\(\int \frac {(a+b \text {arccosh}(c x))^2}{(d-c^2 d x^2)^2} \, dx\) [11]

Optimal result

Integrand size = 24, antiderivative size = 195 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {b (a+b \text {arccosh}(c x))}{c d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x (a+b \text {arccosh}(c x))^2}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(a+b \text {arccosh}(c x))^2 \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c d^2}-\frac {b^2 \text {arctanh}(c x)}{c d^2}+\frac {b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{c d^2}-\frac {b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c d^2}-\frac {b^2 \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(c x)}\right )}{c d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,e^{\text {arccosh}(c x)}\right )}{c d^2} \] Output:

-b*(a+b*arccosh(c*x))/c/d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/2*x*(a+b*arccosh 
(c*x))^2/d^2/(-c^2*x^2+1)+(a+b*arccosh(c*x))^2*arctanh(c*x+(c*x-1)^(1/2)*( 
c*x+1)^(1/2))/c/d^2-b^2*arctanh(c*x)/c/d^2+b*(a+b*arccosh(c*x))*polylog(2, 
-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d^2-b*(a+b*arccosh(c*x))*polylog(2,c*x 
+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d^2-b^2*polylog(3,-c*x-(c*x-1)^(1/2)*(c*x+ 
1)^(1/2))/c/d^2+b^2*polylog(3,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d^2
 

Mathematica [A] (warning: unable to verify)

Time = 5.43 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.84 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {-\frac {4 a^2 x}{-1+c^2 x^2}-\frac {2 a^2 \log (1-c x)}{c}+\frac {2 a^2 \log (1+c x)}{c}+\frac {4 a b \left (-\frac {2 \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)+\text {arccosh}(c x) \left (c x+\left (-1+c^2 x^2\right ) \log \left (1-e^{\text {arccosh}(c x)}\right )+\left (1-c^2 x^2\right ) \log \left (1+e^{\text {arccosh}(c x)}\right )\right )\right )}{-1+c^2 x^2}+2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c}+\frac {b^2 \left (-4 \text {arccosh}(c x) \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )-\text {arccosh}(c x)^2 \text {csch}^2\left (\frac {1}{2} \text {arccosh}(c x)\right )-4 \text {arccosh}(c x)^2 \log \left (1-e^{-\text {arccosh}(c x)}\right )+4 \text {arccosh}(c x)^2 \log \left (1+e^{-\text {arccosh}(c x)}\right )+8 \log \left (\tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-8 \text {arccosh}(c x) \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(c x)}\right )+8 \text {arccosh}(c x) \operatorname {PolyLog}\left (2,e^{-\text {arccosh}(c x)}\right )-8 \operatorname {PolyLog}\left (3,-e^{-\text {arccosh}(c x)}\right )+8 \operatorname {PolyLog}\left (3,e^{-\text {arccosh}(c x)}\right )-\text {arccosh}(c x)^2 \text {sech}^2\left (\frac {1}{2} \text {arccosh}(c x)\right )+4 \text {arccosh}(c x) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{c}}{8 d^2} \] Input:

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

Output:

((-4*a^2*x)/(-1 + c^2*x^2) - (2*a^2*Log[1 - c*x])/c + (2*a^2*Log[1 + c*x]) 
/c + (4*a*b*((-2*(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) + ArcCosh[c*x]*(c*x 
 + (-1 + c^2*x^2)*Log[1 - E^ArcCosh[c*x]] + (1 - c^2*x^2)*Log[1 + E^ArcCos 
h[c*x]])))/(-1 + c^2*x^2) + 2*PolyLog[2, -E^ArcCosh[c*x]] - 2*PolyLog[2, E 
^ArcCosh[c*x]]))/c + (b^2*(-4*ArcCosh[c*x]*Coth[ArcCosh[c*x]/2] - ArcCosh[ 
c*x]^2*Csch[ArcCosh[c*x]/2]^2 - 4*ArcCosh[c*x]^2*Log[1 - E^(-ArcCosh[c*x]) 
] + 4*ArcCosh[c*x]^2*Log[1 + E^(-ArcCosh[c*x])] + 8*Log[Tanh[ArcCosh[c*x]/ 
2]] - 8*ArcCosh[c*x]*PolyLog[2, -E^(-ArcCosh[c*x])] + 8*ArcCosh[c*x]*PolyL 
og[2, E^(-ArcCosh[c*x])] - 8*PolyLog[3, -E^(-ArcCosh[c*x])] + 8*PolyLog[3, 
 E^(-ArcCosh[c*x])] - ArcCosh[c*x]^2*Sech[ArcCosh[c*x]/2]^2 + 4*ArcCosh[c* 
x]*Tanh[ArcCosh[c*x]/2]))/c)/(8*d^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.89 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.95, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {6316, 27, 6318, 3042, 26, 4670, 3011, 2720, 6330, 25, 39, 219, 7143}

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.

