3.3.79 \(\int \frac {e^{\cosh ^{-1}(a+b x)}}{x} \, dx\) [279]
Optimal. Leaf size=100 \[ b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}+2 a \sinh ^{-1}\left (\frac {\sqrt {-1+a+b x}}{\sqrt {2}}\right )+2 \sqrt {1-a^2} \text {ArcTan}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {-1+a+b x}}\right )+a \log (x) \]
[Out]
b*x+2*a*arcsinh(1/2*(b*x+a-1)^(1/2)*2^(1/2))+a*ln(x)+2*arctan((1-a)^(1/2)*(b*x+a+1)^(1/2)/(1+a)^(1/2)/(b*x+a-1
)^(1/2))*(-a^2+1)^(1/2)+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)
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Rubi [A]
time = 0.07, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6020, 14, 103,
163, 65, 221, 95, 211} \begin {gather*} 2 \sqrt {1-a^2} \text {ArcTan}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {a+b x-1}}\right )+\sqrt {a+b x-1} \sqrt {a+b x+1}+2 a \sinh ^{-1}\left (\frac {\sqrt {a+b x-1}}{\sqrt {2}}\right )+a \log (x)+b x \end {gather*}
Antiderivative was successfully verified.
[In]
Int[E^ArcCosh[a + b*x]/x,x]
[Out]
b*x + Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x] + 2*a*ArcSinh[Sqrt[-1 + a + b*x]/Sqrt[2]] + 2*Sqrt[1 - a^2]*ArcTan[
(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[-1 + a + b*x])] + a*Log[x]
Rule 14
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&& !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Rule 65
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 95
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]
Rule 103
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b
*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
Rule 163
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Rule 211
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 221
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]
Rule 6020
Int[E^(ArcCosh[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(u + Sqrt[-1 + u]*Sqrt[1 + u])^n, x] /; RationalQ[m
] && IntegerQ[n] && PolyQ[u, x]
Rubi steps
\begin {align*} \int \frac {e^{\cosh ^{-1}(a+b x)}}{x} \, dx &=\int \frac {a+b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}}{x} \, dx\\ &=\int \left (b+\frac {a}{x}+\frac {\sqrt {-1+a+b x} \sqrt {1+a+b x}}{x}\right ) \, dx\\ &=b x+a \log (x)+\int \frac {\sqrt {-1+a+b x} \sqrt {1+a+b x}}{x} \, dx\\ &=b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}+a \log (x)-\int \frac {1-a^2-a b x}{x \sqrt {-1+a+b x} \sqrt {1+a+b x}} \, dx\\ &=b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}+a \log (x)-\left (1-a^2\right ) \int \frac {1}{x \sqrt {-1+a+b x} \sqrt {1+a+b x}} \, dx+(a b) \int \frac {1}{\sqrt {-1+a+b x} \sqrt {1+a+b x}} \, dx\\ &=b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}+a \log (x)+(2 a) \text {Subst}\left (\int \frac {1}{\sqrt {2+x^2}} \, dx,x,\sqrt {-1+a+b x}\right )-\left (2 \left (1-a^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-a-(1-a) x^2} \, dx,x,\frac {\sqrt {1+a+b x}}{\sqrt {-1+a+b x}}\right )\\ &=b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}+2 a \sinh ^{-1}\left (\frac {\sqrt {-1+a+b x}}{\sqrt {2}}\right )+2 \sqrt {1-a^2} \tan ^{-1}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {-1+a+b