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

Optimal. Leaf size=279 \[ \frac{2 b^2 c \text{PolyLog}\left (2,-\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{2 b^2 c \text{PolyLog}\left (2,-\frac{e e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e \sqrt{c^2 d^2-e^2}}+\frac{2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}+1\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{e e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}+1\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)} \]

[Out]

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

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Rubi [A]  time = 0.610123, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {5802, 5832, 3320, 2264, 2190, 2279, 2391} \[ \frac{2 b^2 c \text{PolyLog}\left (2,-\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{2 b^2 c \text{PolyLog}\left (2,-\frac{e e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e \sqrt{c^2 d^2-e^2}}+\frac{2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}+1\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{e e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}+1\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5802

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

Rule 5832

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_
) + (e2_.)*(x_)]), x_Symbol] :> Dist[1/(c^(m + 1)*Sqrt[-(d1*d2)]), Subst[Int[(a + b*x)^n*(c*f + g*Cosh[x])^m,
x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2
, 0] && IntegerQ[m] && GtQ[d1, 0] && LtQ[d2, 0] && (GtQ[m, 0] || IGtQ[n, 0])

Rule 3320

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol]
:> Dist[2, Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(E^(I*Pi*(k - 1/2))*(b + (2*a*E^(-(I*e) + f*fz*x))/E^(I*Pi*(k
- 1/2)) - (b*E^(2*(-(I*e) + f*fz*x)))/E^(2*I*k*Pi))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{(d+e x)^2} \, dx &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac{(2 b c) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x} (d+e x)} \, dx}{e}\\ &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac{(2 b c) \operatorname{Subst}\left (\int \frac{a+b x}{c d+e \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{e}\\ &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac{(4 b c) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{e+2 c d e^x+e e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{e}\\ &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac{(4 b c) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{2 c d-2 \sqrt{c^2 d^2-e^2}+2 e e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{c^2 d^2-e^2}}-\frac{(4 b c) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{2 c d+2 \sqrt{c^2 d^2-e^2}+2 e e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{c^2 d^2-e^2}}\\ &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac{2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{e e^{\cosh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{\left (2 b^2 c\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e e^x}{2 c d-2 \sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{e \sqrt{c^2 d^2-e^2}}+\frac{\left (2 b^2 c\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e e^x}{2 c d+2 \sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{e \sqrt{c^2 d^2-e^2}}\\ &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac{2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{e e^{\cosh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{\left (2 b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 e x}{2 c d-2 \sqrt{c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{e \sqrt{c^2 d^2-e^2}}+\frac{\left (2 b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 e x}{2 c d+2 \sqrt{c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{e \sqrt{c^2 d^2-e^2}}\\ &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac{2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{e e^{\cosh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}+\frac{2 b^2 c \text{Li}_2\left (-\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{2 b^2 c \text{Li}_2\left (-\frac{e e^{\cosh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}\\ \end{align*}

