arccosh}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {arccosh}(c x))-\frac {i b d^3 \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^3 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+d^3 (a+b \text {arccosh}(c x)) \log (x)-\frac {b d^3 \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b d^3 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.79 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.68 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=\frac {3}{2} a d^2 e x^2+\frac {3}{4} a d e^2 x^4+\frac {1}{6} a e^3 x^6-\frac {3 b d^2 e \left (c x \sqrt {-1+c x} \sqrt {1+c x}-2 c^2 x^2 \text {arccosh}(c x)+2 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{4 c^2}-\frac {3 b d e^2 \left (c x \sqrt {\frac {-1+c x}{1+c x}} \left (3+3 c x+2 c^2 x^2+2 c^3 x^3\right )-8 c^4 x^4 \text {arccosh}(c x)+6 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{32 c^4}-\frac {b e^3 \left (c x \sqrt {\frac {-1+c x}{1+c x}} \left (15+15 c x+10 c^2 x^2+10 c^3 x^3+8 c^4 x^4+8 c^5 x^5\right )-48 c^6 x^6 \text {arccosh}(c x)+30 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{288 c^6}+a d^3 \log (x)+\frac {1}{2} b d^3 \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right ) \]

[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)/(1 + c*x)]
*(3 + 3*c*x + 2*c^2*x^2 + 2*c^3*x^3) - 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*Sqrt[(-1 + c*x)/(1 + c*x)]*(15 + 15*c*x + 10*c^2*x^2 + 10*c^3*x^3 + 8*c^4*x^4 + 8*c^5*x^5) -
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

Maple [A] (verified)

Time = 1.17 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.68

method	result	size
parts	\(a \left (\frac {e^{3} x^{6}}{6}+\frac {3 d \,e^{2} x^{4}}{4}+\frac {3 d^{2} e \,x^{2}}{2}+d^{3} \ln \left (x \right )\right )+\frac {b \,\operatorname {arccosh}\left (c x \right ) e^{3} x^{6}}{6}-\frac {9 b d \,e^{2} \operatorname {arccosh}\left (c x \right )}{32 c^{4}}-\frac {3 b \,d^{2} e \,\operatorname {arccosh}\left (c x \right )}{4 c^{2}}-\frac {b \,e^{3} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}}{36 c}-\frac {5 b \,e^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{144 c^{3}}-\frac {5 b \,e^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{96 c^{5}}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{4}}{4}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{2}}{2}+b \,d^{3} \operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {5 b \,e^{3} \operatorname {arccosh}\left (c x \right )}{96 c^{6}}-\frac {3 b d \,e^{2} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{16 c}-\frac {9 b d \,e^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{32 c^{3}}-\frac {3 b \,d^{2} e x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c}-\frac {b \,d^{3} \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {b \,d^{3} \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\)	\(348\)
derivativedivides	\(\frac {a \left (\frac {3 c^{6} d^{2} e \,x^{2}}{2}+\frac {3 c^{6} d \,e^{2} x^{4}}{4}+\frac {c^{6} e^{3} x^{6}}{6}+c^{6} d^{3} \ln \left (c x \right )\right )}{c^{6}}+\frac {b \,d^{3} \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\frac {b \,\operatorname {arccosh}\left (c x \right ) e^{3} x^{6}}{6}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{2}}{2}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{4}}{4}-\frac {3 b d \,e^{2} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{16 c}-\frac {9 b d \,e^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{32 c^{3}}-\frac {3 b \,d^{2} e x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c}-\frac {3 b \,d^{2} e \,\operatorname {arccosh}\left (c x \right )}{4 c^{2}}+b \,d^{3} \operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {9 b d \,e^{2} \operatorname {arccosh}\left (c x \right )}{32 c^{4}}-\frac {b \,d^{3} \operatorname {arccosh}\left (c x \right )^{2}}{2}-\frac {b \,e^{3} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}}{36 c}-\frac {5 b \,e^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{144 c^{3}}-\frac {5 b \,e^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{96 c^{5}}-\frac {5 b \,e^{3} \operatorname {arccosh}\left (c x \right )}{96 c^{6}}\)	\(365\)
default	\(\frac {a \left (\frac {3 c^{6} d^{2} e \,x^{2}}{2}+\frac {3 c^{6} d \,e^{2} x^{4}}{4}+\frac {c^{6} e^{3} x^{6}}{6}+c^{6} d^{3} \ln \left (c x \right )\right )}{c^{6}}+\frac {b \,d^{3} \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\frac {b \,\operatorname {arccosh}\left (c x \right ) e^{3} x^{6}}{6}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{2}}{2}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{4}}{4}-\frac {3 b d \,e^{2} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{16 c}-\frac {9 b d \,e^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{32 c^{3}}-\frac {3 b \,d^{2} e x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c}-\frac {3 b \,d^{2} e \,\operatorname {arccosh}\left (c x \right )}{4 c^{2}}+b \,d^{3} \operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {9 b d \,e^{2} \operatorname {arccosh}\left (c x \right )}{32 c^{4}}-\frac {b \,d^{3} \operatorname {arccosh}\left (c x \right )^{2}}{2}-\frac {b \,e^{3} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}}{36 c}-\frac {5 b \,e^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{144 c^{3}}-\frac {5 b \,e^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{96 c^{5}}-\frac {5 b \,e^{3} \operatorname {arccosh}\left (c x \right )}{96 c^{6}}\)	\(365\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}}{x}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3}{x} \,d x \]

[In]

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

[Out]

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