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

Optimal. Leaf size=844 \[ -\frac{a}{d^2 x}-\frac{b \text{sech}^{-1}(c x)}{d^2 x}+\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^2 \left (\frac{d}{x}+\sqrt{-d} \sqrt{e}\right )}-\frac{b e \tan ^{-1}\left (\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right )}{2 d^2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{b e \tan ^{-1}\left (\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right )}{2 d^2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{\sqrt{-d} e^{\text{sech}^{-1}(c x)} c}{\sqrt{e}-\sqrt{d c^2+e}}+1\right )}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{\sqrt{-d} e^{\text{sech}^{-1}(c x)} c}{\sqrt{e}+\sqrt{d c^2+e}}+1\right )}{4 (-d)^{5/2}}+\frac{3 b \sqrt{e} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac{3 b \sqrt{e} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac{3 b \sqrt{e} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac{3 b \sqrt{e} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac{b c \sqrt{\frac{1}{c x}-1} \sqrt{1+\frac{1}{c x}}}{d^2} \]

[Out]

(b*c*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/d^2 - a/(d^2*x) - (b*ArcSech[c*x])/(d^2*x) + (e*(a + b*ArcSech[c*x]
))/(4*d^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (e*(a + b*ArcSech[c*x]))/(4*d^2*(Sqrt[-d]*Sqrt[e] + d/x)) - (b*e*ArcTan[
(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(2*d^2*Sq
rt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]) - (b*e*ArcTan[(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[1 +
1/(c*x)])/(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(2*d^2*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + S
qrt[-d]*Sqrt[e]]) - (3*Sqrt[e]*(a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d
+ e])])/(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c
^2*d + e])])/(4*(-d)^(5/2)) - (3*Sqrt[e]*(a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + S
qrt[c^2*d + e])])/(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e
] + Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) + (3*b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[
c^2*d + e]))])/(4*(-d)^(5/2)) - (3*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]
)])/(4*(-d)^(5/2)) + (3*b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(4*(
-d)^(5/2)) - (3*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(5/2))

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Rubi [A]  time = 2.95263, antiderivative size = 844, normalized size of antiderivative = 1., number of steps used = 50, number of rules used = 13, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.619, Rules used = {6303, 5792, 5654, 74, 5707, 5802, 93, 205, 5800, 5562, 2190, 2279, 2391} \[ -\frac{a}{d^2 x}-\frac{b \text{sech}^{-1}(c x)}{d^2 x}+\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^2 \left (\frac{d}{x}+\sqrt{-d} \sqrt{e}\right )}-\frac{b e \tan ^{-1}\left (\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right )}{2 d^2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{b e \tan ^{-1}\left (\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right )}{2 d^2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{\sqrt{-d} e^{\text{sech}^{-1}(c x)} c}{\sqrt{e}-\sqrt{d c^2+e}}+1\right )}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{\sqrt{-d} e^{\text{sech}^{-1}(c x)} c}{\sqrt{e}+\sqrt{d c^2+e}}+1\right )}{4 (-d)^{5/2}}+\frac{3 b \sqrt{e} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac{3 b \sqrt{e} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac{3 b \sqrt{e} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac{3 b \sqrt{e} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac{b c \sqrt{\frac{1}{c x}-1} \sqrt{1+\frac{1}{c x}}}{d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/d^2 - a/(d^2*x) - (b*ArcSech[c*x])/(d^2*x) + (e*(a + b*ArcSech[c*x]
))/(4*d^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (e*(a + b*ArcSech[c*x]))/(4*d^2*(Sqrt[-d]*Sqrt[e] + d/x)) - (b*e*ArcTan[
(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(2*d^2*Sq
rt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]) - (b*e*ArcTan[(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[1 +
1/(c*x)])/(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(2*d^2*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + S
qrt[-d]*Sqrt[e]]) - (3*Sqrt[e]*(a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d
+ e])])/(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c
^2*d + e])])/(4*(-d)^(5/2)) - (3*Sqrt[e]*(a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + S
qrt[c^2*d + e])])/(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e
] + Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) + (3*b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[
c^2*d + e]))])/(4*(-d)^(5/2)) - (3*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]
)])/(4*(-d)^(5/2)) + (3*b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(4*(
-d)^(5/2)) - (3*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(5/2))

