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

Optimal. Leaf size=713 \[ \frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{a+b \text{csch}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{a+b \text{csch}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{b \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d-e}}\right )}{4 d^{3/2} \sqrt{c^2 d-e}}+\frac{b \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d-e}}\right )}{4 d^{3/2} \sqrt{c^2 d-e}} \]

[Out]

-(a + b*ArcCsch[c*x])/(4*d*(Sqrt[-d]*Sqrt[e] - d/x)) + (a + b*ArcCsch[c*x])/(4*d*(Sqrt[-d]*Sqrt[e] + d/x)) + (
b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(4*d^(3/2)*Sqrt[c
^2*d - e]) + (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(4*
d^(3/2)*Sqrt[c^2*d - e]) - ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d)
+ e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c
^2*d) + e])])/(4*(-d)^(3/2)*Sqrt[e]) - ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sq
rt[-(c^2*d) + e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e
] + Sqrt[-(c^2*d) + e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt
[-(c^2*d) + e]))])/(4*(-d)^(3/2)*Sqrt[e]) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d)
+ e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e]))]
)/(4*(-d)^(3/2)*Sqrt[e]) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(4*(-d)^
(3/2)*Sqrt[e])

________________________________________________________________________________________

Rubi [A]  time = 2.18736, antiderivative size = 713, normalized size of antiderivative = 1., number of steps used = 47, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {6294, 5791, 5706, 5801, 725, 206, 5799, 5561, 2190, 2279, 2391} \[ \frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{a+b \text{csch}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{a+b \text{csch}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{b \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d-e}}\right )}{4 d^{3/2} \sqrt{c^2 d-e}}+\frac{b \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d-e}}\right )}{4 d^{3/2} \sqrt{c^2 d-e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a + b*ArcCsch[c*x])/(4*d*(Sqrt[-d]*Sqrt[e] - d/x)) + (a + b*ArcCsch[c*x])/(4*d*(Sqrt[-d]*Sqrt[e] + d/x)) + (
b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(4*d^(3/2)*Sqrt[c
^2*d - e]) + (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(4*
d^(3/2)*Sqrt[c^2*d - e]) - ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d)
+ e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c
^2*d) + e])])/(4*(-d)^(3/2)*Sqrt[e]) - ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sq
rt[-(c^2*d) + e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e
] + Sqrt[-(c^2*d) + e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt
[-(c^2*d) + e]))])/(4*(-d)^(3/2)*Sqrt[e]) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d)
+ e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e]))]
)/(4*(-d)^(3/2)*Sqrt[e]) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(4*(-d)^
(3/2)*Sqrt[e])