\(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx\)

\(\Big \downarrow \) 6316

\(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c x))^2}{d \left (1-c^2 x^2\right )}dx}{2 d}+\frac {b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{d^2}+\frac {x (a+b \text {arccosh}(c x))^2}{2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c x))^2}{1-c^2 x^2}dx}{2 d^2}+\frac {b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{d^2}+\frac {x (a+b \text {arccosh}(c x))^2}{2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 6318

\(\displaystyle \frac {b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{d^2}-\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{2 c d^2}+\frac {x (a+b \text {arccosh}(c x))^2}{2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{d^2}-\frac {\int i (a+b \text {arccosh}(c x))^2 \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c d^2}+\frac {x (a+b \text {arccosh}(c x))^2}{2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{d^2}-\frac {i \int (a+b \text {arccosh}(c x))^2 \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c d^2}+\frac {x (a+b \text {arccosh}(c x))^2}{2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 4670

\(\displaystyle -\frac {i \left (2 i b \int (a+b \text {arccosh}(c x)) \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-2 i b \int (a+b \text {arccosh}(c x)) \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{2 c d^2}+\frac {b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{d^2}+\frac {x (a+b \text {arccosh}(c x))^2}{2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {i \left (-2 i b \left (b \int \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i b \left (b \int \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{2 c d^2}+\frac {b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{d^2}+\frac {x (a+b \text {arccosh}(c x))^2}{2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {i \left (-2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{2 c d^2}+\frac {b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{d^2}+\frac {x (a+b \text {arccosh}(c x))^2}{2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 6330

\(\displaystyle -\frac {i \left (-2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{2 c d^2}+\frac {b c \left (\frac {b \int -\frac {1}{(1-c x) (c x+1)}dx}{c}-\frac {a+b \text {arccosh}(c x)}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2}+\frac {x (a+b \text {arccosh}(c x))^2}{2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {i \left (-2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{2 c d^2}+\frac {b c \left (-\frac {b \int \frac {1}{(1-c x) (c x+1)}dx}{c}-\frac {a+b \text {arccosh}(c x)}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2}+\frac {x (a+b \text {arccosh}(c x))^2}{2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 39

\(\displaystyle -\frac {i \left (-2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{2 c d^2}+\frac {b c \left (-\frac {b \int \frac {1}{1-c^2 x^2}dx}{c}-\frac {a+b \text {arccosh}(c x)}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2}+\frac {x (a+b \text {arccosh}(c x))^2}{2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {i \left (-2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{2 c d^2}+\frac {b c \left (-\frac {a+b \text {arccosh}(c x)}{c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b \text {arctanh}(c x)}{c^2}\right )}{d^2}+\frac {x (a+b \text {arccosh}(c x))^2}{2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {b c \left (-\frac {a+b \text {arccosh}(c x)}{c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b \text {arctanh}(c x)}{c^2}\right )}{d^2}-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2-2 i b \left (b \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i b \left (b \operatorname {PolyLog}\left (3,e^{\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )\right )}{2 c d^2}+\frac {x (a+b \text {arccosh}(c x))^2}{2 d^2 \left (1-c^2 x^2\right )}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

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 39

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( 
a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( 
IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{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 3011

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] + Simp[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 3042

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

rule 4670

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

rule 6316

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x 
_Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 
 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1))   Int[(d + e*x^2)^(p + 1)*(a + b* 
ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/((1 + 
 c*x)^p*(-1 + c*x)^p)]   Int[x*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a 
+ b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2* 
d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
 

rule 6318

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[-(c*d)^(-1)   Subst[Int[(a + b*x)^n*Csch[x], x], x, ArcCosh[c*x 
]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
 

rule 6330

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p 
_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + 
e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[b*(n/(2 
*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^ 
p]   Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 
 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && E 
qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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]
 

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 428, normalized size of antiderivative = 2.19

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \left (-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\operatorname {arccosh}\left (c x \right ) \left (c x \,\operatorname {arccosh}\left (c x \right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\frac {\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \,\operatorname {arctanh}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {c x \,\operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\frac {\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2}}}{c}\)	\(428\)
default	\(\frac {\frac {a^{2} \left (-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\operatorname {arccosh}\left (c x \right ) \left (c x \,\operatorname {arccosh}\left (c x \right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\frac {\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \,\operatorname {arctanh}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {c x \,\operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\frac {\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2}}}{c}\)	\(428\)
parts	\(\frac {a^{2} \left (-\frac {1}{4 c \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4 c}-\frac {1}{4 c \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4 c}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\operatorname {arccosh}\left (c x \right ) \left (c x \,\operatorname {arccosh}\left (c x \right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\frac {\operatorname {arccosh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \,\operatorname {arctanh}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2} c}+\frac {2 a b \left (-\frac {c x \,\operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}-\frac {\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2} c}\)	\(442\)

