\(\int \frac {(a+b \text {arcsinh}(\frac {\sqrt {1-c x}}{\sqrt {1+c x}}))^2}{1-c^2 x^2} \, dx\) [279]

Optimal result

Integrand size = 40, antiderivative size = 195 \[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{3 b c}-\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}-\frac {b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c} \] Output:

1/3*(a+b*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^3/b/c-(a+b*arcsinh((-c*x+1 
)^(1/2)/(c*x+1)^(1/2)))^2*ln(1-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+(1+(-c*x+1)/( 
c*x+1))^(1/2))^2)/c-b*(a+b*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))*polylog( 
2,((-c*x+1)^(1/2)/(c*x+1)^(1/2)+(1+(-c*x+1)/(c*x+1))^(1/2))^2)/c+1/2*b^2*p 
olylog(3,((-c*x+1)^(1/2)/(c*x+1)^(1/2)+(1+(-c*x+1)/(c*x+1))^(1/2))^2)/c
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )-3 b \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )\right )-6 b^2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )+3 b^3 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{6 b c} \] Input:

Integrate[(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2),x]
 

Output:

(2*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*(a + b*ArcSinh[Sqrt[1 - 
c*x]/Sqrt[1 + c*x]] - 3*b*Log[1 - E^(2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x] 
])]) - 6*b^2*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, E^(2* 
ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])] + 3*b^3*PolyLog[3, E^(2*ArcSinh[Sqr 
t[1 - c*x]/Sqrt[1 + c*x]])])/(6*b*c)
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.81 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {7232, 6190, 25, 3042, 26, 4201, 2620, 3011, 2720, 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.

\(\displaystyle \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{1-c^2 x^2} \, dx\)

\(\Big \downarrow \) 7232

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

\(\Big \downarrow \) 6190

\(\displaystyle -\frac {\int -\frac {(1-c x) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{b}\right )}{c x+1}d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {(1-c x) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{b}\right )}{c x+1}d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int -\frac {i (1-c x) \tan \left (\frac {i a}{b}-\frac {i \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}+\frac {\pi }{2}\right )}{c x+1}d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \int \frac {(1-c x) \tan \left (\frac {1}{2} \left (\frac {2 i a}{b}+\pi \right )-\frac {i \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}\right )}{c x+1}d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\)

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {i \left (2 i \left (b \int \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \log \left (1+e^{\frac {2 a}{b}-\frac {2 \sqrt {1-c x}}{b \sqrt {c x+1}}-i \pi }\right )d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {b (1-c x) \log \left (1+\exp \left (-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}+\frac {2 a}{b}-i \pi \right )\right )}{2 (c x+1)}\right )-\frac {i (1-c x)^{3/2}}{3 (c x+1)^{3/2}}\right )}{b c}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {i \left (2 i \left (b \left (\frac {1}{2} b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 \sqrt {1-c x}}{b \sqrt {c x+1}}-i \pi }\right )-\frac {1}{2} b \int \operatorname {PolyLog}\left (2,-\exp \left (\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}-i \pi \right )\right )d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )\right )-\frac {b (1-c x) \log \left (1+\exp \left (-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}+\frac {2 a}{b}-i \pi \right )\right )}{2 (c x+1)}\right )-\frac {i (1-c x)^{3/2}}{3 (c x+1)^{3/2}}\right )}{b c}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \int \frac {\sqrt {c x+1} \operatorname {PolyLog}\left (2,-a-b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{\sqrt {1-c x}}d\exp \left (\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}-i \pi \right )+\frac {1}{2} b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 \sqrt {1-c x}}{b \sqrt {c x+1}}-i \pi }\right )\right )-\frac {b (1-c x) \log \left (1+\exp \left (-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}+\frac {2 a}{b}-i \pi \right )\right )}{2 (c x+1)}\right )-\frac {i (1-c x)^{3/2}}{3 (c x+1)^{3/2}}\right )}{b c}\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \operatorname {PolyLog}\left (3,-a-b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )+\frac {1}{2} b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 \sqrt {1-c x}}{b \sqrt {c x+1}}-i \pi }\right )\right )-\frac {b (1-c x) \log \left (1+\exp \left (-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}+\frac {2 a}{b}-i \pi \right )\right )}{2 (c x+1)}\right )-\frac {i (1-c x)^{3/2}}{3 (c x+1)^{3/2}}\right )}{b c}\)

Input:

Int[(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2),x]
 

Output:

((-I)*(((-1/3*I)*(1 - c*x)^(3/2))/(1 + c*x)^(3/2) + (2*I)*(-1/2*(b*(1 - c* 
x)*Log[1 + E^((2*a)/b - I*Pi - (2*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c* 
x]]))/b)])/(1 + c*x) + b*((b*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])* 
PolyLog[2, -E^((2*a)/b - I*Pi - (2*Sqrt[1 - c*x])/(b*Sqrt[1 + c*x]))])/2 + 
 (b^2*PolyLog[3, -a - b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]]])/4))))/(b*c)
 

