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

Optimal. Leaf size=370 \[ \frac{i b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-d^2 \log \left (\frac{1}{x}\right ) \left (a+b \text{sech}^{-1}(c x)\right )+d e x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{i b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \log \left (\frac{1}{x}\right ) \csc ^{-1}(c x)}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{b e x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \left (6 c^2 d+e\right )}{6 c^3}-\frac{b e^2 x^3 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}{12 c} \]

[Out]

-(b*e*(6*c^2*d + e)*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)/(6*c^3) - (b*e^2*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*
x)]*x^3)/(12*c) + ((I/2)*b*d^2*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]^2)/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + d
*e*x^2*(a + b*ArcSech[c*x]) + (e^2*x^4*(a + b*ArcSech[c*x]))/4 - (b*d^2*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]*Log[
1 - E^((2*I)*ArcCsc[c*x])])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + (b*d^2*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]*
Log[x^(-1)])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - d^2*(a + b*ArcSech[c*x])*Log[x^(-1)] + ((I/2)*b*d^2*Sqrt
[1 - 1/(c^2*x^2)]*PolyLog[2, E^((2*I)*ArcCsc[c*x])])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])

________________________________________________________________________________________

Rubi [A]  time = 1.09776, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 14, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6303, 266, 43, 5790, 6742, 454, 95, 2328, 2326, 4625, 3717, 2190, 2279, 2391} \[ \frac{i b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-d^2 \log \left (\frac{1}{x}\right ) \left (a+b \text{sech}^{-1}(c x)\right )+d e x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{i b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \log \left (\frac{1}{x}\right ) \csc ^{-1}(c x)}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{b e x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \left (6 c^2 d+e\right )}{6 c^3}-\frac{b e^2 x^3 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}{12 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*e*(6*c^2*d + e)*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)/(6*c^3) - (b*e^2*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*
x)]*x^3)/(12*c) + ((I/2)*b*d^2*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]^2)/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + d
*e*x^2*(a + b*ArcSech[c*x]) + (e^2*x^4*(a + b*ArcSech[c*x]))/4 - (b*d^2*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]*Log[
1 - E^((2*I)*ArcCsc[c*x])])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + (b*d^2*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]*
Log[x^(-1)])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - d^2*(a + b*ArcSech[c*x])*Log[x^(-1)] + ((I/2)*b*d^2*Sqrt
[1 - 1/(c^2*x^2)]*PolyLog[2, E^((2*I)*ArcCsc[c*x])])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])

Rule 6303

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Subst[Int
[((e + d*x^2)^p*(a + b*ArcCosh[x/c])^n)/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ
[n, 0] && IntegersQ[m, p]

Rule 266

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

Rule 43

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

Rule 5790

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

Rule 6742

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

Rule 454

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(a1*a2*e*
(m + 1)), x] + Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(a1*a2*e^n*(m + 1)), Int[(e*x)^(m + n)*(a1
+ b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, p}, x] && EqQ[non2, n/2] && Eq
Q[a2*b1 + a1*b2, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1
])) &&  !ILtQ[p, -1]

Rule 95

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))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] /; FreeQ[{a, b, c, d,
e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0
] && NeQ[m, -1]

Rule 2328

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

Rule 2326

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

Rule 4625

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

Rule 3717

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

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^2 \left (a+b \text{sech}^{-1}(c x)\right )}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (e+d x^2\right )^2 \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{x^5} \, dx,x,\frac{1}{x}\right )\\ &=d e x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{sech}^{-1}(c x)\right )-d^2 \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{b \operatorname{Subst}\left (\int \frac{-\frac{e \left (e+4 d x^2\right )}{4 x^4}+d^2 \log (x)}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=d e x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{sech}^{-1}(c x)\right )-d^2 \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{b \operatorname{Subst}\left (\int \left (-\frac{e \left (e+4 d x^2\right )}{4 x^4 \sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}}}+\frac{d^2 \log (x)}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}}}\right ) \, dx,x,\frac{1}{x}\right )}{c}\\ &=d e x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{sech}^{-1}(c x)\right )-d^2 \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{\left (b d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}}} \, dx,x,\frac{1}{x}\right )}{c}-\frac{(b e) \operatorname{Subst}\left (\int \frac{e+4 d x^2}{x^4 \sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}}} \, dx,x,\frac{1}{x}\right )}{4 c}\\ &=-\frac{b e^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x^3}{12 c}+d e x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{sech}^{-1}(c x)\right )-d^2 \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\frac{\left (b e \left (6 c^2 d+e\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}}} \, dx,x,\frac{1}{x}\right )}{6 c^3}+\frac{\left (b d^2 \sqrt{1-\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=-\frac{b e \left (6 c^2 d+e\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{6 c^3}-\frac{b e^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x^3}{12 c}+d e x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-d^2 \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\frac{\left (b d^2 \sqrt{1-\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \frac{\sin ^{-1}\left (\frac{x}{c}\right )}{x} \, dx,x,\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=-\frac{b e \left (6 c^2 d+e\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{6 c^3}-\frac{b e^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x^3}{12 c}+d e x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-d^2 \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\frac{\left (b d^2 \sqrt{1-\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}(c x)\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=-\frac{b e \left (6 c^2 d+e\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{6 c^3}-\frac{b e^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x^3}{12 c}+\frac{i b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+d e x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-d^2 \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{\left (2 i b d^2 \sqrt{1-\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=-\frac{b e \left (6 c^2 d+e\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{6 c^3}-\frac{b e^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x^3}{12 c}+\frac{i b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+d e x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-d^2 \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{\left (b d^2 \sqrt{1-\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=-\frac{b e \left (6 c^2 d+e\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{6 c^3}-\frac{b e^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x^3}{12 c}+\frac{i b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+d e x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-d^2 \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\frac{\left (i b d^2 \sqrt{1-\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=-\frac{b e \left (6 c^2 d+e\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{6 c^3}-\frac{b e^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x^3}{12 c}+\frac{i b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+d e x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-d^2 \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{i b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \text{Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ \end{align*}

