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

Optimal. Leaf size=707 \[ \frac{b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}+1\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}+1\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c \tan ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 x^2+1} \sqrt{c^2 d-e}}\right )}{4 d \sqrt{e} \sqrt{c^2 d-e}}-\frac{b c \tan ^{-1}\left (\frac{c^2 \sqrt{-d} x+\sqrt{e}}{\sqrt{c^2 x^2+1} \sqrt{c^2 d-e}}\right )}{4 d \sqrt{e} \sqrt{c^2 d-e}} \]

[Out]

-(a + b*ArcSinh[c*x])/(4*d*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcSinh[c*x])/(4*d*Sqrt[e]*(Sqrt[-d] + Sqr
t[e]*x)) - (b*c*ArcTan[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d - e]*Sqrt[1 + c^2*x^2])])/(4*d*Sqrt[c^2*d - e]*S
qrt[e]) - (b*c*ArcTan[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d - e]*Sqrt[1 + c^2*x^2])])/(4*d*Sqrt[c^2*d - e]*Sq
rt[e]) - ((a + b*ArcSinh[c*x])*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])])/(4*(-d)^(3
/2)*Sqrt[e]) + ((a + b*ArcSinh[c*x])*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])])/(4*(
-d)^(3/2)*Sqrt[e]) - ((a + b*ArcSinh[c*x])*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])]
)/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcSinh[c*x])*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d)
+ e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e]))])
/(4*(-d)^(3/2)*Sqrt[e]) - (b*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])])/(4*(-d)^(
3/2)*Sqrt[e]) + (b*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e]))])/(4*(-d)^(3/2)*Sq
rt[e]) - (b*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])])/(4*(-d)^(3/2)*Sqrt[e])