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 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 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 4201

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] + Simp[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 6190

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b 
 Subst[Int[x^n*Coth[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a 
, b, c}, x] && IGtQ[n, 0]
 

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]
 

rule 7232

Int[((a_.) + (b_.)*(F_)[((c_.)*Sqrt[(d_.) + (e_.)*(x_)])/Sqrt[(f_.) + (g_.) 
*(x_)]])^(n_.)/((A_.) + (C_.)*(x_)^2), x_Symbol] :> Simp[2*e*(g/(C*(e*f - d 
*g)))   Subst[Int[(a + b*F[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]], 
x] /; FreeQ[{a, b, c, d, e, f, g, A, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && 
EqQ[e*f + d*g, 0] && IGtQ[n, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(623\) vs. \(2(215)=430\).

Time = 1.18 (sec) , antiderivative size = 624, normalized size of antiderivative = 3.20

method	result	size
default	\(-\frac {a^{2} \ln \left (x c -1\right )}{2 c}+\frac {a^{2} \ln \left (x c +1\right )}{2 c}-b^{2} \left (-\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right )^{3}}{3 c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right )^{2} \ln \left (1-\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}-\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}+\frac {2 \,\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right ) \operatorname {polylog}\left (2, \frac {\sqrt {-x c +1}}{\sqrt {x c +1}}+\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}-\frac {2 \operatorname {polylog}\left (3, \frac {\sqrt {-x c +1}}{\sqrt {x c +1}}+\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right )^{2} \ln \left (1+\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}+\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}+\frac {2 \,\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right ) \operatorname {polylog}\left (2, -\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}-\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}-\frac {2 \operatorname {polylog}\left (3, -\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}-\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}\right )-2 a b \left (-\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right )^{2}}{2 c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right ) \ln \left (1-\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}-\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}+\frac {\operatorname {polylog}\left (2, \frac {\sqrt {-x c +1}}{\sqrt {x c +1}}+\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right ) \ln \left (1+\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}+\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}+\frac {\operatorname {polylog}\left (2, -\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}-\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}\right )\)	\(624\)
parts	\(-\frac {a^{2} \ln \left (x c -1\right )}{2 c}+\frac {a^{2} \ln \left (x c +1\right )}{2 c}-b^{2} \left (-\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right )^{3}}{3 c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right )^{2} \ln \left (1-\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}-\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}+\frac {2 \,\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right ) \operatorname {polylog}\left (2, \frac {\sqrt {-x c +1}}{\sqrt {x c +1}}+\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}-\frac {2 \operatorname {polylog}\left (3, \frac {\sqrt {-x c +1}}{\sqrt {x c +1}}+\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right )^{2} \ln \left (1+\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}+\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}+\frac {2 \,\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right ) \operatorname {polylog}\left (2, -\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}-\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}-\frac {2 \operatorname {polylog}\left (3, -\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}-\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}\right )-2 a b \left (-\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right )^{2}}{2 c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right ) \ln \left (1-\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}-\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}+\frac {\operatorname {polylog}\left (2, \frac {\sqrt {-x c +1}}{\sqrt {x c +1}}+\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right ) \ln \left (1+\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}+\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}+\frac {\operatorname {polylog}\left (2, -\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}-\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}\right )\)	\(624\)

Input:

int((a+b*arcsinh((-x*c+1)^(1/2)/(x*c+1)^(1/2)))^2/(-c^2*x^2+1),x,method=_R 
ETURNVERBOSE)
 

Output:

-1/2*a^2/c*ln(c*x-1)+1/2*a^2/c*ln(x*c+1)-b^2*(-1/3/c*arcsinh((-x*c+1)^(1/2 
)/(x*c+1)^(1/2))^3+1/c*arcsinh((-x*c+1)^(1/2)/(x*c+1)^(1/2))^2*ln(1-(-x*c+ 
1)^(1/2)/(x*c+1)^(1/2)-(1+(-x*c+1)/(x*c+1))^(1/2))+2/c*arcsinh((-x*c+1)^(1 
/2)/(x*c+1)^(1/2))*polylog(2,(-x*c+1)^(1/2)/(x*c+1)^(1/2)+(1+(-x*c+1)/(x*c 
+1))^(1/2))-2/c*polylog(3,(-x*c+1)^(1/2)/(x*c+1)^(1/2)+(1+(-x*c+1)/(x*c+1) 
)^(1/2))+1/c*arcsinh((-x*c+1)^(1/2)/(x*c+1)^(1/2))^2*ln(1+(-x*c+1)^(1/2)/( 
x*c+1)^(1/2)+(1+(-x*c+1)/(x*c+1))^(1/2))+2/c*arcsinh((-x*c+1)^(1/2)/(x*c+1 
)^(1/2))*polylog(2,-(-x*c+1)^(1/2)/(x*c+1)^(1/2)-(1+(-x*c+1)/(x*c+1))^(1/2 
))-2/c*polylog(3,-(-x*c+1)^(1/2)/(x*c+1)^(1/2)-(1+(-x*c+1)/(x*c+1))^(1/2)) 
)-2*a*b*(-1/2*arcsinh((-x*c+1)^(1/2)/(x*c+1)^(1/2))^2/c+1/c*arcsinh((-x*c+ 
1)^(1/2)/(x*c+1)^(1/2))*ln(1-(-x*c+1)^(1/2)/(x*c+1)^(1/2)-(1+(-x*c+1)/(x*c 
+1))^(1/2))+1/c*polylog(2,(-x*c+1)^(1/2)/(x*c+1)^(1/2)+(1+(-x*c+1)/(x*c+1) 
)^(1/2))+1/c*arcsinh((-x*c+1)^(1/2)/(x*c+1)^(1/2))*ln(1+(-x*c+1)^(1/2)/(x* 
c+1)^(1/2)+(1+(-x*c+1)/(x*c+1))^(1/2))+1/c*polylog(2,-(-x*c+1)^(1/2)/(x*c+ 
1)^(1/2)-(1+(-x*c+1)/(x*c+1))^(1/2)))
 

