\(\int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x^2} \, dx\) [617]

Optimal result

Integrand size = 20, antiderivative size = 739 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x^2} \, dx=\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}} \]

[Out]

1/2*(a+b*arcsinh(c*x))^2*ln(1-(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1
/2)-1/2*(a+b*arcsinh(c*x))^2*ln(1+(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d+e)^(1/2)))/(-d)^(1/2)/
e^(1/2)+1/2*(a+b*arcsinh(c*x))^2*ln(1-(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d+e)^(1/2)))/(-d)^(1
/2)/e^(1/2)-1/2*(a+b*arcsinh(c*x))^2*ln(1+(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d+e)^(1/2)))/(-d
)^(1/2)/e^(1/2)-b*(a+b*arcsinh(c*x))*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d+e)^(1/2)
))/(-d)^(1/2)/e^(1/2)+b*(a+b*arcsinh(c*x))*polylog(2,(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d+e)^
(1/2)))/(-d)^(1/2)/e^(1/2)-b*(a+b*arcsinh(c*x))*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2
*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)+b*(a+b*arcsinh(c*x))*polylog(2,(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+
(-c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)+b^2*polylog(3,-(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d+e)^
(1/2)))/(-d)^(1/2)/e^(1/2)-b^2*polylog(3,(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d+e)^(1/2)))/(-d)
^(1/2)/e^(1/2)+b^2*polylog(3,-(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1
/2)-b^2*polylog(3,(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 739, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5793, 5827, 5680, 2221, 2611, 2320, 6724} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x^2} \, dx=-\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {-d} c+\sqrt {e-c^2 d}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {-d} c+\sqrt {e-c^2 d}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x))^2 \log \left (\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}+1\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x))^2 \log \left (\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}+1\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {-d} c+\sqrt {e-c^2 d}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {-d} c+\sqrt {e-c^2 d}}\right )}{\sqrt {-d} \sqrt {e}} \]

[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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2320

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 2611

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 5680

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 5793

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 5827

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 6724

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*} \text {integral}& = \int \left (\frac {\sqrt {-d} (a+b \text {arcsinh}(c x))^2}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} (a+b \text {arcsinh}(c x))^2}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx \\ & = -\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 \sqrt {-d}}-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 \sqrt {-d}} \\ & = -\frac {\text {Subst}\left (\int \frac {(a+b x)^2 \cosh (x)}{c \sqrt {-d}-\sqrt {e} \sinh (x)} \, dx,x,\text {arcsinh}(c x)\right )}{2 \sqrt {-d}}-\frac {\text {Subst}\left (\int \frac {(a+b x)^2 \cosh (x)}{c \sqrt {-d}+\sqrt {e} \sinh (x)} \, dx,x,\text {arcsinh}(c x)\right )}{2 \sqrt {-d}} \\ & = -\frac {\text {Subst}\left (\int \frac {e^x (a+b x)^2}{c \sqrt {-d}-\sqrt {-c^2 d+e}-\sqrt {e} e^x} \, dx,x,\text {arcsinh}(c x)\right )}{2 \sqrt {-d}}-\frac {\text {Subst}\left (\int \frac {e^x (a+b x)^2}{c \sqrt {-d}+\sqrt {-c^2 d+e}-\sqrt {e} e^x} \, dx,x,\text {arcsinh}(c x)\right )}{2 \sqrt {-d}}-\frac {\text {Subst}\left (\int \frac {e^x (a+b x)^2}{c \sqrt {-d}-\sqrt {-c^2 d+e}+\sqrt {e} e^x} \, dx,x,\text {arcsinh}(c x)\right )}{2 \sqrt {-d}}-\frac {\text {Subst}\left (\int \frac {e^x (a+b x)^2}{c \sqrt {-d}+\sqrt {-c^2 d+e}+\sqrt {e} e^x} \, dx,x,\text {arcsinh}(c x)\right )}{2 \sqrt {-d}} \\ & = \frac {(a+b \text {arcsinh}(c x))^2 \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \text {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,\text {arcsinh}(c x)\right )}{\sqrt {-d} \sqrt {e}}+\frac {b \text {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,\text {arcsinh}(c x)\right )}{\sqrt {-d} \sqrt {e}}-\frac {b \text {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,\text {arcsinh}(c x)\right )}{\sqrt {-d} \sqrt {e}}+\frac {b \text {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,\text {arcsinh}(c x)\right )}{\sqrt {-d} \sqrt {e}} \\ & = \frac {(a+b \text {arcsinh}(c x))^2 \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \text {Subst}\left (\int \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \text {Subst}\left (\int \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {-d} \sqrt {e}} \\ & = \frac {(a+b \text {arcsinh}(c x))^2 \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{-c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{\sqrt {-d} \sqrt {e}} \\ & = \frac {(a+b \text {arcsinh}(c x))^2 \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 985, normalized size of antiderivative = 1.33 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x^2} \, dx=\frac {2 a^2 \sqrt {-d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-2 a b \sqrt {d} \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )-b^2 \sqrt {d} \text {arcsinh}(c x)^2 \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )+2 a b \sqrt {d} \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )+b^2 \sqrt {d} \text {arcsinh}(c x)^2 \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )+2 a b \sqrt {d} \text {arcsinh}(c x) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )+b^2 \sqrt {d} \text {arcsinh}(c x)^2 \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )-2 a b \sqrt {d} \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )-b^2 \sqrt {d} \text {arcsinh}(c x)^2 \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )+2 b \sqrt {d} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )-2 b \sqrt {d} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )-2 a b \sqrt {d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )-2 b^2 \sqrt {d} \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )+2 a b \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )+2 b^2 \sqrt {d} \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )-2 b^2 \sqrt {d} \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )+2 b^2 \sqrt {d} \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )+2 b^2 \sqrt {d} \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )-2 b^2 \sqrt {d} \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d^2} \sqrt {e}} \]

[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])

Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{e \,x^{2}+d}d x\]

[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)

Fricas [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{e x^{2} + d} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x^2} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{d + e x^{2}}\, dx \]

[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)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{e x^{2} + d} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{e\,x^2+d} \,d x \]

[In]

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

[Out]

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