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

Optimal. Leaf size=542 \[ -\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 d^2}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 d^2}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right )}{2 d^2}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 d^2}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right )}{2 d^2}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{2 d^2 \left (\frac{d}{x^2}+e\right )}+\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{2 b d^2}+\frac{b \sqrt{e} \sqrt{\frac{1}{c^2 x^2}-1} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}-1}}\right )}{2 d^2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \sqrt{c^2 d+e}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.36288, antiderivative size = 542, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 11, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.524, Rules used = {6303, 5792, 5788, 519, 377, 208, 5800, 5562, 2190, 2279, 2391} \[ -\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 d^2}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 d^2}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right )}{2 d^2}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 d^2}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right )}{2 d^2}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{2 d^2 \left (\frac{d}{x^2}+e\right )}+\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{2 b d^2}+\frac{b \sqrt{e} \sqrt{\frac{1}{c^2 x^2}-1} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}-1}}\right )}{2 d^2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \sqrt{c^2 d+e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*(a + b*ArcSech[c*x]))/(2*d^2*(e + d/x^2)) + (a + b*ArcSech[c*x])^2/(2*b*d^2) + (b*Sqrt[e]*Sqrt[-1 + 1/(c^2
*x^2)]*ArcTanh[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*x)])/(2*d^2*Sqrt[c^2*d + e]*Sqrt[-1 + 1/(c*x)
]*Sqrt[1 + 1/(c*x)]) - ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])
/(2*d^2) - ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^2) - (
(a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^2) - ((a + b*ArcSe
ch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^2) - (b*PolyLog[2, -((c*Sqrt[-
d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*d^2) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e]
- Sqrt[c^2*d + e])])/(2*d^2) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*
d^2) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^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 5788

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

Rule 519

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_)*((a2_) + (b2_.)*(x_)^(non2_.))^(
p_), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1*a2 + b1*b2*x^n)^FracP
art[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[
non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(EqQ[n, 2] && IGtQ[q, 0])

Rule 377

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

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[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 \left (d+e x^2\right )^2} \, dx &=-\operatorname{Subst}\left (\int \frac{x^3 \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{e x \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{d \left (e+d x^2\right )^2}+\frac{x \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{d \left (e+d x^2\right )}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{e+d x^2} \, dx,x,\frac{1}{x}\right )}{d}+\frac{e \operatorname{Subst}\left (\int \frac{x \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac{1}{x}\right )}{d}\\ &=-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{2 d^2 \left (e+\frac{d}{x^2}\right )}-\frac{\operatorname{Subst}\left (\int \left (-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{2 d \left (\sqrt{e}-\sqrt{-d} x\right )}+\frac{\sqrt{-d} \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{2 d \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\frac{1}{x}\right )}{d}+\frac{(b e) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}} \left (e+d x^2\right )} \, dx,x,\frac{1}{x}\right )}{2 c d^2}\\ &=-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{2 d^2 \left (e+\frac{d}{x^2}\right )}+\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 )}{2 (-d)^{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 )}{2 (-d)^{3/2}}+\frac{\left (b e \sqrt{-1+\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+\frac{x^2}{c^2}} \left (e+d x^2\right )} \, dx,x,\frac{1}{x}\right )}{2 c d^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{2 d^2 \left (e+\frac{d}{x^2}\right )}+\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 )}{2 (-d)^{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 )}{2 (-d)^{3/2}}+\frac{\left (b e \sqrt{-1+\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{e-\left (d+\frac{e}{c^2}\right ) x^2} \, dx,x,\frac{1}{\sqrt{-1+\frac{1}{c^2 x^2}} x}\right )}{2 c d^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{2 d^2 \left (e+\frac{d}{x^2}\right )}+\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{2 b d^2}+\frac{b \sqrt{e} \sqrt{-1+\frac{1}{c^2 x^2}} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} \sqrt{-1+\frac{1}{c^2 x^2}} x}\right )}{2 d^2 \sqrt{c^2 d+e} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\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 )}{2 (-d)^{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 )}{2 (-d)^{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 )}{2 (-d)^{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 )}{2 (-d)^{3/2}}\\ &=-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{2 d^2 \left (e+\frac{d}{x^2}\right )}+\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{2 b d^2}+\frac{b \sqrt{e} \sqrt{-1+\frac{1}{c^2 x^2}} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} \sqrt{-1+\frac{1}{c^2 x^2}} x}\right )}{2 d^2 \sqrt{c^2 d+e} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-\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 )}{2 d^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 )}{2 d^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 )}{2 d^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 )}{2 d^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 )}{2 d^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 )}{2 d^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 )}{2 d^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 )}{2 d^2}\\ &=-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{2 d^2 \left (e+\frac{d}{x^2}\right )}+\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{2 b d^2}+\frac{b \sqrt{e} \sqrt{-1+\frac{1}{c^2 x^2}} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} \sqrt{-1+\frac{1}{c^2 x^2}} x}\right )}{2 d^2 \sqrt{c^2 d+e} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-\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 )}{2 d^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 )}{2 d^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 )}{2 d^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 )}{2 d^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 )}{2 d^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 )}{2 d^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 )}{2 d^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 )}{2 d^2}\\ &=-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{2 d^2 \left (e+\frac{d}{x^2}\right )}+\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{2 b d^2}+\frac{b \sqrt{e} \sqrt{-1+\frac{1}{c^2 x^2}} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} \sqrt{-1+\frac{1}{c^2 x^2}} x}\right )}{2 d^2 \sqrt{c^2 d+e} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-\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 )}{2 d^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 )}{2 d^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 )}{2 d^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 )}{2 d^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 )}{2 d^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 )}{2 d^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 )}{2 d^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 )}{2 d^2}\\ \end{align*}