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 5792

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

Rule 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 5707

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p
] && (p > 0 || IGtQ[n, 0])

Rule 5802

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 5800

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Subst[Int[((a + b*x)^n*Sinh[x
])/(c*d + e*Cosh[x]), x], x, ArcCosh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rule 5562

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 - b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 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{a+b \text{sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx &=-\operatorname{Subst}\left (\int \frac{x^4 \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{d^2}+\frac{e^2 \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{d^2 \left (e+d x^2\right )^2}-\frac{2 e \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{d^2 \left (e+d x^2\right )}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right ) \, dx,x,\frac{1}{x}\right )}{d^2}+\frac{(2 e) \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{e+d x^2} \, dx,x,\frac{1}{x}\right )}{d^2}-\frac{e^2 \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac{1}{x}\right )}{d^2}\\ &=-\frac{a}{d^2 x}-\frac{b \operatorname{Subst}\left (\int \cosh ^{-1}\left (\frac{x}{c}\right ) \, dx,x,\frac{1}{x}\right )}{d^2}+\frac{(2 e) \operatorname{Subst}\left (\int \left (\frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{2 \sqrt{e} \left (\sqrt{e}-\sqrt{-d} x\right )}+\frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{2 \sqrt{e} \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\frac{1}{x}\right )}{d^2}-\frac{e^2 \operatorname{Subst}\left (\int \left (-\frac{d \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{4 e \left (\sqrt{-d} \sqrt{e}-d x\right )^2}-\frac{d \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{4 e \left (\sqrt{-d} \sqrt{e}+d x\right )^2}-\frac{d \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{2 e \left (-d e-d^2 x^2\right )}\right ) \, dx,x,\frac{1}{x}\right )}{d^2}\\ &=-\frac{a}{d^2 x}-\frac{b \text{sech}^{-1}(c x)}{d^2 x}+\frac{b \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}}} \, dx,x,\frac{1}{x}\right )}{c d^2}+\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{d^2}+\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{d^2}+\frac{e \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{\left (\sqrt{-d} \sqrt{e}-d x\right )^2} \, dx,x,\frac{1}{x}\right )}{4 d}+\frac{e \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{\left (\sqrt{-d} \sqrt{e}+d x\right )^2} \, dx,x,\frac{1}{x}\right )}{4 d}+\frac{e \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{-d e-d^2 x^2} \, dx,x,\frac{1}{x}\right )}{2 d}\\ &=\frac{b c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{d^2}-\frac{a}{d^2 x}-\frac{b \text{sech}^{-1}(c x)}{d^2 x}+\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{\frac{\sqrt{e}}{c}-\sqrt{-d} \cosh (x)} \, dx,x,\text{sech}^{-1}(c x)\right )}{d^2}+\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{\frac{\sqrt{e}}{c}+\sqrt{-d} \cosh (x)} \, dx,x,\text{sech}^{-1}(c x)\right )}{d^2}-\frac{(b e) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}} \left (\sqrt{-d} \sqrt{e}-d x\right )} \, dx,x,\frac{1}{x}\right )}{4 c d^2}+\frac{(b e) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}} \left (\sqrt{-d} \sqrt{e}+d x\right )} \, dx,x,\frac{1}{x}\right )}{4 c d^2}+\frac{e \operatorname{Subst}\left (\int \left (-\frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{2 d \sqrt{e} \left (\sqrt{e}-\sqrt{-d} x\right )}-\frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{2 d \sqrt{e} \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\frac{1}{x}\right )}{2 d}\\ &=\frac{b c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{d^2}-\frac{a}{d^2 x}-\frac{b \text{sech}^{-1}(c x)}{d^2 x}+\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{4 d^2}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{4 d^2}+\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}-\sqrt{-d} e^x} \, dx,x,\text{sech}^{-1}(c x)\right )}{d^2}+\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}-\sqrt{-d} e^x} \, dx,x,\text{sech}^{-1}(c x)\right )}{d^2}+\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}+\sqrt{-d} e^x} \, dx,x,\text{sech}^{-1}(c x)\right )}{d^2}+\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}+\sqrt{-d} e^x} \, dx,x,\text{sech}^{-1}(c x)\right )}{d^2}-\frac{(b e) \operatorname{Subst}\left (\int \frac{1}{d+\frac{\sqrt{-d} \sqrt{e}}{c}-\left (-d+\frac{\sqrt{-d} \sqrt{e}}{c}\right ) x^2} \, dx,x,\frac{\sqrt{1+\frac{1}{c x}}}{\sqrt{-1+\frac{1}{c x}}}\right )}{2 c d^2}+\frac{(b e) \operatorname{Subst}\left (\int \frac{1}{-d+\frac{\sqrt{-d} \sqrt{e}}{c}-\left (d+\frac{\sqrt{-d} \sqrt{e}}{c}\right ) x^2} \, dx,x,\frac{\sqrt{1+\frac{1}{c x}}}{\sqrt{-1+\frac{1}{c x}}}\right )}{2 c d^2}\\ &=\frac{b c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{d^2}-\frac{a}{d^2 x}-\frac{b \text{sech}^{-1}(c x)}{d^2 x}+\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b e \tan ^{-1}\left (\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d^2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{b e \tan ^{-1}\left (\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d^2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{\sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{(-d)^{5/2}}+\frac{\sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{(-d)^{5/2}}-\frac{\sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{(-d)^{5/2}}+\frac{\sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{(-d)^{5/2}}+\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\text{sech}^{-1}(c x)\right )}{(-d)^{5/2}}-\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\text{sech}^{-1}(c x)\right )}{(-d)^{5/2}}+\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\text{sech}^{-1}(c x)\right )}{(-d)^{5/2}}-\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\text{sech}^{-1}(c x)\right )}{(-d)^{5/2}}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{\frac{\sqrt{e}}{c}-\sqrt{-d} \cosh (x)} \, dx,x,\text{sech}^{-1}(c x)\right )}{4 d^2}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{\frac{\sqrt{e}}{c}+\sqrt{-d} \cosh (x)} \, dx,x,\text{sech}^{-1}(c x)\right )}{4 d^2}\\ &=\frac{b c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{d^2}-\frac{a}{d^2 x}-\frac{b \text{sech}^{-1}(c x)}{d^2 x}+\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b e \tan ^{-1}\left (\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d^2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{b e \tan ^{-1}\left (\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d^2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{\sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{(-d)^{5/2}}+\frac{\sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{(-d)^{5/2}}-\frac{\sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{(-d)^{5/2}}+\frac{\sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{(-d)^{5/2}}+\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(c x)}\right )}{(-d)^{5/2}}-\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(c x)}\right )}{(-d)^{5/2}}+\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(c x)}\right )}{(-d)^{5/2}}-\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(c x)}\right )}{(-d)^{5/2}}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}-\sqrt{-d} e^x} \, dx,x,\text{sech}^{-1}(c x)\right )}{4 d^2}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}-\sqrt{-d} e^x} \, dx,x,\text{sech}^{-1}(c x)\right )}{4 d^2}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}+\sqrt{-d} e^x} \, dx,x,\text{sech}^{-1}(c x)\right )}{4 d^2}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}+\sqrt{-d} e^x} \, dx,x,\text{sech}^{-1}(c x)\right )}{4 d^2}\\ &=\frac{b c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{d^2}-\frac{a}{d^2 x}-\frac{b \text{sech}^{-1}(c x)}{d^2 x}+\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b e \tan ^{-1}\left (\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d^2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{b e \tan ^{-1}\left (\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d^2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{b \sqrt{e} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{(-d)^{5/2}}-\frac{b \sqrt{e} \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{(-d)^{5/2}}+\frac{b \sqrt{e} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{(-d)^{5/2}}-\frac{b \sqrt{e} \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{(-d)^{5/2}}-\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\text{sech}^{-1}(c x)\right )}{4 (-d)^{5/2}}+\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\text{sech}^{-1}(c x)\right )}{4 (-d)^{5/2}}-\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\text{sech}^{-1}(c x)\right )}{4 (-d)^{5/2}}+\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\text{sech}^{-1}(c x)\right )}{4 (-d)^{5/2}}\\ &=\frac{b c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{d^2}-\frac{a}{d^2 x}-\frac{b \text{sech}^{-1}(c x)}{d^2 x}+\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b e \tan ^{-1}\left (\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d^2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{b e \tan ^{-1}\left (\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d^2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{b \sqrt{e} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{(-d)^{5/2}}-\frac{b \sqrt{e} \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{(-d)^{5/2}}+\frac{b \sqrt{e} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{(-d)^{5/2}}-\frac{b \sqrt{e} \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{(-d)^{5/2}}-\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(c x)}\right )}{4 (-d)^{5/2}}+\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(c x)}\right )}{4 (-d)^{5/2}}-\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(c x)}\right )}{4 (-d)^{5/2}}+\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(c x)}\right )}{4 (-d)^{5/2}}\\ &=\frac{b c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{d^2}-\frac{a}{d^2 x}-\frac{b \text{sech}^{-1}(c x)}{d^2 x}+\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b e \tan ^{-1}\left (\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d^2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{b e \tan ^{-1}\left (\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d^2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{3 b \sqrt{e} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac{3 b \sqrt{e} \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{3 b \sqrt{e} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac{3 b \sqrt{e} \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 (-d)^{5/2}}\\ \end{align*}

