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

Optimal. Leaf size=741 \[ -\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 d^3}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 d^3}-\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 d^3}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 d^3}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 d^3}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right )}{2 d^3}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 d^3}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right )}{2 d^3}+\frac{e^2 \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^3 \left (\frac{d}{x^2}+e\right )^2}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{d^3 \left (\frac{d}{x^2}+e\right )}+\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{2 b d^3}-\frac{b e \left (c^2-\frac{1}{x^2}\right )}{8 c d^2 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \left (c^2 d+e\right ) \left (\frac{d}{x^2}+e\right )}-\frac{b \sqrt{e} \sqrt{\frac{1}{c^2 x^2}-1} \left (c^2 d+2 e\right ) \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}-1}}\right )}{8 d^3 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \left (c^2 d+e\right )^{3/2}}+\frac{b \sqrt{e} \sqrt{\frac{1}{c^2 x^2}-1} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}-1}}\right )}{d^3 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \sqrt{c^2 d+e}} \]

[Out]

-(b*e*(c^2 - x^(-2)))/(8*c*d^2*(c^2*d + e)*(e + d/x^2)*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) + (e^2*(a + b*A
rcSech[c*x]))/(4*d^3*(e + d/x^2)^2) - (e*(a + b*ArcSech[c*x]))/(d^3*(e + d/x^2)) + (a + b*ArcSech[c*x])^2/(2*b
*d^3) + (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)])/(d^3*
Sqrt[c^2*d + e]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - (b*Sqrt[e]*(c^2*d + 2*e)*Sqrt[-1 + 1/(c^2*x^2)]*ArcTan
h[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*x)])/(8*d^3*(c^2*d + e)^(3/2)*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^3) -
((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcS
ech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcSech[c*x])*Lo
g[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^3) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSec
h[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*d^3) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2
*d + e])])/(2*d^3) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*d^3) - (b*P
olyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^3)

________________________________________________________________________________________

Rubi [A]  time = 1.54211, antiderivative size = 741, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 12, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {6303, 5792, 5788, 519, 382, 377, 208, 5800, 5562, 2190, 2279, 2391} \[ -\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 d^3}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 d^3}-\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 d^3}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 d^3}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 d^3}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right )}{2 d^3}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 d^3}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right )}{2 d^3}+\frac{e^2 \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^3 \left (\frac{d}{x^2}+e\right )^2}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{d^3 \left (\frac{d}{x^2}+e\right )}+\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{2 b d^3}-\frac{b e \left (c^2-\frac{1}{x^2}\right )}{8 c d^2 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \left (c^2 d+e\right ) \left (\frac{d}{x^2}+e\right )}-\frac{b \sqrt{e} \sqrt{\frac{1}{c^2 x^2}-1} \left (c^2 d+2 e\right ) \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}-1}}\right )}{8 d^3 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \left (c^2 d+e\right )^{3/2}}+\frac{b \sqrt{e} \sqrt{\frac{1}{c^2 x^2}-1} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}-1}}\right )}{d^3 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \sqrt{c^2 d+e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*e*(c^2 - x^(-2)))/(8*c*d^2*(c^2*d + e)*(e + d/x^2)*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) + (e^2*(a + b*A
rcSech[c*x]))/(4*d^3*(e + d/x^2)^2) - (e*(a + b*ArcSech[c*x]))/(d^3*(e + d/x^2)) + (a + b*ArcSech[c*x])^2/(2*b
*d^3) + (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)])/(d^3*
Sqrt[c^2*d + e]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - (b*Sqrt[e]*(c^2*d + 2*e)*Sqrt[-1 + 1/(c^2*x^2)]*ArcTan
h[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*x)])/(8*d^3*(c^2*d + e)^(3/2)*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^3) -
((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcS
ech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcSech[c*x])*Lo
g[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^3) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSec
h[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*d^3) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2
*d + e])])/(2*d^3) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*d^3) - (b*P
olyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^3)

