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

Optimal. Leaf size=598 \[ -\frac{b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 d^2}-\frac{b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 d^2}-\frac{b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 d^2}-\frac{b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 d^2}-\frac{b \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )}{2 d^2}-\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 d^2}-\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right )}{2 d^2}-\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 d^2}-\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right )}{2 d^2}+\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b d^2}+\frac{\log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2}+\frac{a+b \cosh ^{-1}(c x)}{2 d \left (d+e x^2\right )}-\frac{b c \sqrt{c^2 x^2-1} \tanh ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{2 d^{3/2} \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 d+e}} \]

[Out]

(a + b*ArcCosh[c*x])/(2*d*(d + e*x^2)) + (a + b*ArcCosh[c*x])^2/(b*d^2) - (b*c*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqr
t[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(2*d^(3/2)*Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((a
+ b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/d^2 - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(
c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*d^2) - ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d
] - Sqrt[-(c^2*d) - e])])/(2*d^2) - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[
-(c^2*d) - e])])/(2*d^2) - ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d)
- e])])/(2*d^2) - (b*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(2*d^2) - (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*S
qrt[-d] - Sqrt[-(c^2*d) - e]))])/(2*d^2) - (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d)
- e])])/(2*d^2) - (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(2*d^2) - (b*P
olyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*d^2)

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Rubi [A]  time = 1.06196, antiderivative size = 581, normalized size of antiderivative = 0.97, number of steps used = 29, number of rules used = 12, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {5792, 5660, 3718, 2190, 2279, 2391, 5788, 519, 377, 208, 5800, 5562} \[ -\frac{b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 d^2}-\frac{b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 d^2}-\frac{b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 d^2}-\frac{b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 d^2}+\frac{b \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{2 d^2}-\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 d^2}-\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right )}{2 d^2}-\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 d^2}-\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right )}{2 d^2}+\frac{\log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2}+\frac{a+b \cosh ^{-1}(c x)}{2 d \left (d+e x^2\right )}-\frac{b c \sqrt{c^2 x^2-1} \tanh ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{2 d^{3/2} \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 d+e}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Rule 5792

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int
[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]

Rule 5660

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

Rule 3718

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

Rule 2190

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

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 5788

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^(p +
1)*(a + b*ArcCosh[c*x]))/(2*e*(p + 1)), x] - Dist[(b*c)/(2*e*(p + 1)), Int[(d + e*x^2)^(p + 1)/(Sqrt[1 + c*x]*
Sqrt[-1 + c*x]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[c^2*d + e, 0] && NeQ[p, -1]

Rule 519

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_)*((a2_) + (b2_.)*(x_)^(non2_.))^(
p_), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1*a2 + b1*b2*x^n)^FracP
art[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[
non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(EqQ[n, 2] && IGtQ[q, 0])

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

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 5800

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Subst[Int[((a + b*x)^n*Sinh[x
])/(c*d + e*Cosh[x]), x], x, ArcCosh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rule 5562

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 - b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0]

Rubi steps

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

Mathematica [F]  time = 4.8704, size = 0, normalized size = 0. \[ \int \frac{a+b \cosh ^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [C]  time = 0.203, size = 529, normalized size = 0.9 \begin{align*}{\frac{a{c}^{2}}{2\,d \left ({x}^{2}{c}^{2}e+{c}^{2}d \right ) }}-{\frac{a\ln \left ({x}^{2}{c}^{2}e+{c}^{2}d \right ) }{2\,{d}^{2}}}+{\frac{a\ln \left ( cx \right ) }{{d}^{2}}}+{\frac{b{c}^{2}{\rm arccosh} \left (cx\right )}{2\,d \left ({x}^{2}{c}^{2}e+{c}^{2}d \right ) }}+{\frac{b}{2\,{d}^{2} \left ({c}^{2}d+e \right ) }\sqrt{{c}^{2}d \left ({c}^{2}d+e \right ) }{\it Artanh} \left ({\frac{1}{4} \left ( 2\, \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}e+4\,{c}^{2}d+2\,e \right ){\frac{1}{\sqrt{{d}^{2}{c}^{4}+{c}^{2}de}}}} \right ) }-{\frac{b}{4\,{d}^{2}}\sum _{{\it \_R1}={\it RootOf} \left ( e{{\it \_Z}}^{4}+ \left ( 4\,{c}^{2}d+2\,e \right ){{\it \_Z}}^{2}+e \right ) }{\frac{{{\it \_R1}}^{2}e+4\,{c}^{2}d+e}{{{\it \_R1}}^{2}e+2\,{c}^{2}d+e} \left ({\rm arccosh} \left (cx\right )\ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) } \right ) \right ) }}+{\frac{b{\rm arccosh} \left (cx\right )}{{d}^{2}}\ln \left ( 1+i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \right ) }+{\frac{b{\rm arccosh} \left (cx\right )}{{d}^{2}}\ln \left ( 1-i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \right ) }+{\frac{b}{{d}^{2}}{\it dilog} \left ( 1+i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \right ) }+{\frac{b}{{d}^{2}}{\it dilog} \left ( 1-i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \right ) }-{\frac{be}{4\,{d}^{2}}\sum _{{\it \_R1}={\it RootOf} \left ( e{{\it \_Z}}^{4}+ \left ( 4\,{c}^{2}d+2\,e \right ){{\it \_Z}}^{2}+e \right ) }{\frac{{{\it \_R1}}^{2}+1}{{{\it \_R1}}^{2}e+2\,{c}^{2}d+e} \left ({\rm arccosh} \left (cx\right )\ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) } \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*c^2/d/(c^2*e*x^2+c^2*d)-1/2*a/d^2*ln(c^2*e*x^2+c^2*d)+a/d^2*ln(c*x)+1/2*b*c^2*arccosh(c*x)/d/(c^2*e*x^2+
c^2*d)+1/2*b*(c^2*d*(c^2*d+e))^(1/2)/d^2/(c^2*d+e)*arctanh(1/4*(2*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2*e+4*c^2*
d+2*e)/(c^4*d^2+c^2*d*e)^(1/2))-1/4*b/d^2*sum((_R1^2*e+4*c^2*d+e)/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x)*ln((_R1-c*
x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2
*d+2*e)*_Z^2+e))+b/d^2*arccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+b/d^2*arccosh(c*x)*ln(1-I*(c*x+(
c*x-1)^(1/2)*(c*x+1)^(1/2)))+b/d^2*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+b/d^2*dilog(1-I*(c*x+(c*x-1)^(
1/2)*(c*x+1)^(1/2)))-1/4*b/d^2*e*sum((_R1^2+1)/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x)*ln((_R1-c*x-(c*x-1)^(1/2)*(c*
x+1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a{\left (\frac{1}{d e x^{2} + d^{2}} - \frac{\log \left (e x^{2} + d\right )}{d^{2}} + \frac{2 \, \log \left (x\right )}{d^{2}}\right )} + b \int \frac{\log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{e^{2} x^{5} + 2 \, d e x^{3} + d^{2} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*(1/(d*e*x^2 + d^2) - log(e*x^2 + d)/d^2 + 2*log(x)/d^2) + b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x -
1))/(e^2*x^5 + 2*d*e*x^3 + d^2*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{arcosh}\left (c x\right ) + a}{e^{2} x^{5} + 2 \, d e x^{3} + d^{2} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arccosh(c*x) + a)/(e^2*x^5 + 2*d*e*x^3 + d^2*x), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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