Mathematica [C]  time = 4.46334, size = 959, normalized size = 3.44 \[ -\frac{a^2}{e (d+e x)}+2 b c \left (\frac{2 \tan ^{-1}\left (\frac{\sqrt{e (e-c d)} \sqrt{\frac{c x-1}{c x+1}}}{\sqrt{e (c d+e)}}\right )}{\sqrt{e (e-c d)} \sqrt{e (c d+e)}}-\frac{\cosh ^{-1}(c x)}{e (c d+c e x)}\right ) a-\frac{b^2 c \left (\frac{\cosh ^{-1}(c x)^2}{c d+c e x}+\frac{2 \left (2 \cosh ^{-1}(c x) \tan ^{-1}\left (\frac{(c d+e) \coth \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{e^2-c^2 d^2}}\right )-2 i \cos ^{-1}\left (-\frac{c d}{e}\right ) \tan ^{-1}\left (\frac{(e-c d) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{e^2-c^2 d^2}}\right )+\left (\cos ^{-1}\left (-\frac{c d}{e}\right )+2 \left (\tan ^{-1}\left (\frac{(c d+e) \coth \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{e^2-c^2 d^2}}\right )+\tan ^{-1}\left (\frac{(e-c d) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{e^2-c^2 d^2}}\right )\right )\right ) \log \left (\frac{\sqrt{e^2-c^2 d^2} e^{-\frac{1}{2} \cosh ^{-1}(c x)}}{\sqrt{2} \sqrt{e} \sqrt{c d+c e x}}\right )+\left (\cos ^{-1}\left (-\frac{c d}{e}\right )-2 \left (\tan ^{-1}\left (\frac{(c d+e) \coth \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{e^2-c^2 d^2}}\right )+\tan ^{-1}\left (\frac{(e-c d) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{e^2-c^2 d^2}}\right )\right )\right ) \log \left (\frac{\sqrt{e^2-c^2 d^2} e^{\frac{1}{2} \cosh ^{-1}(c x)}}{\sqrt{2} \sqrt{e} \sqrt{c d+c e x}}\right )-\left (\cos ^{-1}\left (-\frac{c d}{e}\right )+2 \tan ^{-1}\left (\frac{(e-c d) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{e^2-c^2 d^2}}\right )\right ) \log \left (\frac{(c d+e) \left (c d-e+i \sqrt{e^2-c^2 d^2}\right ) \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )-1\right )}{e \left (c d+e+i \sqrt{e^2-c^2 d^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\left (\cos ^{-1}\left (-\frac{c d}{e}\right )-2 \tan ^{-1}\left (\frac{(e-c d) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{e^2-c^2 d^2}}\right )\right ) \log \left (\frac{(c d+e) \left (-c d+e+i \sqrt{e^2-c^2 d^2}\right ) \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )+1\right )}{e \left (c d+e+i \sqrt{e^2-c^2 d^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}\right )+i \left (\text{PolyLog}\left (2,\frac{\left (c d-i \sqrt{e^2-c^2 d^2}\right ) \left (c d+e-i \sqrt{e^2-c^2 d^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}{e \left (c d+e+i \sqrt{e^2-c^2 d^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\text{PolyLog}\left (2,\frac{\left (c d+i \sqrt{e^2-c^2 d^2}\right ) \left (c d+e-i \sqrt{e^2-c^2 d^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}{e \left (c d+e+i \sqrt{e^2-c^2 d^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}\right )\right )\right )}{\sqrt{e^2-c^2 d^2}}\right )}{e} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(a^2/(e*(d + e*x))) + 2*a*b*c*(-(ArcCosh[c*x]/(e*(c*d + c*e*x))) + (2*ArcTan[(Sqrt[e*(-(c*d) + e)]*Sqrt[(-1 +
c*x)/(1 + c*x)])/Sqrt[e*(c*d + e)]])/(Sqrt[e*(-(c*d) + e)]*Sqrt[e*(c*d + e)])) - (b^2*c*(ArcCosh[c*x]^2/(c*d
+ c*e*x) + (2*(2*ArcCosh[c*x]*ArcTan[((c*d + e)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]] - (2*I)*ArcCos[-
((c*d)/e)]*ArcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]] + (ArcCos[-((c*d)/e)] + 2*(ArcTa
n[((c*d + e)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]] + ArcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqrt[-
(c^2*d^2) + e^2]]))*Log[Sqrt[-(c^2*d^2) + e^2]/(Sqrt[2]*Sqrt[e]*E^(ArcCosh[c*x]/2)*Sqrt[c*d + c*e*x])] + (ArcC
os[-((c*d)/e)] - 2*(ArcTan[((c*d + e)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]] + ArcTan[((-(c*d) + e)*Tan
h[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]]))*Log[(Sqrt[-(c^2*d^2) + e^2]*E^(ArcCosh[c*x]/2))/(Sqrt[2]*Sqrt[e]*
Sqrt[c*d + c*e*x])] - (ArcCos[-((c*d)/e)] + 2*ArcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2
]])*Log[((c*d + e)*(c*d - e + I*Sqrt[-(c^2*d^2) + e^2])*(-1 + Tanh[ArcCosh[c*x]/2]))/(e*(c*d + e + I*Sqrt[-(c^
2*d^2) + e^2]*Tanh[ArcCosh[c*x]/2]))] - (ArcCos[-((c*d)/e)] - 2*ArcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqr
t[-(c^2*d^2) + e^2]])*Log[((c*d + e)*(-(c*d) + e + I*Sqrt[-(c^2*d^2) + e^2])*(1 + Tanh[ArcCosh[c*x]/2]))/(e*(c
*d + e + I*Sqrt[-(c^2*d^2) + e^2]*Tanh[ArcCosh[c*x]/2]))] + I*(PolyLog[2, ((c*d - I*Sqrt[-(c^2*d^2) + e^2])*(c
*d + e - I*Sqrt[-(c^2*d^2) + e^2]*Tanh[ArcCosh[c*x]/2]))/(e*(c*d + e + I*Sqrt[-(c^2*d^2) + e^2]*Tanh[ArcCosh[c
*x]/2]))] - PolyLog[2, ((c*d + I*Sqrt[-(c^2*d^2) + e^2])*(c*d + e - I*Sqrt[-(c^2*d^2) + e^2]*Tanh[ArcCosh[c*x]
/2]))/(e*(c*d + e + I*Sqrt[-(c^2*d^2) + e^2]*Tanh[ArcCosh[c*x]/2]))])))/Sqrt[-(c^2*d^2) + e^2]))/e