________________________________________________________________________________________

Rubi [A]  time = 1.07483, antiderivative size = 707, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5706, 5801, 725, 204, 5799, 5561, 2190, 2279, 2391} \[ \frac{b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}+1\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}+1\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c \tan ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 x^2+1} \sqrt{c^2 d-e}}\right )}{4 d \sqrt{e} \sqrt{c^2 d-e}}-\frac{b c \tan ^{-1}\left (\frac{c^2 \sqrt{-d} x+\sqrt{e}}{\sqrt{c^2 x^2+1} \sqrt{c^2 d-e}}\right )}{4 d \sqrt{e} \sqrt{c^2 d-e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a + b*ArcSinh[c*x])/(4*d*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcSinh[c*x])/(4*d*Sqrt[e]*(Sqrt[-d] + Sqr
t[e]*x)) - (b*c*ArcTan[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d - e]*Sqrt[1 + c^2*x^2])])/(4*d*Sqrt[c^2*d - e]*S
qrt[e]) - (b*c*ArcTan[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d - e]*Sqrt[1 + c^2*x^2])])/(4*d*Sqrt[c^2*d - e]*Sq
rt[e]) - ((a + b*ArcSinh[c*x])*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])])/(4*(-d)^(3
/2)*Sqrt[e]) + ((a + b*ArcSinh[c*x])*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])])/(4*(
-d)^(3/2)*Sqrt[e]) - ((a + b*ArcSinh[c*x])*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])]
)/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcSinh[c*x])*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d)
+ e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e]))])
/(4*(-d)^(3/2)*Sqrt[e]) - (b*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])])/(4*(-d)^(
3/2)*Sqrt[e]) + (b*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e]))])/(4*(-d)^(3/2)*Sq
rt[e]) - (b*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])])/(4*(-d)^(3/2)*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 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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[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 \sinh ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac{e \left (a+b \sinh ^{-1}(c x)\right )}{4 d \left (\sqrt{-d} \sqrt{e}-e x\right )^2}-\frac{e \left (a+b \sinh ^{-1}(c x)\right )}{4 d \left (\sqrt{-d} \sqrt{e}+e x\right )^2}-\frac{e \left (a+b \sinh ^{-1}(c x)\right )}{2 d \left (-d e-e^2 x^2\right )}\right ) \, dx\\ &=-\frac{e \int \frac{a+b \sinh ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}-e x\right )^2} \, dx}{4 d}-\frac{e \int \frac{a+b \sinh ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}+e x\right )^2} \, dx}{4 d}-\frac{e \int \frac{a+b \sinh ^{-1}(c x)}{-d e-e^2 x^2} \, dx}{2 d}\\ &=-\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{(b c) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}-e x\right ) \sqrt{1+c^2 x^2}} \, dx}{4 d}-\frac{(b c) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}+e x\right ) \sqrt{1+c^2 x^2}} \, dx}{4 d}-\frac{e \int \left (-\frac{\sqrt{-d} \left (a+b \sinh ^{-1}(c x)\right )}{2 d e \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{-d} \left (a+b \sinh ^{-1}(c x)\right )}{2 d e \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{2 d}\\ &=-\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{\int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{4 (-d)^{3/2}}+\frac{\int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{4 (-d)^{3/2}}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{-c^2 d e+e^2-x^2} \, dx,x,\frac{-e-c^2 \sqrt{-d} \sqrt{e} x}{\sqrt{1+c^2 x^2}}\right )}{4 d}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{-c^2 d e+e^2-x^2} \, dx,x,\frac{e-c^2 \sqrt{-d} \sqrt{e} x}{\sqrt{1+c^2 x^2}}\right )}{4 d}\\ &=-\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c \tan ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d-e} \sqrt{1+c^2 x^2}}\right )}{4 d \sqrt{c^2 d-e} \sqrt{e}}-\frac{b c \tan ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d-e} \sqrt{1+c^2 x^2}}\right )}{4 d \sqrt{c^2 d-e} \sqrt{e}}+\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \cosh (x)}{c \sqrt{-d}-\sqrt{e} \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \cosh (x)}{c \sqrt{-d}+\sqrt{e} \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2}}\\ &=-\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c \tan ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d-e} \sqrt{1+c^2 x^2}}\right )}{4 d \sqrt{c^2 d-e} \sqrt{e}}-\frac{b c \tan ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d-e} \sqrt{1+c^2 x^2}}\right )}{4 d \sqrt{c^2 d-e} \sqrt{e}}+\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}-\sqrt{-c^2 d+e}-\sqrt{e} e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}+\sqrt{-c^2 d+e}-\sqrt{e} e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}-\sqrt{-c^2 d+e}+\sqrt{e} e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}+\sqrt{-c^2 d+e}+\sqrt{e} e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2}}\\ &=-\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c \tan ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d-e} \sqrt{1+c^2 x^2}}\right )}{4 d \sqrt{c^2 d-e} \sqrt{e}}-\frac{b c \tan ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d-e} \sqrt{1+c^2 x^2}}\right )}{4 d \sqrt{c^2 d-e} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt{e}}\\ &=-\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c \tan ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d-e} \sqrt{1+c^2 x^2}}\right )}{4 d \sqrt{c^2 d-e} \sqrt{e}}-\frac{b c \tan ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d-e} \sqrt{1+c^2 x^2}}\right )}{4 d \sqrt{c^2 d-e} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt{e}}\\ &=-\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b c \tan ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d-e} \sqrt{1+c^2 x^2}}\right )}{4 d \sqrt{c^2 d-e} \sqrt{e}}-\frac{b c \tan ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d-e} \sqrt{1+c^2 x^2}}\right )}{4 d \sqrt{c^2 d-e} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{Li}_2\left (-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{Li}_2\left (-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left (\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}\\ \end{align*}