Fricas [F]

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

integrate((a+b*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2/(-c^2*x^2+1),x, al 
gorithm="fricas")
 

Output:

integral(-(b^2*arcsinh(sqrt(-c*x + 1)/sqrt(c*x + 1))^2 + 2*a*b*arcsinh(sqr 
t(-c*x + 1)/sqrt(c*x + 1)) + a^2)/(c^2*x^2 - 1), x)
 

Sympy [F]

\[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=- \int \frac {a^{2}}{c^{2} x^{2} - 1}\, dx - \int \frac {b^{2} \operatorname {asinh}^{2}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {2 a b \operatorname {asinh}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \] Input:

integrate((a+b*asinh((-c*x+1)**(1/2)/(c*x+1)**(1/2)))**2/(-c**2*x**2+1),x)
 

Output:

-Integral(a**2/(c**2*x**2 - 1), x) - Integral(b**2*asinh(sqrt(-c*x + 1)/sq 
rt(c*x + 1))**2/(c**2*x**2 - 1), x) - Integral(2*a*b*asinh(sqrt(-c*x + 1)/ 
sqrt(c*x + 1))/(c**2*x**2 - 1), x)
 

Maxima [F]

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

integrate((a+b*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2/(-c^2*x^2+1),x, al 
gorithm="maxima")
 

Output:

1/2*a^2*(log(c*x + 1)/c - log(c*x - 1)/c) + 1/2*(b^2*log(c*x + 1) - b^2*lo 
g(-c*x + 1))*log(sqrt(2) + sqrt(-c*x + 1))^2/c + integrate(-1/4*((sqrt(2)* 
b^2 + sqrt(-c*x + 1)*b^2)*log(c*x + 1)^2 - 4*(sqrt(2)*a*b + sqrt(-c*x + 1) 
*a*b)*log(c*x + 1) + 2*(4*sqrt(2)*a*b - 2*(sqrt(2)*b^2 + sqrt(-c*x + 1)*b^ 
2)*log(c*x + 1) + (4*a*b + (b^2*c*x + b^2)*log(c*x + 1) - (b^2*c*x + b^2)* 
log(-c*x + 1))*sqrt(-c*x + 1))*log(sqrt(2) + sqrt(-c*x + 1)))/(sqrt(2)*c^2 
*x^2 + (c^2*x^2 - 1)*sqrt(-c*x + 1) - sqrt(2)), x)
 

Giac [F]

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

integrate((a+b*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2/(-c^2*x^2+1),x, al 
gorithm="giac")
 

Output:

integrate(-(b*arcsinh(sqrt(-c*x + 1)/sqrt(c*x + 1)) + a)^2/(c^2*x^2 - 1), 
x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int -\frac {{\left (a+b\,\mathrm {asinh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^2}{c^2\,x^2-1} \,d x \] Input:

int(-(a + b*asinh((1 - c*x)^(1/2)/(c*x + 1)^(1/2)))^2/(c^2*x^2 - 1),x)
 

Output:

int(-(a + b*asinh((1 - c*x)^(1/2)/(c*x + 1)^(1/2)))^2/(c^2*x^2 - 1), x)
 

Reduce [F]

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

int((a+b*asinh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2/(-c^2*x^2+1),x)
 

Output:

( - 4*int(asinh(sqrt( - c*x + 1)/sqrt(c*x + 1))/(c**2*x**2 - 1),x)*a*b*c - 
 2*int(asinh(sqrt( - c*x + 1)/sqrt(c*x + 1))**2/(c**2*x**2 - 1),x)*b**2*c 
- log(c**2*x - c)*a**2 + log(c**2*x + c)*a**2)/(2*c)