Mathematica [F]  time = 40.2766, size = 0, normalized size = 0. \[ \int \frac{a+b \text{sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.757, size = 3326, normalized size = 6.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b/d^2*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*arcsech(c
*x)-3*b/c^2/d^3*e/(c^2*d+e)*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2
)-2*e))*arcsech(c*x)*(e*(c^2*d+e))^(1/2)-2*b/c^4/d^4*e^2/(c^2*d+e)*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x
)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*arcsech(c*x)*(e*(c^2*d+e))^(1/2)-1/4*b*c^2/d/e/(c^2*d+e)*ln(1-c
^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*arcsech(c*x)*(e*(c^2*d+e))
^(1/2)+1/4*b*c^2*(e*(c^2*d+e))^(1/2)/d/e/(c^2*d+e)*arcsech(c*x)*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(
1/2))^2/(-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e))+a/d^2*ln(c*x)+1/2*a*c^2/d/(c^2*e*x^2+c^2*d)-1/2*b/d^2*sum((_R1^2*c
^2*d+2*c^2*d+4*e)/(_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)+d
ilog((_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))+b*arc
sech(c*x)^2/d^2-1/4*b/d^2*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(
1/2)-2*e))-1/2*a/d^2*ln(c^2*e*x^2+c^2*d)-b/c^2/d^3*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/
(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*e+2*b/c^2/d^3*e*arcsech(c*x)^2+2*b/c^4/d^4*e^2*arcsech(c*x)^2-b/c^2/d^3*ar
csech(c*x)^2*(e*(c^2*d+e))^(1/2)-b/c^4/d^4*e^2*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^
2*d-2*(e*(c^2*d+e))^(1/2)-2*e))+1/4*b*c^2/d/(c^2*d+e)*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))
^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))-1/2*b*c^2/d/(c^2*d+e)*arcsech(c*x)^2+1/4*b*(e*(c^2*d+e))^(1/2)/d^2/(c^2
*d+e)*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e))-1/2*b*(e*
(c^2*d+e))^(1/2)/d^2/(c^2*d+e)*arctanh(1/4*(2*c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2+2*c^2*d+4*e)/(c
^2*d*e+e^2)^(1/2))+b*(e*(c^2*d+e))^(1/2)/d^2/(c^2*d+e)*arcsech(c*x)^2-5/2*b/d^2/(c^2*d+e)*arcsech(c*x)^2*e+1/8
*b*c^2*(e*(c^2*d+e))^(1/2)/d/e/(c^2*d+e)*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d+2*
(e*(c^2*d+e))^(1/2)-2*e))-3/2*b/c^2/d^3/(c^2*d+e)*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(
-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*(e*(c^2*d+e))^(1/2)*e-1/2*b*c^2*x^2*e*arcsech(c*x)/(c^2*e*x^2+c^2*d)/d^2+4*
b/c^2/d^3*e^2/(c^2*d+e)*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*
e))*arcsech(c*x)-1/8*b*c^2/d/e/(c^2*d+e)*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*
(e*(c^2*d+e))^(1/2)-2*e))*(e*(c^2*d+e))^(1/2)+2*b/c^4/d^4*e^2/(c^2*d+e)*arcsech(c*x)^2*(e*(c^2*d+e))^(1/2)-b/c