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Maple [A]  time = 0.063, size = 536, normalized size = 1.9 \begin{align*} -{\frac{c{a}^{2}}{ \left ( cxe+cd \right ) e}}-{\frac{c{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{ \left ( cxe+cd \right ) e}}+2\,{\frac{c{b}^{2}{\rm arccosh} \left (cx\right )}{e\sqrt{{c}^{2}{d}^{2}-{e}^{2}}}\ln \left ({\frac{- \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) e-cd+\sqrt{{c}^{2}{d}^{2}-{e}^{2}}}{-cd+\sqrt{{c}^{2}{d}^{2}-{e}^{2}}}} \right ) }-2\,{\frac{c{b}^{2}{\rm arccosh} \left (cx\right )}{e\sqrt{{c}^{2}{d}^{2}-{e}^{2}}}\ln \left ({\frac{ \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) e+cd+\sqrt{{c}^{2}{d}^{2}-{e}^{2}}}{cd+\sqrt{{c}^{2}{d}^{2}-{e}^{2}}}} \right ) }+2\,{\frac{c{b}^{2}}{e\sqrt{{c}^{2}{d}^{2}-{e}^{2}}}{\it dilog} \left ({\frac{- \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) e-cd+\sqrt{{c}^{2}{d}^{2}-{e}^{2}}}{-cd+\sqrt{{c}^{2}{d}^{2}-{e}^{2}}}} \right ) }-2\,{\frac{c{b}^{2}}{e\sqrt{{c}^{2}{d}^{2}-{e}^{2}}}{\it dilog} \left ({\frac{ \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) e+cd+\sqrt{{c}^{2}{d}^{2}-{e}^{2}}}{cd+\sqrt{{c}^{2}{d}^{2}-{e}^{2}}}} \right ) }-2\,{\frac{cab{\rm arccosh} \left (cx\right )}{ \left ( cxe+cd \right ) e}}-2\,{\frac{cab\sqrt{cx-1}\sqrt{cx+1}}{{e}^{2}\sqrt{{c}^{2}{x}^{2}-1}}\ln \left ( -2\,{\frac{1}{cxe+cd} \left ({c}^{2}dx-\sqrt{{c}^{2}{x}^{2}-1}\sqrt{{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}}e+e \right ) } \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c*a^2/(c*e*x+c*d)/e-c*b^2*arccosh(c*x)^2/e/(c*e*x+c*d)+2*c*b^2/e*arccosh(c*x)/(c^2*d^2-e^2)^(1/2)*ln((-(c*x+(
c*x-1)^(1/2)*(c*x+1)^(1/2))*e-c*d+(c^2*d^2-e^2)^(1/2))/(-c*d+(c^2*d^2-e^2)^(1/2)))-2*c*b^2/e*arccosh(c*x)/(c^2
*d^2-e^2)^(1/2)*ln(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e+c*d+(c^2*d^2-e^2)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))+2*
c*b^2/e/(c^2*d^2-e^2)^(1/2)*dilog((-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e-c*d+(c^2*d^2-e^2)^(1/2))/(-c*d+(c^2*d^
2-e^2)^(1/2)))-2*c*b^2/e/(c^2*d^2-e^2)^(1/2)*dilog(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e+c*d+(c^2*d^2-e^2)^(1/2
))/(c*d+(c^2*d^2-e^2)^(1/2)))-2*c*a*b/(c*e*x+c*d)/e*arccosh(c*x)-2*c*a*b/e^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(-2
*(c^2*d*x-(c^2*x^2-1)^(1/2)*((c^2*d^2-e^2)/e^2)^(1/2)*e+e)/(c*e*x+c*d))/((c^2*d^2-e^2)/e^2)^(1/2)/(c^2*x^2-1)^
(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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