\(\int \frac {(a+b \text {arcsinh}(\frac {\sqrt {1-c x}}{\sqrt {1+c x}}))^2}{1-c^2 x^2} \, dx\) [344]

Optimal result

Integrand size = 40, antiderivative size = 194 \[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=-\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{3 b c}-\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \log \left (1-e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {b^2 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c} \]

[Out]

-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(1-1/((
-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)))*pol
ylog(2,1/((-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*polylog(3,1/((-c*x+1)^(1/2)/(c
*x+1)^(1/2)+(1+(-c*x+1)/(c*x+1))^(1/2))^2)/c

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {6813, 5775, 3797, 2221, 2611, 2320, 6724} \[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\frac {b \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{c}-\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3}{3 b c}-\frac {\log \left (1-e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{c}+\frac {b^2 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )}{2 c} \]

[In]

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

[Out]

-1/3*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(b*c) - ((a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*Lo
g[1 - E^(-2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c + (b*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLo
g[2, E^(-2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c + (b^2*PolyLog[3, E^(-2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*
x]])])/(2*c)

Rule 2221

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]
- 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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{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 2611

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] + Dist[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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5775

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[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 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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 6813

Int[((a_.) + (b_.)*(F_)[((c_.)*Sqrt[(d_.) + (e_.)*(x_)])/Sqrt[(f_.) + (g_.)*(x_)]])^(n_.)/((A_.) + (C_.)*(x_)^
2), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^2}{x} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c} \\ & = \frac {\text {Subst}\left (\int x^2 \coth \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{b c} \\ & = -\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{3 b c}-\frac {2 \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x^2}{1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{b c} \\ & = -\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 {2 \text {Subst}\left (\int x \log \left (1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c} \\ & = -\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 \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c} \\ & = -\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 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{b}\right )}\right )}{2 c} \\ & = -\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 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{b}\right )}\right )}{2 c} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

(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[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(6*b*
c)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(623\) vs. \(2(214)=428\).

Time = 0.40 (sec) , antiderivative size = 624, normalized size of antiderivative = 3.22

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

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

[In]

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

[Out]

1/2*a^2*(log(c*x + 1)/c - log(c*x - 1)/c) + 1/2*(b^2*log(c*x + 1) - b^2*log(-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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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