Mathematica [A]  time = 0.420813, size = 176, normalized size = 0.48 \[ \frac{1}{2} b d^2 \text{PolyLog}\left (2,-e^{-2 \text{sech}^{-1}(c x)}\right )+a d^2 \log (x)+a d e x^2+\frac{1}{4} a e^2 x^4-\frac{b d e \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{c^2}-\frac{b e^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (c^2 x^2+2\right )}{12 c^4}-\frac{1}{2} b d^2 \text{sech}^{-1}(c x) \left (\text{sech}^{-1}(c x)+2 \log \left (e^{-2 \text{sech}^{-1}(c x)}+1\right )\right )+b d e x^2 \text{sech}^{-1}(c x)+\frac{1}{4} b e^2 x^4 \text{sech}^{-1}(c x) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

a*d*e*x^2 + (a*e^2*x^4)/4 - (b*d*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/c^2 - (b*e^2*Sqrt[(1 - c*x)/(1 + c*x)]
*(1 + c*x)*(2 + c^2*x^2))/(12*c^4) + b*d*e*x^2*ArcSech[c*x] + (b*e^2*x^4*ArcSech[c*x])/4 - (b*d^2*ArcSech[c*x]
*(ArcSech[c*x] + 2*Log[1 + E^(-2*ArcSech[c*x])]))/2 + a*d^2*Log[x] + (b*d^2*PolyLog[2, -E^(-2*ArcSech[c*x])])/
2

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Maple [A]  time = 0.485, size = 286, normalized size = 0.8 \begin{align*}{\frac{a{e}^{2}{x}^{4}}{4}}+a{x}^{2}de+a{d}^{2}\ln \left ( cx \right ) +{\frac{b \left ({\rm arcsech} \left (cx\right ) \right ) ^{2}{d}^{2}}{2}}-{\frac{b{x}^{3}{e}^{2}}{12\,c}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}-{\frac{bxde}{c}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}+{\frac{b{\rm arcsech} \left (cx\right ){e}^{2}{x}^{4}}{4}}+b{\rm arcsech} \left (cx\right ){x}^{2}de-{\frac{bx{e}^{2}}{6\,{c}^{3}}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}+{\frac{bed}{{c}^{2}}}+{\frac{b{e}^{2}}{6\,{c}^{4}}}-b{d}^{2}{\rm arcsech} \left (cx\right )\ln \left ( 1+ \left ({\frac{1}{cx}}+\sqrt{-1+{\frac{1}{cx}}}\sqrt{1+{\frac{1}{cx}}} \right ) ^{2} \right ) -{\frac{b{d}^{2}}{2}{\it polylog} \left ( 2,- \left ({\frac{1}{cx}}+\sqrt{-1+{\frac{1}{cx}}}\sqrt{1+{\frac{1}{cx}}} \right ) ^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*e^2*x^4+a*x^2*d*e+a*d^2*ln(c*x)+1/2*b*arcsech(c*x)^2*d^2-1/12*b/c*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/
2)*x^3*e^2-b/c*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)*x*d*e+1/4*b*arcsech(c*x)*e^2*x^4+b*arcsech(c*x)*x^2*d*
e-1/6*b/c^3*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)*x*e^2+b/c^2*d*e+1/6*b/c^4*e^2-b*d^2*arcsech(c*x)*ln(1+(1/
c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2)-1/2*b*d^2*polylog(2,-(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a*e^2*x^4 + 2*a*d*e*x^2 + a*d^2 + (b*e^2*x^4 + 2*b*d*e*x^2 + b*d^2)*arcsech(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{asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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