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 382

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

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 )^3} \, dx &=-\operatorname{Subst}\left (\int \frac{x^5 \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^2 x \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{d^2 \left (e+d x^2\right )^3}-\frac{2 e x \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{d^2 \left (e+d x^2\right )^2}+\frac{x \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 \frac{x \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{e+d x^2} \, dx,x,\frac{1}{x}\right )}{d^2}+\frac{(2 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^2}-\frac{e^2 \operatorname{Subst}\left (\int \frac{x \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{\left (e+d x^2\right )^3} \, dx,x,\frac{1}{x}\right )}{d^2}\\ &=\frac{e^2 \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^3 \left (e+\frac{d}{x^2}\right )^2}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{d^3 \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^2}+\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 )}{c d^3}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}} \left (e+d x^2\right )^2} \, dx,x,\frac{1}{x}\right )}{4 c d^3}\\ &=\frac{e^2 \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^3 \left (e+\frac{d}{x^2}\right )^2}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{d^3 \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)^{5/2}}+\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)^{5/2}}+\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 )}{c d^3 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-\frac{\left (b e^2 \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 )^2} \, dx,x,\frac{1}{x}\right )}{4 c d^3 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=-\frac{b e \left (c^2-\frac{1}{x^2}\right )}{8 c d^2 \left (c^2 d+e\right ) \left (e+\frac{d}{x^2}\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}+\frac{e^2 \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^3 \left (e+\frac{d}{x^2}\right )^2}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{d^3 \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)^{5/2}}+\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)^{5/2}}+\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 )}{c d^3 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-\frac{\left (b e \left (c^2 d+2 e\right ) \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 )}{8 c d^3 \left (c^2 d+e\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=-\frac{b e \left (c^2-\frac{1}{x^2}\right )}{8 c d^2 \left (c^2 d+e\right ) \left (e+\frac{d}{x^2}\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}+\frac{e^2 \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^3 \left (e+\frac{d}{x^2}\right )^2}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{d^3 \left (e+\frac{d}{x^2}\right )}+\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{2 b d^3}+\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 )}{d^3 \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)^{5/2}}-\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)^{5/2}}+\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)^{5/2}}+\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)^{5/2}}-\frac{\left (b e \left (c^2 d+2 e\right ) \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 )}{8 c d^3 \left (c^2 d+e\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=-\frac{b e \left (c^2-\frac{1}{x^2}\right )}{8 c d^2 \left (c^2 d+e\right ) \left (e+\frac{d}{x^2}\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}+\frac{e^2 \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^3 \left (e+\frac{d}{x^2}\right )^2}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{d^3 \left (e+\frac{d}{x^2}\right )}+\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{2 b d^3}+\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 )}{d^3 \sqrt{c^2 d+e} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-\frac{b \sqrt{e} \left (c^2 d+2 e\right ) \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 )}{8 d^3 \left (c^2 d+e\right )^{3/2} \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^3}-\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^3}-\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^3}-\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^3}+\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^3}+\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^3}+\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^3}+\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^3}\\ &=-\frac{b e \left (c^2-\frac{1}{x^2}\right )}{8 c d^2 \left (c^2 d+e\right ) \left (e+\frac{d}{x^2}\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}+\frac{e^2 \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^3 \left (e+\frac{d}{x^2}\right )^2}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{d^3 \left (e+\frac{d}{x^2}\right )}+\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{2 b d^3}+\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 )}{d^3 \sqrt{c^2 d+e} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-\frac{b \sqrt{e} \left (c^2 d+2 e\right ) \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 )}{8 d^3 \left (c^2 d+e\right )^{3/2} \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^3}-\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^3}-\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^3}-\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^3}+\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^3}+\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^3}+\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^3}+\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^3}\\ &=-\frac{b e \left (c^2-\frac{1}{x^2}\right )}{8 c d^2 \left (c^2 d+e\right ) \left (e+\frac{d}{x^2}\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}+\frac{e^2 \left (a+b \text{sech}^{-1}(c x)\right )}{4 d^3 \left (e+\frac{d}{x^2}\right )^2}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{d^3 \left (e+\frac{d}{x^2}\right )}+\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{2 b d^3}+\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 )}{d^3 \sqrt{c^2 d+e} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-\frac{b \sqrt{e} \left (c^2 d+2 e\right ) \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 )}{8 d^3 \left (c^2 d+e\right )^{3/2} \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^3}-\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^3}-\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^3}-\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^3}-\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^3}-\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^3}-\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^3}-\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^3}\\ \end{align*}

Mathematica [F]  time = 62.3782, size = 0, normalized size = 0. \[ \int \frac{a+b \text{sech}^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [C]  time = 1.322, size = 5713, normalized size = 7.7 \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)^3,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \, a{\left (\frac{2 \, e x^{2} + 3 \, d}{d^{2} e^{2} x^{4} + 2 \, d^{3} e x^{2} + d^{4}} - \frac{2 \, \log \left (e x^{2} + d\right )}{d^{3}} + \frac{4 \, \log \left (x\right )}{d^{3}}\right )} + b \int \frac{\log \left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right )}{e^{3} x^{7} + 3 \, d e^{2} x^{5} + 3 \, d^{2} e x^{3} + d^{3} 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)^3,x, algorithm="maxima")

[Out]

1/4*a*((2*e*x^2 + 3*d)/(d^2*e^2*x^4 + 2*d^3*e*x^2 + d^4) - 2*log(e*x^2 + d)/d^3 + 4*log(x)/d^3) + b*integrate(
log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))/(e^3*x^7 + 3*d*e^2*x^5 + 3*d^2*e*x^3 + d^3*x), x)

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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^{3} x^{7} + 3 \, d e^{2} x^{5} + 3 \, d^{2} e x^{3} + d^{3} 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)^3,x, algorithm="fricas")

[Out]

integral((b*arcsech(c*x) + a)/(e^3*x^7 + 3*d*e^2*x^5 + 3*d^2*e*x^3 + d^3*x), 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/(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{b \operatorname{arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} 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)^3,x, algorithm="giac")

[Out]

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