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

Optimal. Leaf size=512 \[ -\frac{b \sqrt{-d} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{2 e^{3/2}}-\frac{b \sqrt{-d} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{2 e^{3/2}}+\frac{\sqrt{-d} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{2 e^{3/2}}-\frac{\sqrt{-d} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right )}{2 e^{3/2}}+\frac{\sqrt{-d} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{2 e^{3/2}}-\frac{\sqrt{-d} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right )}{2 e^{3/2}}+\frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{b \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{c e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.19779, antiderivative size = 512, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 12, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {6304, 5791, 5661, 266, 63, 208, 5706, 5799, 5561, 2190, 2279, 2391} \[ -\frac{b \sqrt{-d} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{2 e^{3/2}}-\frac{b \sqrt{-d} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{2 e^{3/2}}+\frac{\sqrt{-d} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{2 e^{3/2}}-\frac{\sqrt{-d} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right )}{2 e^{3/2}}+\frac{\sqrt{-d} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{2 e^{3/2}}-\frac{\sqrt{-d} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right )}{2 e^{3/2}}+\frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{b \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{c e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 5661

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 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 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{x^2 \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^2 \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^2}-\frac{d \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^2} \, dx,x,\frac{1}{x}\right )}{e}+\frac{d \operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{e+d x^2} \, dx,x,\frac{1}{x}\right )}{e}\\ &=\frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{e}-\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{c e}+\frac{d \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 )}{e}\\ &=\frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{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^{3/2}}+\frac{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^{3/2}}-\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 c e}\\ &=\frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{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^{3/2}}+\frac{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^{3/2}}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{-c^2+c^2 x^2} \, dx,x,\sqrt{1+\frac{1}{c^2 x^2}}\right )}{e}\\ &=\frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{b \tanh ^{-1}\left (\sqrt{1+\frac{1}{c^2 x^2}}\right )}{c e}+\frac{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^{3/2}}+\frac{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^{3/2}}+\frac{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^{3/2}}+\frac{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^{3/2}}\\ &=\frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{b \tanh ^{-1}\left (\sqrt{1+\frac{1}{c^2 x^2}}\right )}{c e}+\frac{\sqrt{-d} \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^{3/2}}-\frac{\sqrt{-d} \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^{3/2}}+\frac{\sqrt{-d} \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^{3/2}}-\frac{\sqrt{-d} \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^{3/2}}-\frac{\left (b \sqrt{-d}\right ) \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^{3/2}}+\frac{\left (b \sqrt{-d}\right ) \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^{3/2}}-\frac{\left (b \sqrt{-d}\right ) \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^{3/2}}+\frac{\left (b \sqrt{-d}\right ) \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^{3/2}}\\ &=\frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{b \tanh ^{-1}\left (\sqrt{1+\frac{1}{c^2 x^2}}\right )}{c e}+\frac{\sqrt{-d} \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^{3/2}}-\frac{\sqrt{-d} \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^{3/2}}+\frac{\sqrt{-d} \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^{3/2}}-\frac{\sqrt{-d} \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^{3/2}}-\frac{\left (b \sqrt{-d}\right ) \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^{3/2}}+\frac{\left (b \sqrt{-d}\right ) \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^{3/2}}-\frac{\left (b \sqrt{-d}\right ) \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^{3/2}}+\frac{\left (b \sqrt{-d}\right ) \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^{3/2}}\\ &=\frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{b \tanh ^{-1}\left (\sqrt{1+\frac{1}{c^2 x^2}}\right )}{c e}+\frac{\sqrt{-d} \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^{3/2}}-\frac{\sqrt{-d} \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^{3/2}}+\frac{\sqrt{-d} \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^{3/2}}-\frac{\sqrt{-d} \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^{3/2}}-\frac{b \sqrt{-d} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{2 e^{3/2}}-\frac{b \sqrt{-d} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{2 e^{3/2}}\\ \end{align*}

Mathematica [C]  time = 1.61707, size = 1221, normalized size = 2.38 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4*a*c*Sqrt[e]*x + 4*b*c*Sqrt[e]*x*ArcCsch[c*x] - 4*a*c*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - (8*I)*b*c*Sqrt[d
]*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*I)*b*c*Sqrt[d]*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + S
qrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] + b*c*Sqrt[d]*Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c
^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (2*I)*b*c*Sqrt[d]*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d)
+ e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*c*Sqrt[d]*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])] - b*c*Sqrt[d]*Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2
*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*c*Sqrt[d]*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) +
e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - 4*b*c*Sqrt[d]*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-S
qrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - b*c*Sqrt[d]*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d)
+ e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*c*Sqrt[d]*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])
*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*c*Sqrt[d]*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])] + b*c*Sqrt[d]*Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e]
)*E^ArcCsch[c*x])/(c*Sqrt[d])] - (2*I)*b*c*Sqrt[d]*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^Ar
cCsch[c*x])/(c*Sqrt[d])] - 4*b*c*Sqrt[d]*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + S
qrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + b*c*Sqrt[d]*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] - b*c*Sqrt[d]*Pi
*Log[Sqrt[e] + (I*Sqrt[d])/x] - 4*b*Sqrt[e]*Log[Tanh[ArcCsch[c*x]/2]] + (2*I)*b*c*Sqrt[d]*PolyLog[2, ((-I)*(-S
qrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (2*I)*b*c*Sqrt[d]*PolyLog[2, (I*(-Sqrt[e] + Sqrt[-
(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (2*I)*b*c*Sqrt[d]*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])
*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*c*Sqrt[d]*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])
/(c*Sqrt[d])])/(4*c*e^(3/2))

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Maple [F]  time = 1.688, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2} \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^2*(a+b*arccsch(c*x))/(e*x^2+d),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Integral(x**2*(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^{2}}{e x^{2} + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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