x}}\right )+a \log (x)\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.07, size = 141, normalized size = 1.41 \begin {gather*} b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}+a \log (x)+a \log \left (a+b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}\right )+i \sqrt {1-a^2} \log \left (\frac {2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{\left (-1+a^2\right ) x}+\frac {2 i \left (-1+a^2+a b x\right )}{\sqrt {1-a^2} \left (-1+a^2\right ) x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[E^ArcCosh[a + b*x]/x,x]
[Out]
b*x + Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x] + a*Log[x] + a*Log[a + b*x + Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]]
+ I*Sqrt[1 - a^2]*Log[(2*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/((-1 + a^2)*x) + ((2*I)*(-1 + a^2 + a*b*x))/(Sq
rt[1 - a^2]*(-1 + a^2)*x)]
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.07, size = 156, normalized size = 1.56
| | |
method |
result |
size |
| | |
default |
\(\frac {\left (-\mathrm {csgn}\left (b \right ) \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) \sqrt {a^{2}-1}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\left (b \right )+\ln \left (\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\left (b \right )+b x +a \right ) \mathrm {csgn}\left (b \right )\right ) a \right ) \sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \mathrm {csgn}\left (b \right )}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}+b x +a \ln \left (x \right )\) |
\(156\) |
| | |
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x,x,method=_RETURNVERBOSE)
[Out]
(-csgn(b)*ln(2*(a*b*x+(a^2-1)^(1/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+a^2-1)/x)*(a^2-1)^(1/2)+(b^2*x^2+2*a*b*x+a^2
-1)^(1/2)*csgn(b)+ln(((b^2*x^2+2*a*b*x+a^2-1)^(1/2)*csgn(b)+b*x+a)*csgn(b))*a)*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)
*csgn(b)/(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+b*x+a*ln(x)
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x,x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a-1>0)', see `assume?` for mor
e details)Is
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Fricas [A]
time = 0.38, size = 246, normalized size = 2.46 \begin {gather*} \left [b x - a \log \left (-b x + \sqrt {b x + a + 1} \sqrt {b x + a - 1} - a\right ) + a \log \left (x\right ) + \sqrt {a^{2} - 1} \log \left (\frac {a^{2} b x + a^{3} + {\left (a^{2} - \sqrt {a^{2} - 1} a - 1\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} - {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) + \sqrt {b x + a + 1} \sqrt {b x + a - 1}, b x - a \log \left (-b x + \sqrt {b x + a + 1} \sqrt {b x + a - 1} - a\right ) + a \log \left (x\right ) + 2 \, \sqrt {-a^{2} + 1} \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {-a^{2} + 1} \sqrt {b x + a + 1} \sqrt {b x + a - 1}}{a^{2} - 1}\right ) + \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x,x, algorithm="fricas")
[Out]
[b*x - a*log(-b*x + sqrt(b*x + a + 1)*sqrt(b*x + a - 1) - a) + a*log(x) + sqrt(a^2 - 1)*log((a^2*b*x + a^3 + (
