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

Optimal. Leaf size=467 \[ \frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{2 e}+\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{2 e}+\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{2 e}+\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{2 e}+\frac{b \text{PolyLog}\left (2,e^{-2 \text{csch}^{-1}(c x)}\right )}{2 e}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{2 e}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right )}{2 e}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{2 e}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right )}{2 e}-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{b e}-\frac{\log \left (1-e^{-2 \text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{e} \]

[Out]

-((a + b*ArcCsch[c*x])^2/(b*e)) - ((a + b*ArcCsch[c*x])*Log[1 - E^(-2*ArcCsch[c*x])])/e + ((a + b*ArcCsch[c*x]
)*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(2*e) + ((a + b*ArcCsch[c*x])*Log[1 + (
c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(2*e) + ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*
E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(2*e) + ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c
*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(2*e) + (b*PolyLog[2, E^(-2*ArcCsch[c*x])])/(2*e) + (b*PolyLog[2, -((c*S
qrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e]))])/(2*e) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(S
qrt[e] - Sqrt[-(c^2*d) + e])])/(2*e) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) +
e]))])/(2*e) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(2*e)

________________________________________________________________________________________

Rubi [A]  time = 1.10582, antiderivative size = 449, normalized size of antiderivative = 0.96, number of steps used = 26, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {6304, 5791, 5659, 3716, 2190, 2279, 2391, 5799, 5561} \[ \frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{2 e}+\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{2 e}+\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{2 e}+\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{2 e}-\frac{b \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right )}{2 e}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{2 e}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right )}{2 e}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{2 e}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right )}{2 e}-\frac{\log \left (1-e^{2 \text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{e} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(2*e) + ((a + b*Arc
Csch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(2*e) + ((a + b*ArcCsch[c*x])*
Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(2*e) + ((a + b*ArcCsch[c*x])*Log[1 + (c*
Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(2*e) - ((a + b*ArcCsch[c*x])*Log[1 - E^(2*ArcCsch[c
*x])])/e + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e]))])/(2*e) + (b*PolyLog[2,
(c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(2*e) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*
x])/(Sqrt[e] + Sqrt[-(c^2*d) + e]))])/(2*e) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[-(c^2*
d) + e])])/(2*e) - (b*PolyLog[2, E^(2*ArcCsch[c*x])])/(2*e)

Rule 6304

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Subst[Int
[((e + d*x^2)^p*(a + b*ArcSinh[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 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 5659

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

Rule 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c
+ d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 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]

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]

Rubi steps

\begin{align*} \int \frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{d+e x^2} \, dx &=-\operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{x \left (e+d x^2\right )} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{e x}-\frac{d x \left (a+b \sinh ^{-1}\left (\frac{x}{c}\right )\right )}{e \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 )}{x} \, dx,x,\frac{1}{x}\right )}{e}+\frac{d \operatorname{Subst}\left (\int \frac{x \left (a+b \sinh ^{-1}\left (\frac{x}{c}\right )\right )}{e+d x^2} \, dx,x,\frac{1}{x}\right )}{e}\\ &=-\frac{\operatorname{Subst}\left (\int (a+b x) \coth (x) \, dx,x,\text{csch}^{-1}(c x)\right )}{e}+\frac{d \operatorname{Subst}\left (\int \left (-\frac{\sqrt{-d} \left (a+b \sinh ^{-1}\left (\frac{x}{c}\right )\right )}{2 d \left (\sqrt{e}-\sqrt{-d} x\right )}+\frac{\sqrt{-d} \left (a+b \sinh ^{-1}\left (\frac{x}{c}\right )\right )}{2 d \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\frac{1}{x}\right )}{e}\\ &=\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{2 b e}+\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\text{csch}^{-1}(c x)\right )}{e}-\frac{\sqrt{-d} \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 e}+\frac{\sqrt{-d} \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 e}\\ &=\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{2 b e}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )}{e}+\frac{b \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{e}-\frac{\sqrt{-d} \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 e}+\frac{\sqrt{-d} \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 e}\\ &=-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )}{e}+\frac{b \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \text{csch}^{-1}(c x)}\right )}{2 e}-\frac{\sqrt{-d} \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 e}-\frac{\sqrt{-d} \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 e}+\frac{\sqrt{-d} \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 e}+\frac{\sqrt{-d} \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 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 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 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 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 e}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )}{e}-\frac{b \text{Li}_2\left (e^{2 \text{csch}^{-1}(c x)}\right )}{2 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 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 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 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 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 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 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 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 e}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )}{e}-\frac{b \text{Li}_2\left (e^{2 \text{csch}^{-1}(c x)}\right )}{2 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 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 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 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 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 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 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 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 e}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )}{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 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 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 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 e}-\frac{b \text{Li}_2\left (e^{2 \text{csch}^{-1}(c x)}\right )}{2 e}\\ \end{align*}

Mathematica [C]  time = 0.340055, size = 1103, normalized size = 2.36 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(b*Pi^2 - (4*I)*b*Pi*ArcCsch[c*x] - 8*b*ArcCsch[c*x]^2 + 16*b*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Ar
cTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 16*b*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]] -
8*b*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (2*I)*b*Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCs
ch[c*x])/(c*Sqrt[d])] + 4*b*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d]
)] + (8*I)*b*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsc
h[c*x])/(c*Sqrt[d])] + (2*I)*b*Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*
b*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (8*I)*b*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])] + (2*
I)*b*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*ArcCsch[c*x]*Log[1 - (I*(
Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (8*I)*b*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])] + (2*I)*b*Pi*Log[1 + (I*(Sqrt[e] +
Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])
*E^ArcCsch[c*x])/(c*Sqrt[d])] - (8*I)*b*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sq
rt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (2*I)*b*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] - (2*I)*b*Pi*Log[Sqrt
[e] + (I*Sqrt[d])/x] + 4*a*Log[d + e*x^2] + 4*b*PolyLog[2, E^(-2*ArcCsch[c*x])] + 4*b*PolyLog[2, ((-I)*(-Sqrt[
e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*PolyLog[2, (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^Ar
cCsch[c*x])/(c*Sqrt[d])] + 4*b*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] +
4*b*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/(8*e)

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Maple [F]  time = 0.552, size = 0, normalized size = 0. \begin{align*} \int{\frac{x \left ( a+b{\rm arccsch} \left (cx\right ) \right ) }{e{x}^{2}+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} b \int \frac{x \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + \frac{1}{c x}\right )}{e x^{2} + d}\,{d x} + \frac{a \log \left (e x^{2} + d\right )}{2 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*integrate(x*log(sqrt(1/(c^2*x^2) + 1) + 1/(c*x))/(e*x^2 + d), x) + 1/2*a*log(e*x^2 + d)/e

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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