^4/d^4*e^2/(c^2*d+e)*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-
2*e))*(e*(c^2*d+e))^(1/2)-3/4*b/d^2/(c^2*d+e)*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2
*d-2*(e*(c^2*d+e))^(1/2)-2*e))*(e*(c^2*d+e))^(1/2)+5/4*b/d^2/(c^2*d+e)*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)
*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*e+1/2*b/c^2/d^3*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2
)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*(e*(c^2*d+e))^(1/2)+b/c^2/d^3*ln(1-c^2*d*(1/c/x+(-1+1
/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*arcsech(c*x)*(e*(c^2*d+e))^(1/2)-2*b/c^2/d^
3*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*arcsech(c*x)*e-2*b
/c^4/d^4*e^2*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*arcsech
(c*x)+1/2*b*c^2/d/(c^2*d+e)*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2
)-2*e))*arcsech(c*x)-2*b/c^4/d^4*e*arcsech(c*x)^2*(e*(c^2*d+e))^(1/2)+b/c^4/d^4*e*polylog(2,c^2*d*(1/c/x+(-1+1
/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*(e*(c^2*d+e))^(1/2)-4*b/c^2/d^3/(c^2*d+e)*a
rcsech(c*x)^2*e^2-3/2*b/d^2/(c^2*d+e)*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*
d+e))^(1/2)-2*e))*arcsech(c*x)*(e*(c^2*d+e))^(1/2)+1/2*b*(e*(c^2*d+e))^(1/2)/d^2/(c^2*d+e)*arcsech(c*x)*ln(1-c
^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e))+5/2*b/d^2*e/(c^2*d+e)*ln(1
-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*arcsech(c*x)+b/c^4/d^4*e
^3/(c^2*d+e)*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))+2*
b/c^2/d^3/(c^2*d+e)*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2
*e))*e^2-2*b/c^4/d^4*e^3/(c^2*d+e)*arcsech(c*x)^2+2*b/c^4/d^4*e*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(
1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*arcsech(c*x)*(e*(c^2*d+e))^(1/2)+3*b/c^2/d^3/(c^2*d+e)*arcsech(c*x
)^2*(e*(c^2*d+e))^(1/2)*e+2*b/c^4/d^4*e^3/(c^2*d+e)*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^
2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*arcsech(c*x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

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^{5} + 2 \, d e x^{3} + d^{2} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arcsech(c*x) + a)/(e^2*x^5 + 2*d*e*x^3 + d^2*x), x)

________________________________________________________________________________________

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/(e*x**2+d)**2,x)

[Out]

Timed out

________________________________________________________________________________________

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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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