Mathematica [C]  time = 1.60643, size = 622, normalized size = 0.88 \[ \frac{1}{2} \left (\frac{b \left (i \left (2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}-i c \sqrt{d}}\right )+2 \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+i c \sqrt{d}}\right )+\sinh ^{-1}(c x) \left (-\sinh ^{-1}(c x)+2 \left (\log \left (1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{-\sqrt{e-c^2 d}+i c \sqrt{d}}\right )+\log \left (1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+i c \sqrt{d}}\right )\right )\right )\right )-i \left (2 \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}-i c \sqrt{d}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+i c \sqrt{d}}\right )+\sinh ^{-1}(c x) \left (-\sinh ^{-1}(c x)+2 \left (\log \left (1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}-i c \sqrt{d}}\right )+\log \left (1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+i c \sqrt{d}}\right )\right )\right )\right )-2 \sqrt{d} \left (\frac{c \tan ^{-1}\left (\frac{\sqrt{e}-i c^2 \sqrt{d} x}{\sqrt{c^2 x^2+1} \sqrt{c^2 d-e}}\right )}{\sqrt{c^2 d-e}}-\frac{\sinh ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}\right )+2 i \sqrt{d} \left (\frac{c \tanh ^{-1}\left (\frac{-c^2 \sqrt{d} x+i \sqrt{e}}{\sqrt{c^2 x^2+1} \sqrt{c^2 d-e}}\right )}{\sqrt{c^2 d-e}}+\frac{\sinh ^{-1}(c x)}{\sqrt{d}+i \sqrt{e} x}\right )\right )}{4 d^{3/2} \sqrt{e}}+\frac{a x}{d^2+d e x^2}+\frac{a \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{3/2} \sqrt{e}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcSinh[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]*(-(ArcSinh[c*x]/(I
*Sqrt[d] + Sqrt[e]*x)) + (c*ArcTan[(Sqrt[e] - I*c^2*Sqrt[d]*x)/(Sqrt[c^2*d - e]*Sqrt[1 + c^2*x^2])])/Sqrt[c^2*
d - e]) + (2*I)*Sqrt[d]*(ArcSinh[c*x]/(Sqrt[d] + I*Sqrt[e]*x) + (c*ArcTanh[(I*Sqrt[e] - c^2*Sqrt[d]*x)/(Sqrt[c
^2*d - e]*Sqrt[1 + c^2*x^2])])/Sqrt[c^2*d - e]) + I*(ArcSinh[c*x]*(-ArcSinh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcSi
nh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) + e])] + Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) +
e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) + e])] + 2*PolyLog[2, -((Sqrt[
e]*E^ArcSinh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]))]) - I*(ArcSinh[c*x]*(-ArcSinh[c*x] + 2*(Log[1 + (Sqrt[e
]*E^ArcSinh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) + e])] + Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(I*c*Sqrt[d] + Sqr
t[-(c^2*d) + e])])) + 2*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) + e]))] + 2*Poly
Log[2, (Sqrt[e]*E^ArcSinh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e])])))/(4*d^(3/2)*Sqrt[e]))/2

