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

Optimal. Leaf size=1276 \[ \text{result too large to display} \]

[Out]

(b*c*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/(16*Sqrt[-d]*Sqrt[e]*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] - d/x)) + (b*c*S
qrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/(16*Sqrt[-d]*Sqrt[e]*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] + d/x)) + (a + b*ArcSe
ch[c*x])/(16*Sqrt[-d]*Sqrt[e]*(Sqrt[-d]*Sqrt[e] - d/x)^2) + (a + b*ArcSech[c*x])/(16*d*e*(Sqrt[-d]*Sqrt[e] - d
/x)) - (a + b*ArcSech[c*x])/(16*Sqrt[-d]*Sqrt[e]*(Sqrt[-d]*Sqrt[e] + d/x)^2) - (a + b*ArcSech[c*x])/(16*d*e*(S
qrt[-d]*Sqrt[e] + d/x)) + (b*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)])])/(8*(c*d - Sqrt[-d]*Sqrt[e])^(3/2)*(c*d + Sqrt[-d]*Sqrt[e])^(3/2)) - (b*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)])])/(8*d*Sqrt[c*d
- Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*e) + (b*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)])])/(8*(c*d - Sqrt[-d]*Sqrt[e])^(3/2)*(c*d + Sqrt[-d]*Sqrt[
e])^(3/2)) - (b*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)])])/(8*d*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*e) - ((a + b*ArcSech[c*x])*Log[1
- (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcSech[c*x])*Lo
g[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) - ((a + b*ArcSech[c*x]
)*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcSech[
c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + (b*PolyLog[2
, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(16*(-d)^(3/2)*e^(3/2)) - (b*PolyLog[2, (c*Sqrt
[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + (b*PolyLog[2, -((c*Sqrt[-d]*E^Arc
Sech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(16*(-d)^(3/2)*e^(3/2)) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/
(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 4.04015, antiderivative size = 1276, normalized size of antiderivative = 1., number of steps used = 63, number of rules used = 12, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {6303, 5792, 5707, 5802, 96, 93, 205, 5800, 5562, 2190, 2279, 2391} \[ \frac{b \sqrt{\frac{1}{c x}-1} \sqrt{1+\frac{1}{c x}} c}{16 \sqrt{-d} \sqrt{e} \left (d c^2+e\right ) \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{b \sqrt{\frac{1}{c x}-1} \sqrt{1+\frac{1}{c x}} c}{16 \sqrt{-d} \sqrt{e} \left (d c^2+e\right ) \left (\frac{d}{x}+\sqrt{-d} \sqrt{e}\right )}+\frac{a+b \text{sech}^{-1}(c x)}{16 d e \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{a+b \text{sech}^{-1}(c x)}{16 d e \left (\frac{d}{x}+\sqrt{-d} \sqrt{e}\right )}+\frac{a+b \text{sech}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )^2}-\frac{a+b \text{sech}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left (\frac{d}{x}+\sqrt{-d} \sqrt{e}\right )^2}-\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} e}+\frac{b \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 )}{8 \left (c d-\sqrt{-d} \sqrt{e}\right )^{3/2} \left (c d+\sqrt{-d} \sqrt{e}\right )^{3/2}}-\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} e}+\frac{b \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 )}{8 \left (c d-\sqrt{-d} \sqrt{e}\right )^{3/2} \left (c d+\sqrt{-d} \sqrt{e}\right )^{3/2}}-\frac{\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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{\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 )}{16 (-d)^{3/2} e^{3/2}}-\frac{\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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{\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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/(16*Sqrt[-d]*Sqrt[e]*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] - d/x)) + (b*c*S
qrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/(16*Sqrt[-d]*Sqrt[e]*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] + d/x)) + (a + b*ArcSe
ch[c*x])/(16*Sqrt[-d]*Sqrt[e]*(Sqrt[-d]*Sqrt[e] - d/x)^2) + (a + b*ArcSech[c*x])/(16*d*e*(Sqrt[-d]*Sqrt[e] - d
/x)) - (a + b*ArcSech[c*x])/(16*Sqrt[-d]*Sqrt[e]*(Sqrt[-d]*Sqrt[e] + d/x)^2) - (a + b*ArcSech[c*x])/(16*d*e*(S
qrt[-d]*Sqrt[e] + d/x)) + (b*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)])])/(8*(c*d - Sqrt[-d]*Sqrt[e])^(3/2)*(c*d + Sqrt[-d]*Sqrt[e])^(3/2)) - (b*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)])])/(8*d*Sqrt[c*d
- Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*e) + (b*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)])])/(8*(c*d - Sqrt[-d]*Sqrt[e])^(3/2)*(c*d + Sqrt[-d]*Sqrt[
e])^(3/2)) - (b*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)])])/(8*d*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*e) - ((a + b*ArcSech[c*x])*Log[1
- (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcSech[c*x])*Lo
g[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) - ((a + b*ArcSech[c*x]
)*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcSech[
c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + (b*PolyLog[2
, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(16*(-d)^(3/2)*e^(3/2)) - (b*PolyLog[2, (c*Sqrt
[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + (b*PolyLog[2, -((c*Sqrt[-d]*E^Arc
Sech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(16*(-d)^(3/2)*e^(3/2)) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/
(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2))