Rule 6294

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

Rule 5791

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

Rule 5706

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

Rule 5801

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

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 5799

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

Rule 5561

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), 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{csch}^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2 \left (a+b \sinh ^{-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 \left (a+b \sinh ^{-1}\left (\frac{x}{c}\right )\right )}{d \left (e+d x^2\right )^2}+\frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{d \left (e+d x^2\right )}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{e+d x^2} \, dx,x,\frac{1}{x}\right )}{d}+\frac{e \operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac{1}{x}\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{2 \sqrt{e} \left (\sqrt{e}-\sqrt{-d} x\right )}+\frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{2 \sqrt{e} \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\frac{1}{x}\right )}{d}+\frac{e \operatorname{Subst}\left (\int \left (-\frac{d \left (a+b \sinh ^{-1}\left (\frac{x}{c}\right )\right )}{4 e \left (\sqrt{-d} \sqrt{e}-d x\right )^2}-\frac{d \left (a+b \sinh ^{-1}\left (\frac{x}{c}\right )\right )}{4 e \left (\sqrt{-d} \sqrt{e}+d x\right )^2}-\frac{d \left (a+b \sinh ^{-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}\\ &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{\left (\sqrt{-d} \sqrt{e}-d x\right )^2} \, dx,x,\frac{1}{x}\right )\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{\left (\sqrt{-d} \sqrt{e}+d x\right )^2} \, dx,x,\frac{1}{x}\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{-d e-d^2 x^2} \, dx,x,\frac{1}{x}\right )-\frac{\operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{2 d \sqrt{e}}-\frac{\operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{2 d \sqrt{e}}\\ &=-\frac{a+b \text{csch}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{a+b \text{csch}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{2 d \sqrt{e} \left (\sqrt{e}-\sqrt{-d} x\right )}-\frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{2 d \sqrt{e} \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\frac{1}{x}\right )+\frac{b \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-d} \sqrt{e}-d x\right ) \sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{4 c d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-d} \sqrt{e}+d x\right ) \sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{4 c d}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \cosh (x)}{\frac{\sqrt{e}}{c}-\sqrt{-d} \sinh (x)} \, dx,x,\text{csch}^{-1}(c x)\right )}{2 d \sqrt{e}}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \cosh (x)}{\frac{\sqrt{e}}{c}+\sqrt{-d} \sinh (x)} \, dx,x,\text{csch}^{-1}(c x)\right )}{2 d \sqrt{e}}\\ &=-\frac{a+b \text{csch}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{a+b \text{csch}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b \operatorname{Subst}\left (\int \frac{1}{d^2-\frac{d e}{c^2}-x^2} \, dx,x,\frac{-d-\frac{\sqrt{-d} \sqrt{e}}{c^2 x}}{\sqrt{1+\frac{1}{c^2 x^2}}}\right )}{4 c d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{d^2-\frac{d e}{c^2}-x^2} \, dx,x,\frac{d-\frac{\sqrt{-d} \sqrt{e}}{c^2 x}}{\sqrt{1+\frac{1}{c^2 x^2}}}\right )}{4 c d}+\frac{\operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{4 d \sqrt{e}}+\frac{\operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{4 d \sqrt{e}}-\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{csch}^{-1}(c x)\right )}{2 d \sqrt{e}}-\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{csch}^{-1}(c x)\right )}{2 d \sqrt{e}}-\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{csch}^{-1}(c x)\right )}{2 d \sqrt{e}}-\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{csch}^{-1}(c x)\right )}{2 d \sqrt{e}}\\ &=-\frac{a+b \text{csch}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{a+b \text{csch}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{b \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right )}{4 d^{3/2} \sqrt{c^2 d-e}}+\frac{b \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right )}{4 d^{3/2} \sqrt{c^2 d-e}}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\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{csch}^{-1}(c x)\right )}{2 (-d)^{3/2} \sqrt{e}}-\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{csch}^{-1}(c x)\right )}{2 (-d)^{3/2} \sqrt{e}}+\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{csch}^{-1}(c x)\right )}{2 (-d)^{3/2} \sqrt{e}}-\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{csch}^{-1}(c x)\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \cosh (x)}{\frac{\sqrt{e}}{c}-\sqrt{-d} \sinh (x)} \, dx,x,\text{csch}^{-1}(c x)\right )}{4 d \sqrt{e}}+\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \cosh (x)}{\frac{\sqrt{e}}{c}+\sqrt{-d} \sinh (x)} \, dx,x,\text{csch}^{-1}(c x)\right )}{4 d \sqrt{e}}\\ &=-\frac{a+b \text{csch}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{a+b \text{csch}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{b \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right )}{4 d^{3/2} \sqrt{c^2 d-e}}+\frac{b \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right )}{4 d^{3/2} \sqrt{c^2 d-e}}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\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{csch}^{-1}(c x)}\right )}{2 (-d)^{3/2} \sqrt{e}}-\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{csch}^{-1}(c x)}\right )}{2 (-d)^{3/2} \sqrt{e}}+\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{csch}^{-1}(c x)}\right )}{2 (-d)^{3/2} \sqrt{e}}-\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{csch}^{-1}(c x)}\right )}{2 (-d)^{3/2} \sqrt{e}}+\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{csch}^{-1}(c x)\right )}{4 d \sqrt{e}}+\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{csch}^{-1}(c x)\right )}{4 d \sqrt{e}}+\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{csch}^{-1}(c x)\right )}{4 d \sqrt{e}}+\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{csch}^{-1}(c x)\right )}{4 d \sqrt{e}}\\ &=-\frac{a+b \text{csch}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{a+b \text{csch}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{b \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right )}{4 d^{3/2} \sqrt{c^2 d-e}}+\frac{b \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right )}{4 d^{3/2} \sqrt{c^2 d-e}}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\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{csch}^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt{e}}+\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{csch}^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt{e}}-\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{csch}^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt{e}}+\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{csch}^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt{e}}\\ &=-\frac{a+b \text{csch}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{a+b \text{csch}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{b \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right )}{4 d^{3/2} \sqrt{c^2 d-e}}+\frac{b \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right )}{4 d^{3/2} \sqrt{c^2 d-e}}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\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{csch}^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt{e}}+\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{csch}^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt{e}}-\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{csch}^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt{e}}+\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{csch}^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt{e}}\\ &=-\frac{a+b \text{csch}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{a+b \text{csch}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{b \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right )}{4 d^{3/2} \sqrt{c^2 d-e}}+\frac{b \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right )}{4 d^{3/2} \sqrt{c^2 d-e}}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}\\ \end{align*}

Mathematica [C]  time = 3.48739, size = 1437, normalized size = 2.02 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a*x)/(d^2 + d*e*x^2) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*Sqrt[e]) + (b*((2*Sqrt[d]*ArcCsch[c*x])/((-I
)*Sqrt[d]*Sqrt[e] + e*x) + (2*Sqrt[d]*ArcCsch[c*x])/(I*Sqrt[d]*Sqrt[e] + e*x) + ((8*I)*ArcSin[Sqrt[1 + Sqrt[e]
/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]])/Sq
rt[e] + ((8*I)*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + Sqrt[e])*Cot[(Pi + (2*I)*Arc
Csch[c*x])/4])/Sqrt[-(c^2*d) + e]])/Sqrt[e] - (Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(
c*Sqrt[d])])/Sqrt[e] + ((2*I)*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[
d])])/Sqrt[e] - (4*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^
ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] + (Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d
])])/Sqrt[e] - ((2*I)*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sq
rt[e] + (4*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[
c*x])/(c*Sqrt[d])])/Sqrt[e] + (Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt
[e] - ((2*I)*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] - (4
*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sq
rt[d])])/Sqrt[e] - (Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] + ((2*I
)*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] + (4*ArcSin[Sqr
t[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sq
rt[e] - (Pi*Log[Sqrt[e] - (I*Sqrt[d])/x])/Sqrt[e] + (Pi*Log[Sqrt[e] + (I*Sqrt[d])/x])/Sqrt[e] - ((2*I)*Log[(2*
Sqrt[d]*Sqrt[e]*(I*Sqrt[e] + c*(c*Sqrt[d] + I*Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e
]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) + e] + ((2*I)*Log[(-2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] + S
qrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) + e]
- ((2*I)*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] + ((2*I)*Poly
Log[2, (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] + ((2*I)*PolyLog[2, ((-I)*(Sqr
t[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] - ((2*I)*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*
d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e]))/(4*d^(3/2)))/2

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Maple [F]  time = 1.997, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arccsch} \left (cx\right )}{ \left ( e{x}^{2}+d \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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