Mathematica [C]  time = 1.08703, size = 1305, normalized size = 1.55 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-4*a*Sqrt[d])/x + 4*b*c*Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)] + (4*b*Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)])/x - (2*a
*Sqrt[d]*e*x)/(d + e*x^2) - (4*b*Sqrt[d]*ArcSech[c*x])/x - (b*Sqrt[d]*e*ArcSech[c*x])/((-I)*Sqrt[d]*Sqrt[e] +
e*x) - (b*Sqrt[d]*e*ArcSech[c*x])/(I*Sqrt[d]*Sqrt[e] + e*x) - 6*a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + 12*b*S
qrt[e]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]
/2])/Sqrt[c^2*d + e]] - 12*b*Sqrt[e]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sqrt[d] +
Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + (3*I)*b*Sqrt[e]*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2
*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 6*b*Sqrt[e]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 +
(I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (3*I)*b*Sqrt[e]*ArcSech[c*x]*Log[1 + (I*(-Sqrt[e
] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - 6*b*Sqrt[e]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[
2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (3*I)*b*Sqrt[e]*ArcSech[c*x]*Log[1
- (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 6*b*Sqrt[e]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqr
t[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (3*I)*b*Sqrt[e]*ArcSech[
c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - 6*b*Sqrt[e]*ArcSin[Sqrt[1 + (I*Sqrt
[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (I*b*e*Log[((
2*I)*Sqrt[e]*(Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (Sqrt[d]*Sqrt[e] + I*c^2*d*x)/Sqrt[c^2*d + e]))/(I
*Sqrt[d] + Sqrt[e]*x)])/Sqrt[c^2*d + e] - (I*b*e*Log[(2*Sqrt[e]*(I*Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)
+ (I*Sqrt[d]*Sqrt[e] + c^2*d*x)/Sqrt[c^2*d + e]))/((-I)*Sqrt[d] + Sqrt[e]*x)])/Sqrt[c^2*d + e] + (3*I)*b*Sqrt
[e]*PolyLog[2, (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (3*I)*b*Sqrt[e]*PolyLog[2, (I*(-S
qrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (3*I)*b*Sqrt[e]*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*
d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (3*I)*b*Sqrt[e]*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E
^ArcSech[c*x])])/(4*d^(5/2))