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 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 96

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] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[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{x^2 \left (a+b \text{sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2 \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{\left (e+d x^2\right )^3} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{e \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{d \left (e+d x^2\right )^3}+\frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{d \left (e+d x^2\right )^2}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\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}+\frac{e \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{\left (e+d x^2\right )^3} \, dx,x,\frac{1}{x}\right )}{d}\\ &=-\frac{\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}+\frac{e \operatorname{Subst}\left (\int \left (-\frac{d^3 \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{8 (-d)^{3/2} e^{3/2} \left (\sqrt{-d} \sqrt{e}-d x\right )^3}-\frac{3 d \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{16 e^2 \left (\sqrt{-d} \sqrt{e}-d x\right )^2}-\frac{d^3 \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{8 (-d)^{3/2} e^{3/2} \left (\sqrt{-d} \sqrt{e}+d x\right )^3}-\frac{3 d \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{16 e^2 \left (\sqrt{-d} \sqrt{e}+d x\right )^2}-\frac{3 d \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{8 e^2 \left (-d e-d^2 x^2\right )}\right ) \, dx,x,\frac{1}{x}\right )}{d}\\ &=-\frac{3 \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 )}{16 e}-\frac{3 \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 )}{16 e}+\frac{\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 e}+\frac{\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 e}-\frac{3 \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 )}{8 e}+\frac{\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 e}-\frac{\sqrt{-d} \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{\left (\sqrt{-d} \sqrt{e}-d x\right )^3} \, dx,x,\frac{1}{x}\right )}{8 \sqrt{e}}-\frac{\sqrt{-d} \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{\left (\sqrt{-d} \sqrt{e}+d x\right )^3} \, dx,x,\frac{1}{x}\right )}{8 \sqrt{e}}\\ &=\frac{a+b \text{sech}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )^2}+\frac{a+b \text{sech}^{-1}(c x)}{16 d e \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{a+b \text{sech}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )^2}-\frac{a+b \text{sech}^{-1}(c x)}{16 d e \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{3 \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 )}{8 e}+\frac{\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 e}+\frac{(3 b) \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 )}{16 c d e}-\frac{(3 b) \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 )}{16 c d e}-\frac{b \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 e}+\frac{b \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 e}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}} \left (\sqrt{-d} \sqrt{e}-d x\right )^2} \, dx,x,\frac{1}{x}\right )}{16 c \sqrt{-d} \sqrt{e}}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}} \left (\sqrt{-d} \sqrt{e}+d x\right )^2} \, dx,x,\frac{1}{x}\right )}{16 c \sqrt{-d} \sqrt{e}}\\ &=\frac{b c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{16 \sqrt{-d} \sqrt{e} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{b c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{16 \sqrt{-d} \sqrt{e} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{a+b \text{sech}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )^2}+\frac{a+b \text{sech}^{-1}(c x)}{16 d e \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{a+b \text{sech}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )^2}-\frac{a+b \text{sech}^{-1}(c x)}{16 d e \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{3 \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 )}{16 d e^{3/2}}+\frac{3 \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 )}{16 d e^{3/2}}-\frac{\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 e^{3/2}}-\frac{\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 e^{3/2}}+\frac{(3 b) \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 )}{8 c d e}-\frac{(3 b) \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 )}{8 c d e}-\frac{b \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 e}+\frac{b \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 e}+\frac{b \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 )}{16 c d \left (c^2 d+e\right )}-\frac{b \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 )}{16 c d \left (c^2 d+e\right )}\\ &=\frac{b c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{16 \sqrt{-d} \sqrt{e} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{b c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{16 \sqrt{-d} \sqrt{e} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{a+b \text{sech}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )^2}+\frac{a+b \text{sech}^{-1}(c x)}{16 d e \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{a+b \text{sech}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )^2}-\frac{a+b \text{sech}^{-1}(c x)}{16 d e \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} e}-\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} e}+\frac{3 \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 )}{16 d e^{3/2}}+\frac{3 \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 )}{16 d e^{3/2}}-\frac{\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 e^{3/2}}-\frac{\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 e^{3/2}}+\frac{b \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 )}{8 c d \left (c^2 d+e\right )}-\frac{b \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 )}{8 c d \left (c^2 d+e\right )}\\ &=\frac{b c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{16 \sqrt{-d} \sqrt{e} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{b c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{16 \sqrt{-d} \sqrt{e} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{a+b \text{sech}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )^2}+\frac{a+b \text{sech}^{-1}(c x)}{16 d e \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{a+b \text{sech}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )^2}-\frac{a+b \text{sech}^{-1}(c x)}{16 d e \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} e}+\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} \left (c^2 d+e\right )}-\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} e}+\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} \left (c^2 d+e\right )}+\frac{3 \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 )}{16 d e^{3/2}}+\frac{3 \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 )}{16 d e^{3/2}}+\frac{3 \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 )}{16 d e^{3/2}}+\frac{3 \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 )}{16 d e^{3/2}}-\frac{\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 e^{3/2}}-\frac{\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 e^{3/2}}-\frac{\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 e^{3/2}}-\frac{\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 e^{3/2}}\\ &=\frac{b c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{16 \sqrt{-d} \sqrt{e} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{b c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{16 \sqrt{-d} \sqrt{e} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{a+b \text{sech}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )^2}+\frac{a+b \text{sech}^{-1}(c x)}{16 d e \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{a+b \text{sech}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )^2}-\frac{a+b \text{sech}^{-1}(c x)}{16 d e \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} e}+\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} \left (c^2 d+e\right )}-\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} e}+\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} \left (c^2 d+e\right )}-\frac{\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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{\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 )}{16 (-d)^{3/2} e^{3/2}}-\frac{\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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{\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 )}{16 (-d)^{3/2} e^{3/2}}-\frac{(3 b) \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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{(3 b) \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 )}{16 (-d)^{3/2} e^{3/2}}-\frac{(3 b) \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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{(3 b) \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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{b \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)^{3/2} e^{3/2}}-\frac{b \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)^{3/2} e^{3/2}}+\frac{b \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)^{3/2} e^{3/2}}-\frac{b \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)^{3/2} e^{3/2}}\\ &=\frac{b c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{16 \sqrt{-d} \sqrt{e} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{b c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{16 \sqrt{-d} \sqrt{e} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{a+b \text{sech}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )^2}+\frac{a+b \text{sech}^{-1}(c x)}{16 d e \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{a+b \text{sech}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )^2}-\frac{a+b \text{sech}^{-1}(c x)}{16 d e \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} e}+\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} \left (c^2 d+e\right )}-\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} e}+\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} \left (c^2 d+e\right )}-\frac{\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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{\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 )}{16 (-d)^{3/2} e^{3/2}}-\frac{\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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{\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 )}{16 (-d)^{3/2} e^{3/2}}-\frac{(3 b) \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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{(3 b) \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 )}{16 (-d)^{3/2} e^{3/2}}-\frac{(3 b) \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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{(3 b) \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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{b \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)^{3/2} e^{3/2}}-\frac{b \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)^{3/2} e^{3/2}}+\frac{b \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)^{3/2} e^{3/2}}-\frac{b \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)^{3/2} e^{3/2}}\\ &=\frac{b c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{16 \sqrt{-d} \sqrt{e} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{b c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{16 \sqrt{-d} \sqrt{e} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{a+b \text{sech}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )^2}+\frac{a+b \text{sech}^{-1}(c x)}{16 d e \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{a+b \text{sech}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )^2}-\frac{a+b \text{sech}^{-1}(c x)}{16 d e \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} e}+\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} \left (c^2 d+e\right )}-\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} e}+\frac{b \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 )}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} \left (c^2 d+e\right )}-\frac{\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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{\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 )}{16 (-d)^{3/2} e^{3/2}}-\frac{\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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{\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 )}{16 (-d)^{3/2} e^{3/2}}+\frac{b \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac{b \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac{b \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac{b \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}\\ \end{align*}

