3.1.12 \(\int \frac {\text {arccosh}(c x)^2}{(d+e x)^2} \, dx\) [12]

3.1.12.1 Optimal result

Integrand size = 14, antiderivative size = 259 \[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^2} \, dx=-\frac {\text {arccosh}(c x)^2}{e (d+e x)}+\frac {2 c \text {arccosh}(c x) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 c \text {arccosh}(c x) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 c \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 c \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}} \]

output

-arccosh(c*x)^2/e/(e*x+d)+2*c*arccosh(c*x)*ln(1+e*(c*x+(c*x-1)^(1/2)*(c*x+ 
1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e/(c^2*d^2-e^2)^(1/2)-2*c*arccosh(c*x 
)*ln(1+e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e/(c 
^2*d^2-e^2)^(1/2)+2*c*polylog(2,-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*d- 
(c^2*d^2-e^2)^(1/2)))/e/(c^2*d^2-e^2)^(1/2)-2*c*polylog(2,-e*(c*x+(c*x-1)^ 
(1/2)*(c*x+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e/(c^2*d^2-e^2)^(1/2)
 

3.1.12.2 Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.39 (sec) , antiderivative size = 848, normalized size of antiderivative = 3.27 \[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^2} \, dx=-\frac {c \left (\frac {\text {arccosh}(c x)^2}{c d+c e x}+\frac {2 \left (2 \text {arccosh}(c x) \arctan \left (\frac {(c d+e) \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )-2 i \arccos \left (-\frac {c d}{e}\right ) \arctan \left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )+\left (\arccos \left (-\frac {c d}{e}\right )+2 \left (\arctan \left (\frac {(c d+e) \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )+\arctan \left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right )\right ) \log \left (\frac {\sqrt {-c^2 d^2+e^2} e^{-\frac {1}{2} \text {arccosh}(c x)}}{\sqrt {2} \sqrt {e} \sqrt {c d+c e x}}\right )+\left (\arccos \left (-\frac {c d}{e}\right )-2 \left (\arctan \left (\frac {(c d+e) \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )+\arctan \left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right )\right ) \log \left (\frac {\sqrt {-c^2 d^2+e^2} e^{\frac {1}{2} \text {arccosh}(c x)}}{\sqrt {2} \sqrt {e} \sqrt {c d+c e x}}\right )-\left (\arccos \left (-\frac {c d}{e}\right )+2 \arctan \left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right ) \log \left (\frac {(c d+e) \left (c d-e+i \sqrt {-c^2 d^2+e^2}\right ) \left (-1+\tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}\right )-\left (\arccos \left (-\frac {c d}{e}\right )-2 \arctan \left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right ) \log \left (\frac {(c d+e) \left (-c d+e+i \sqrt {-c^2 d^2+e^2}\right ) \left (1+\tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (c d-i \sqrt {-c^2 d^2+e^2}\right ) \left (c d+e-i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (c d+i \sqrt {-c^2 d^2+e^2}\right ) \left (c d+e-i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}\right )\right )\right )}{\sqrt {-c^2 d^2+e^2}}\right )}{e} \]

input

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

output

-((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)]*A 
rcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]] + (ArcCo 
s[-((c*d)/e)] + 2*(ArcTan[((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])] + (ArcCos[-((c*d)/e)] - 2*(ArcTan[((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]*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])/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]))] 
+ 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...
 

3.1.12.3 Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.91, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6378, 6395, 3042, 3801, 2694, 27, 2620, 2715, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\text {arccosh}(c x)^2}{(d+e x)^2} \, dx\)

\(\Big \downarrow \) 6378

\(\displaystyle \frac {2 c \int \frac {\text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1} (d+e x)}dx}{e}-\frac {\text {arccosh}(c x)^2}{e (d+e x)}\)

\(\Big \downarrow \) 6395

\(\displaystyle \frac {2 c \int \frac {\text {arccosh}(c x)}{c d+c e x}d\text {arccosh}(c x)}{e}-\frac {\text {arccosh}(c x)^2}{e (d+e x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\text {arccosh}(c x)^2}{e (d+e x)}+\frac {2 c \int \frac {\text {arccosh}(c x)}{c d+e \sin \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{e}\)

\(\Big \downarrow \) 3801

\(\displaystyle \frac {4 c \int \frac {e^{\text {arccosh}(c x)} \text {arccosh}(c x)}{2 c e^{\text {arccosh}(c x)} d+e e^{2 \text {arccosh}(c x)}+e}d\text {arccosh}(c x)}{e}-\frac {\text {arccosh}(c x)^2}{e (d+e x)}\)