Input:

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

Output:

1/c*(a^2/d^2*(-1/4/(c*x-1)-1/4*ln(c*x-1)-1/4/(c*x+1)+1/4*ln(c*x+1))+b^2/d^ 
2*(-1/2/(c^2*x^2-1)*arccosh(c*x)*(c*x*arccosh(c*x)+2*(c*x-1)^(1/2)*(c*x+1) 
^(1/2))-1/2*arccosh(c*x)^2*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-arccosh(c 
*x)*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+polylog(3,c*x+(c*x-1)^(1/2) 
*(c*x+1)^(1/2))+1/2*arccosh(c*x)^2*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+a 
rccosh(c*x)*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-polylog(3,-c*x-(c* 
x-1)^(1/2)*(c*x+1)^(1/2))-2*arctanh(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+2*a* 
b/d^2*(-1/2*(c*x*arccosh(c*x)+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c^2*x^2-1)-1/2 
*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-1/2*polylog(2,c*x+(c*x 
-1)^(1/2)*(c*x+1)^(1/2))+1/2*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^( 
1/2))+1/2*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))))
 

Fricas [F]

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

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

Output:

integral((b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)/(c^4*d^2*x^4 - 2* 
c^2*d^2*x^2 + d^2), x)
 

Sympy [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a^{2}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} \operatorname {acosh}^{2}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {2 a b \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \] Input:

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

Output:

(Integral(a**2/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b**2*acosh(c*x 
)**2/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(2*a*b*acosh(c*x)/(c**4*x 
**4 - 2*c**2*x**2 + 1), x))/d**2
 

Maxima [F]

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

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

Output:

-1/4*a^2*(2*x/(c^2*d^2*x^2 - d^2) - log(c*x + 1)/(c*d^2) + log(c*x - 1)/(c 
*d^2)) - 1/4*(2*b^2*c*x - (b^2*c^2*x^2 - b^2)*log(c*x + 1) + (b^2*c^2*x^2 
- b^2)*log(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/(c^3*d^2*x^2 
 - c*d^2) - integrate(-1/2*(2*b^2*c^3*x^3 + (2*b^2*c^2*x^2 + 4*a*b - (b^2* 
c^3*x^3 - b^2*c*x)*log(c*x + 1) + (b^2*c^3*x^3 - b^2*c*x)*log(c*x - 1))*sq 
rt(c*x + 1)*sqrt(c*x - 1) + 2*(2*a*b*c - b^2*c)*x - (b^2*c^4*x^4 - 2*b^2*c 
^2*x^2 + b^2)*log(c*x + 1) + (b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*log(c*x - 
 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c^5*d^2*x^5 - 2*c^3*d^2*x^3 + 
 c*d^2*x + (c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*sqrt(c*x + 1)*sqrt(c*x - 1) 
), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int((a + b*acosh(c*x))^2/(d - c^2*d*x^2)^2,x)
 

Output:

int((a + b*acosh(c*x))^2/(d - c^2*d*x^2)^2, x)
 

Reduce [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {8 \left (\int \frac {\mathit {acosh} \left (c x \right )}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) a b \,c^{3} x^{2}-8 \left (\int \frac {\mathit {acosh} \left (c x \right )}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) a b c +4 \left (\int \frac {\mathit {acosh} \left (c x \right )^{2}}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) b^{2} c^{3} x^{2}-4 \left (\int \frac {\mathit {acosh} \left (c x \right )^{2}}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) b^{2} c -\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{2} x^{2}+\mathrm {log}\left (c^{2} x -c \right ) a^{2}+\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{2} x^{2}-\mathrm {log}\left (c^{2} x +c \right ) a^{2}-2 a^{2} c x}{4 c \,d^{2} \left (c^{2} x^{2}-1\right )} \] Input:

int((a+b*acosh(c*x))^2/(-c^2*d*x^2+d)^2,x)
                                                                                    
                                                                                    
 

Output:

(8*int(acosh(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1),x)*a*b*c**3*x**2 - 8*int(a 
cosh(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1),x)*a*b*c + 4*int(acosh(c*x)**2/(c* 
*4*x**4 - 2*c**2*x**2 + 1),x)*b**2*c**3*x**2 - 4*int(acosh(c*x)**2/(c**4*x 
**4 - 2*c**2*x**2 + 1),x)*b**2*c - log(c**2*x - c)*a**2*c**2*x**2 + log(c* 
*2*x - c)*a**2 + log(c**2*x + c)*a**2*c**2*x**2 - log(c**2*x + c)*a**2 - 2 
*a**2*c*x)/(4*c*d**2*(c**2*x**2 - 1))