Mathematica [C]  time = 6.15685, size = 2030, normalized size = 1.59 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(a*x)/(4*e*(d + e*x^2)^2) + (a*x)/(8*d*e*(d + e*x^2)) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(3/2)) +
b*(((-I/16)*(((-I)*Sqrt[e]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(Sqrt[d]*(c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*
x)) - ArcSech[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) + Log[x]/(d*Sqrt[e]) - Log[1 + Sqrt[(1 - c*x)/(1 + c
*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/(d*Sqrt[e]) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e]
- I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)])
)/((2*c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/(Sqrt[d]*e) + ((I/16)*((I*Sqrt[e]*Sqrt[
(1 - c*x)/(1 + c*x)]*(1 + c*x))/(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcSech[c*x]/(Sqrt[e]*(I*Sqrt[
d] + Sqrt[e]*x)^2) + Log[x]/(d*Sqrt[e]) - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/(
d*Sqrt[e]) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqr
t[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x
))])/(d*(c^2*d + e)^(3/2))))/(Sqrt[d]*e) - (-(ArcSech[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) + (I*(Log[x]/Sqrt[e] - L
og[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/Sqrt[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]))/Sqrt[d])/(16*d*e) - (-(ArcSech[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) - (I*(Log[x]/Sqrt[e] - Log[1 +
Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/Sqrt[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]))/Sqrt[d])/(16*d*e) - ((I/32)*(PolyLog[2, -E^(-2*ArcSech[c*x])] - 2*((-4*I)*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]] + ArcSech[c*x]*Lo
g[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])]
+ (2*I)*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])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*Arc
Sin[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])] + PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, ((-I)*(Sqrt[e] + S
qrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])))/(d^(3/2)*e^(3/2)) - ((I/32)*(-PolyLog[2, -E^(-2*ArcSech[c*x])]
+ 2*((-4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSec
h[c*x]/2])/Sqrt[c^2*d + e]] + ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] +
Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*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])] - ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d
+ e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*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])] + PolyLog[2, (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^
ArcSech[c*x])] + PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])))/(d^(3/2)*e^(3/2)))

________________________________________________________________________________________

Maple [C]  time = 2.858, size = 2537, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*c^4*a/(c^2*e*x^2+c^2*d)^2/d*x^3-1/8*c^4*a/(c^2*e*x^2+c^2*d)^2/e*x+1/8*a/d/e/(d*e)^(1/2)*arctan(x*e/(d*e)^(
1/2))+1/4/c^2*b*((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))/(c^2*d+e)/d^3+1/4/c^2*b*(-(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))/(c^2*
d+e)/d^3+1/8*b*((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))/e/(c^2*d+e)/d^2+1/8*b*(-(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))/e/(c^2*d
+e)/d^2-1/8*c^4*b*x/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arcsech(c*x)+1/8*c^6*b*x^3/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arc
sech(c*x)-1/8*c^5*b*x^2/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)+1/4/c^2*b*((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))/(c^2*d+e)^2/d^3*(e*(c^2*d+e))^(1/2)-1/4/c^2*b*(-(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))/(c^2
*d+e)^2/d^3*(e*(c^2*d+e))^(1/2)-1/4/c^2*b*((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))/(c^2*d+e)^2*e/d^3-1/4/c^2*b*(-(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))/(c^2*d+e)^2*e/d^3+1/8*b*((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))/(c^2*d+e)^2/e/d^2*(e*(c^2*d+e)
)^(1/2)-1/8*b*(-(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))/(c^2*d+e)^2/e/d^2*(e*(c^2*d+e))^(1/2)-1/4*b*(-(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))/(c^2*d+e)^2/d^2-1/4*b*((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))/(c^2*d+e)^2/d^2+1/16*c^3*b/e/(c^2*d+e)*su
m(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))-1/16*c^3*b/e/(c
^2*d+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)+dil
og((_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))-1/16*c*
b/d/(c^2*d+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))/_R
1)+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
/16*c*b/d/(c^2*d+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))-1/8*c^5*b*x^4/d*e/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)-1/4/c^2*b*((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))/e/(c^2*d+e)/d^3*(e*(c^2*d+e))^(1/2)+1/4/c^2*b*(-(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))/e/(
c^2*d+e)/d^3*(e*(c^2*d+e))^(1/2)-1/8*c^6*b*x/e/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arcsech(c*x)*d+1/8*c^4*b*x^3/d*e/
(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arcsech(c*x)

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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(x^2*(a+b*arcsech(c*x))/(e*x^2+d)^3,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 x^{2} \operatorname{arsech}\left (c x\right ) + a x^{2}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arcsech(c*x))/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arcsech(c*x))/(e*x^2+d)^3,x, algorithm="giac")

[Out]

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