\(\Big \downarrow \) 2694

\(\displaystyle \frac {4 c \left (\frac {e \int \frac {e^{\text {arccosh}(c x)} \text {arccosh}(c x)}{2 \left (c d+e e^{\text {arccosh}(c x)}-\sqrt {c^2 d^2-e^2}\right )}d\text {arccosh}(c x)}{\sqrt {c^2 d^2-e^2}}-\frac {e \int \frac {e^{\text {arccosh}(c x)} \text {arccosh}(c x)}{2 \left (c d+e e^{\text {arccosh}(c x)}+\sqrt {c^2 d^2-e^2}\right )}d\text {arccosh}(c x)}{\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {\text {arccosh}(c x)^2}{e (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 c \left (\frac {e \int \frac {e^{\text {arccosh}(c x)} \text {arccosh}(c x)}{c d+e e^{\text {arccosh}(c x)}-\sqrt {c^2 d^2-e^2}}d\text {arccosh}(c x)}{2 \sqrt {c^2 d^2-e^2}}-\frac {e \int \frac {e^{\text {arccosh}(c x)} \text {arccosh}(c x)}{c d+e e^{\text {arccosh}(c x)}+\sqrt {c^2 d^2-e^2}}d\text {arccosh}(c x)}{2 \sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {\text {arccosh}(c x)^2}{e (d+e x)}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {4 c \left (\frac {e \left (\frac {\text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e}-\frac {\int \log \left (\frac {e^{\text {arccosh}(c x)} e}{c d-\sqrt {c^2 d^2-e^2}}+1\right )d\text {arccosh}(c x)}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}-\frac {e \left (\frac {\text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e}-\frac {\int \log \left (\frac {e^{\text {arccosh}(c x)} e}{c d+\sqrt {c^2 d^2-e^2}}+1\right )d\text {arccosh}(c x)}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {\text {arccosh}(c x)^2}{e (d+e x)}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {4 c \left (\frac {e \left (\frac {\text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e}-\frac {\int e^{-\text {arccosh}(c x)} \log \left (\frac {e^{\text {arccosh}(c x)} e}{c d-\sqrt {c^2 d^2-e^2}}+1\right )de^{\text {arccosh}(c x)}}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}-\frac {e \left (\frac {\text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e}-\frac {\int e^{-\text {arccosh}(c x)} \log \left (\frac {e^{\text {arccosh}(c x)} e}{c d+\sqrt {c^2 d^2-e^2}}+1\right )de^{\text {arccosh}(c x)}}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {\text {arccosh}(c x)^2}{e (d+e x)}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {4 c \left (\frac {e \left (\frac {\operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}-\frac {e \left (\frac {\operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {\text {arccosh}(c x)^2}{e (d+e x)}\)

input

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

output

-(ArcCosh[c*x]^2/(e*(d + e*x))) + (4*c*((e*((ArcCosh[c*x]*Log[1 + (e*E^Arc 
Cosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2])])/e + PolyLog[2, -((e*E^ArcCosh[c*x 
])/(c*d - Sqrt[c^2*d^2 - e^2]))]/e))/(2*Sqrt[c^2*d^2 - e^2]) - (e*((ArcCos 
h[c*x]*Log[1 + (e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2])])/e + PolyLo 
g[2, -((e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2]))]/e))/(2*Sqrt[c^2*d^ 
2 - e^2])))/e
 

3.1.12.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2620

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] - Si 
mp[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 2694

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]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[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 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[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 2838

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3801

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

rule 6378

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] 
 - Simp[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 6395

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/( 
Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[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])
 

3.1.12.4 Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.50

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arccosh}\left (c x \right )^{2} c^{2}}{e \left (e c x +c d \right )}+\frac {2 c^{2} \operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 c^{2} \operatorname {arccosh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}+\frac {2 c^{2} \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 c^{2} \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}}{c}\)	\(388\)
default	\(\frac {-\frac {\operatorname {arccosh}\left (c x \right )^{2} c^{2}}{e \left (e c x +c d \right )}+\frac {2 c^{2} \operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 c^{2} \operatorname {arccosh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}+\frac {2 c^{2} \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 c^{2} \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}}{c}\)	\(388\)

input

int(arccosh(c*x)^2/(e*x+d)^2,x,method=_RETURNVERBOSE)
 

output

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

3.1.12.5 Fricas [F]

\[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^2} \, dx=\int { \frac {\operatorname {arcosh}\left (c x\right )^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]

input

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

output

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

3.1.12.6 Sympy [F]

\[ \int \frac {\text {arccosh}(c x)^2}{(d+e x)^2} \, dx=\int \frac {\operatorname {acosh}^{2}{\left (c x \right )}}{\left (d + e x\right )^{2}}\, dx \]

input

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

output

Integral(acosh(c*x)**2/(d + e*x)**2, x)
 

3.1.12.7 Maxima [F(-2)]

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

input

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

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume((e-c*d)*(e+c*d)>0)', see `assume 
?` for mor
 

3.1.12.8 Giac [F(-2)]

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

input

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

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

3.1.12.9 Mupad [F(-1)]

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

input

int(acosh(c*x)^2/(d + e*x)^2,x)
 

output

int(acosh(c*x)^2/(d + e*x)^2, x)