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Maple [C]  time = 0.668, size = 1745, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c^2*a*x/d/(c^2*e*x^2+c^2*d)+1/2*a/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+1/2*c^2*b*arcsinh(c*x)*x/d/(c^2*e*
x^2+c^2*d)+1/4*c*b/d*sum(1/_R1/(_R1^2*e+2*c^2*d-e)*(arcsinh(c*x)*ln((_R1-c*x-(c^2*x^2+1)^(1/2))/_R1)+dilog((_R
1-c*x-(c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d-2*e)*_Z^2+e))+1/4*c*b/d*sum(_R1/(_R1^2*e+2*c^2*d-e)*
(arcsinh(c*x)*ln((_R1-c*x-(c^2*x^2+1)^(1/2))/_R1)+dilog((_R1-c*x-(c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4
*c^2*d-2*e)*_Z^2+e))+c^5*b*(-(2*c^2*d-2*(d*c^2*(c^2*d-e))^(1/2)-e)*e)^(1/2)*arctanh((c*x+(c^2*x^2+1)^(1/2))*e/
((-2*c^2*d+2*(d*c^2*(c^2*d-e))^(1/2)+e)*e)^(1/2))*d/(c^2*d-e)/e^3+c^3*b*(-(2*c^2*d-2*(d*c^2*(c^2*d-e))^(1/2)-e
)*e)^(1/2)*arctanh((c*x+(c^2*x^2+1)^(1/2))*e/((-2*c^2*d+2*(d*c^2*(c^2*d-e))^(1/2)+e)*e)^(1/2))/(c^2*d-e)/e^3*(
d*c^2*(c^2*d-e))^(1/2)-c^3*b*(-(2*c^2*d-2*(d*c^2*(c^2*d-e))^(1/2)-e)*e)^(1/2)*arctanh((c*x+(c^2*x^2+1)^(1/2))*
e/((-2*c^2*d+2*(d*c^2*(c^2*d-e))^(1/2)+e)*e)^(1/2))/(c^2*d-e)/e^2-1/2*c*b*(-(2*c^2*d-2*(d*c^2*(c^2*d-e))^(1/2)
-e)*e)^(1/2)*arctanh((c*x+(c^2*x^2+1)^(1/2))*e/((-2*c^2*d+2*(d*c^2*(c^2*d-e))^(1/2)+e)*e)^(1/2))/d/(c^2*d-e)/e
^2*(d*c^2*(c^2*d-e))^(1/2)-c^3*b*(-(2*c^2*d-2*(d*c^2*(c^2*d-e))^(1/2)-e)*e)^(1/2)*arctanh((c*x+(c^2*x^2+1)^(1/
2))*e/((-2*c^2*d+2*(d*c^2*(c^2*d-e))^(1/2)+e)*e)^(1/2))/e^3-c*b*(-(2*c^2*d-2*(d*c^2*(c^2*d-e))^(1/2)-e)*e)^(1/
2)*arctanh((c*x+(c^2*x^2+1)^(1/2))*e/((-2*c^2*d+2*(d*c^2*(c^2*d-e))^(1/2)+e)*e)^(1/2))/d/e^3*(d*c^2*(c^2*d-e))
^(1/2)+1/2*c*b*(-(2*c^2*d-2*(d*c^2*(c^2*d-e))^(1/2)-e)*e)^(1/2)*arctanh((c*x+(c^2*x^2+1)^(1/2))*e/((-2*c^2*d+2
*(d*c^2*(c^2*d-e))^(1/2)+e)*e)^(1/2))/d/e^2+c^5*b*((2*c^2*d+2*(d*c^2*(c^2*d-e))^(1/2)-e)*e)^(1/2)*arctan((c*x+
(c^2*x^2+1)^(1/2))*e/((2*c^2*d+2*(d*c^2*(c^2*d-e))^(1/2)-e)*e)^(1/2))*d/(c^2*d-e)/e^3-c^3*b*((2*c^2*d+2*(d*c^2
*(c^2*d-e))^(1/2)-e)*e)^(1/2)*arctan((c*x+(c^2*x^2+1)^(1/2))*e/((2*c^2*d+2*(d*c^2*(c^2*d-e))^(1/2)-e)*e)^(1/2)
)/(c^2*d-e)/e^3*(d*c^2*(c^2*d-e))^(1/2)-c^3*b*((2*c^2*d+2*(d*c^2*(c^2*d-e))^(1/2)-e)*e)^(1/2)*arctan((c*x+(c^2
*x^2+1)^(1/2))*e/((2*c^2*d+2*(d*c^2*(c^2*d-e))^(1/2)-e)*e)^(1/2))/(c^2*d-e)/e^2+1/2*c*b*((2*c^2*d+2*(d*c^2*(c^
2*d-e))^(1/2)-e)*e)^(1/2)*arctan((c*x+(c^2*x^2+1)^(1/2))*e/((2*c^2*d+2*(d*c^2*(c^2*d-e))^(1/2)-e)*e)^(1/2))/d/
(c^2*d-e)/e^2*(d*c^2*(c^2*d-e))^(1/2)-c^3*b*((2*c^2*d+2*(d*c^2*(c^2*d-e))^(1/2)-e)*e)^(1/2)*arctan((c*x+(c^2*x
^2+1)^(1/2))*e/((2*c^2*d+2*(d*c^2*(c^2*d-e))^(1/2)-e)*e)^(1/2))/e^3+c*b*((2*c^2*d+2*(d*c^2*(c^2*d-e))^(1/2)-e)
*e)^(1/2)*arctan((c*x+(c^2*x^2+1)^(1/2))*e/((2*c^2*d+2*(d*c^2*(c^2*d-e))^(1/2)-e)*e)^(1/2))/d/e^3*(d*c^2*(c^2*
d-e))^(1/2)+1/2*c*b*((2*c^2*d+2*(d*c^2*(c^2*d-e))^(1/2)-e)*e)^(1/2)*arctan((c*x+(c^2*x^2+1)^(1/2))*e/((2*c^2*d
+2*(d*c^2*(c^2*d-e))^(1/2)-e)*e)^(1/2))/d/e^2

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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))/(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{arsinh}\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*arcsinh(c*x))/(e*x^2+d)^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arsinh}\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*arcsinh(c*x))/(e*x^2+d)^2,x, algorithm="giac")

[Out]

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