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

Optimal. Leaf size=739 \[ -\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \text{PolyLog}\left (3,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \text{PolyLog}\left (3,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \text{PolyLog}\left (3,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \text{PolyLog}\left (3,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}+1\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}+1\right )}{2 \sqrt{-d} \sqrt{e}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.31507, antiderivative size = 739, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {5706, 5799, 5561, 2190, 2531, 2282, 6589} \[ -\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \text{PolyLog}\left (3,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \text{PolyLog}\left (3,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \text{PolyLog}\left (3,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \text{PolyLog}\left (3,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}+1\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}+1\right )}{2 \sqrt{-d} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d+e x^2} \, dx &=\int \left (\frac{\sqrt{-d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx\\ &=-\frac{\int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 \sqrt{-d}}-\frac{\int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 \sqrt{-d}}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^2 \cosh (x)}{c \sqrt{-d}-\sqrt{e} \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt{-d}}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^2 \cosh (x)}{c \sqrt{-d}+\sqrt{e} \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt{-d}}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)^2}{c \sqrt{-d}-\sqrt{-c^2 d+e}-\sqrt{e} e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt{-d}}-\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)^2}{c \sqrt{-d}+\sqrt{-c^2 d+e}-\sqrt{e} e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt{-d}}-\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)^2}{c \sqrt{-d}-\sqrt{-c^2 d+e}+\sqrt{e} e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt{-d}}-\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)^2}{c \sqrt{-d}+\sqrt{-c^2 d+e}+\sqrt{e} e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt{-d}}\\ &=\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{b \operatorname{Subst}\left (\int (a+b x) \log \left (1-\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \operatorname{Subst}\left (\int (a+b x) \log \left (1+\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}-\frac{b \operatorname{Subst}\left (\int (a+b x) \log \left (1-\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \operatorname{Subst}\left (\int (a+b x) \log \left (1+\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}\\ &=\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \operatorname{Subst}\left (\int \text{Li}_2\left (\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \operatorname{Subst}\left (\int \text{Li}_2\left (\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}\\ &=\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{\sqrt{e} x}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{\sqrt{e} x}{-c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{\sqrt{e} x}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{\sqrt{e} x}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{-d} \sqrt{e}}\\ &=\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \text{Li}_3\left (-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \text{Li}_3\left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \text{Li}_3\left (-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \text{Li}_3\left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{\sqrt{-d} \sqrt{e}}\\ \end{align*}

Mathematica [A]  time = 0.681279, size = 985, normalized size = 1.33 \[ \frac{2 \sqrt{-d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) a^2-2 b \sqrt{d} \sinh ^{-1}(c x) \log \left (\frac{e^{\sinh ^{-1}(c x)} \sqrt{e}}{c \sqrt{-d}-\sqrt{e-c^2 d}}+1\right ) a+2 b \sqrt{d} \sinh ^{-1}(c x) \log \left (\frac{e^{\sinh ^{-1}(c x)} \sqrt{e}}{\sqrt{e-c^2 d}-c \sqrt{-d}}+1\right ) a+2 b \sqrt{d} \sinh ^{-1}(c x) \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right ) a-2 b \sqrt{d} \sinh ^{-1}(c x) \log \left (\frac{e^{\sinh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d} c+\sqrt{e-c^2 d}}+1\right ) a-2 b \sqrt{d} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right ) a+2 b \sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right ) a-b^2 \sqrt{d} \sinh ^{-1}(c x)^2 \log \left (\frac{e^{\sinh ^{-1}(c x)} \sqrt{e}}{c \sqrt{-d}-\sqrt{e-c^2 d}}+1\right )+b^2 \sqrt{d} \sinh ^{-1}(c x)^2 \log \left (\frac{e^{\sinh ^{-1}(c x)} \sqrt{e}}{\sqrt{e-c^2 d}-c \sqrt{-d}}+1\right )+b^2 \sqrt{d} \sinh ^{-1}(c x)^2 \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right )-b^2 \sqrt{d} \sinh ^{-1}(c x)^2 \log \left (\frac{e^{\sinh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d} c+\sqrt{e-c^2 d}}+1\right )+2 b \sqrt{d} \left (a+b \sinh ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )-2 b \sqrt{d} \left (a+b \sinh ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}-c \sqrt{-d}}\right )-2 b^2 \sqrt{d} \sinh ^{-1}(c x) \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right )+2 b^2 \sqrt{d} \sinh ^{-1}(c x) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right )-2 b^2 \sqrt{d} \text{PolyLog}\left (3,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )+2 b^2 \sqrt{d} \text{PolyLog}\left (3,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}-c \sqrt{-d}}\right )+2 b^2 \sqrt{d} \text{PolyLog}\left (3,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right )-2 b^2 \sqrt{d} \text{PolyLog}\left (3,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right )}{2 \sqrt{-d^2} \sqrt{e}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*a^2*Sqrt[-d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - 2*a*b*Sqrt[d]*ArcSinh[c*x]*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*S
qrt[-d] - Sqrt[-(c^2*d) + e])] - b^2*Sqrt[d]*ArcSinh[c*x]^2*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqr
t[-(c^2*d) + e])] + 2*a*b*Sqrt[d]*ArcSinh[c*x]*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d)
+ e])] + b^2*Sqrt[d]*ArcSinh[c*x]^2*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) + e])] +
2*a*b*Sqrt[d]*ArcSinh[c*x]*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])] + b^2*Sqrt[d]*A
rcSinh[c*x]^2*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])] - 2*a*b*Sqrt[d]*ArcSinh[c*x]
*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])] - b^2*Sqrt[d]*ArcSinh[c*x]^2*Log[1 + (Sqr
t[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])] + 2*b*Sqrt[d]*(a + b*ArcSinh[c*x])*PolyLog[2, (Sqrt[e]
*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])] - 2*b*Sqrt[d]*(a + b*ArcSinh[c*x])*PolyLog[2, (Sqrt[e]*E^A
rcSinh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) + e])] - 2*a*b*Sqrt[d]*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sq
rt[-d] + Sqrt[-(c^2*d) + e]))] - 2*b^2*Sqrt[d]*ArcSinh[c*x]*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d]
+ Sqrt[-(c^2*d) + e]))] + 2*a*b*Sqrt[d]*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])]
+ 2*b^2*Sqrt[d]*ArcSinh[c*x]*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])] - 2*b^2*S
qrt[d]*PolyLog[3, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])] + 2*b^2*Sqrt[d]*PolyLog[3, (Sqrt
[e]*E^ArcSinh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) + e])] + 2*b^2*Sqrt[d]*PolyLog[3, -((Sqrt[e]*E^ArcSinh[c*x]
)/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e]))] - 2*b^2*Sqrt[d]*PolyLog[3, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-
(c^2*d) + e])])/(2*Sqrt[-d^2]*Sqrt[e])

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Maple [F]  time = 0.296, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{e{x}^{2}+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a+b*arcsinh(c*x))^2/(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((a+b*arcsinh(c*x))^2/(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^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}}{e x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*asinh(c*x))**2/(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{arsinh}\left (c x\right ) + a\right )}^{2}}{e x^{2} + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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