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Maple [C]  time = 11.697, size = 1952, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a/d^2*e*c^2*x/(c^2*e*x^2+c^2*d)-3/2*a/d^2*e/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))-a/d^2/x-b*arcsech(c*x)/d^
2/x-3/4*c*b/d^2*e*sum(1/_R1/(_R1^2*c^2*d+c^2*d+2*e)*(arcsech(c*x)*ln((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/
2))/_R1)+dilog((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)),_R1=RootOf(c^2*d*_Z^4+(2*c^2*d+4*e)*_Z^2+c^2
*d))+3/4*c*b/d^2*e*sum(_R1/(_R1^2*c^2*d+c^2*d+2*e)*(arcsech(c*x)*ln((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2
))/_R1)+dilog((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)),_R1=RootOf(c^2*d*_Z^4+(2*c^2*d+4*e)*_Z^2+c^2*
d))+1/2*b/c^2*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctan(c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/
((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^4*e+b/c^4*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^2*arcta
n(c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^5+1/2*b/c^2*(-(c
^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctanh(c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/((-c^2*d+2*(e*(c
^2*d+e))^(1/2)-2*e)*d)^(1/2))/d^4*e+b/c^4*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^2*arctanh(c*d*(1/c/x+
(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/d^5+c*b/d^2*(-(c*x-1)/c/x)^(1/
2)*((c*x+1)/c/x)^(1/2)-1/2*b/c^2*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctanh(c*d*(1/c/x+(-1+1/c/x)^(1
/2)*(1+1/c/x)^(1/2))/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/d^4/(c^2*d+e)*(e*(c^2*d+e))^(1/2)*e-b/c^4*(
-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^2*arctanh(c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/((-c^2*d+
2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/d^5/(c^2*d+e)*(e*(c^2*d+e))^(1/2)+b/c^4*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*
e)*d)^(1/2)*e*arctanh(c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2
))/d^5*(e*(c^2*d+e))^(1/2)-b/c^2*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctanh(c*d*(1/c/x+(-1+1/c/x)^(1
/2)*(1+1/c/x)^(1/2))/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/d^4/(c^2*d+e)*e^2-b/c^4*(-(c^2*d-2*(e*(c^2*
d+e))^(1/2)+2*e)*d)^(1/2)*e^3*arctanh(c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/((-c^2*d+2*(e*(c^2*d+e))^(1
/2)-2*e)*d)^(1/2))/d^5/(c^2*d+e)+1/2*b/c^2*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctan(c*d*(1/c/x+(-1+1
/c/x)^(1/2)*(1+1/c/x)^(1/2))/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^4/(c^2*d+e)*(e*(c^2*d+e))^(1/2)*e+
b/c^4*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^2*arctan(c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/((c^
2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^5/(c^2*d+e)*(e*(c^2*d+e))^(1/2)-b/c^4*((c^2*d+2*(e*(c^2*d+e))^(1/2)
+2*e)*d)^(1/2)*e*arctan(c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/
2))/d^5*(e*(c^2*d+e))^(1/2)-b/c^2*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctan(c*d*(1/c/x+(-1+1/c/x)^(1/
2)*(1+1/c/x)^(1/2))/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^4/(c^2*d+e)*e^2-b/c^4*((c^2*d+2*(e*(c^2*d+e
))^(1/2)+2*e)*d)^(1/2)*e^3*arctan(c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2
*e)*d)^(1/2))/d^5/(c^2*d+e)-1/2*b*c^2*x*e*arcsech(c*x)/d^2/(c^2*e*x^2+c^2*d)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{arsech}\left (c x\right ) + a}{e^{2} x^{6} + 2 \, d e x^{4} + d^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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