a^2 - sqrt(a^2 - 1)*a - 1)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) - (a*b*x + a^2 - 1)*sqrt(a^2 - 1) - a)/x) + sqr
t(b*x + a + 1)*sqrt(b*x + a - 1), b*x - a*log(-b*x + sqrt(b*x + a + 1)*sqrt(b*x + a - 1) - a) + a*log(x) + 2*s
qrt(-a^2 + 1)*arctan(-(sqrt(-a^2 + 1)*b*x - sqrt(-a^2 + 1)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1))/(a^2 - 1)) + s
qrt(b*x + a + 1)*sqrt(b*x + a - 1)]
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b x + \sqrt {a + b x - 1} \sqrt {a + b x + 1}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a+(b*x+a-1)**(1/2)*(b*x+a+1)**(1/2))/x,x)
[Out]
Integral((a + b*x + sqrt(a + b*x - 1)*sqrt(a + b*x + 1))/x, x)
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Giac [A]
time = 0.40, size = 103, normalized size = 1.03 \begin {gather*} b x - a \log \left ({\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2}\right ) + a \log \left ({\left | b x \right |}\right ) + 2 \, \sqrt {-a^{2} + 1} \arctan \left (\frac {{\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} - 2 \, a}{2 \, \sqrt {-a^{2} + 1}}\right ) + \sqrt {b x + a + 1} \sqrt {b x + a - 1} + a + 1 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x,x, algorithm="giac")
[Out]
b*x - a*log((sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^2) + a*log(abs(b*x)) + 2*sqrt(-a^2 + 1)*arctan(1/2*((sqrt(
b*x + a + 1) - sqrt(b*x + a - 1))^2 - 2*a)/sqrt(-a^2 + 1)) + sqrt(b*x + a + 1)*sqrt(b*x + a - 1) + a + 1
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Mupad [B]
time = 23.41, size = 2500, normalized size = 25.00 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + (a + b*x - 1)^(1/2)*(a + b*x + 1)^(1/2) + b*x)/x,x)
[Out]
b*x + a*log(x) - a*atan(-(a*(2*a*((32*(a*(a - 1)^(3/2)*(a + 1)^(3/2) - 2*a*(a - 1)^(1/2)*(a + 1)^(1/2) + 22*a^
3*(a - 1)^(1/2)*(a + 1)^(1/2) - 68*a^5*(a - 1)^(1/2)*(a + 1)^(1/2) - 4*a^3*(a - 1)^(3/2)*(a + 1)^(3/2) + 92*a^
7*(a - 1)^(1/2)*(a + 1)^(1/2) - 22*a^5*(a - 1)^(3/2)*(a + 1)^(3/2) - 58*a^9*(a - 1)^(1/2)*(a + 1)^(1/2) - 5*a^
3*(a - 1)^(5/2)*(a + 1)^(5/2) + 52*a^7*(a - 1)^(3/2)*(a + 1)^(3/2) + 14*a^11*(a - 1)^(1/2)*(a + 1)^(1/2) + 12*
a^5*(a - 1)^(5/2)*(a + 1)^(5/2) - 27*a^9*(a - 1)^(3/2)*(a + 1)^(3/2) + 9*a^7*(a - 1)^(5/2)*(a + 1)^(5/2) + 4*a
^5*(a - 1)^(7/2)*(a + 1)^(7/2)))/(5*a^2 - 10*a^4 + 10*a^6 - 5*a^8 + a^10 - 1) - 2*a*((32*(2*a - 10*a^3 + 20*a^
5 - 20*a^7 + 10*a^9 - 2*a^11 - 2*a*(a - 1)*(a + 1) + 2*a^3*(a - 1)^2*(a + 1)^2 - 6*a^5*(a - 1)^2*(a + 1)^2 + 4
*a^7*(a - 1)^2*(a + 1)^2 + 8*a^3*(a - 1)*(a + 1) - 12*a^5*(a - 1)*(a + 1) + 8*a^7*(a - 1)*(a + 1) - 2*a^9*(a -
1)*(a + 1)))/(5*a^2 - 10*a^4 + 10*a^6 - 5*a^8 + a^10 - 1) + 2*a*((32*(a*(a - 1)^(1/2)*(a + 1)^(1/2) - 4*a^3*(
a - 1)^(1/2)*(a + 1)^(1/2) + 6*a^5*(a - 1)^(1/2)*(a + 1)^(1/2) - 5*a^3*(a - 1)^(3/2)*(a + 1)^(3/2) - 4*a^7*(a
- 1)^(1/2)*(a + 1)^(1/2) + 10*a^5*(a - 1)^(3/2)*(a + 1)^(3/2) + a^9*(a - 1)^(1/2)*(a + 1)^(1/2) - 5*a^7*(a - 1
)^(3/2)*(a + 1)^(3/2) + 4*a^5*(a - 1)^(5/2)*(a + 1)^(5/2)))/(5*a^2 - 10*a^4 + 10*a^6 - 5*a^8 + a^10 - 1) - (32
*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))*(60*a^2 - 150*a^4 + 200*a^6 - 150*a^8 + 60*a^10 - 10*a^12 - 39*a^4*(a -
1)^2*(a + 1)^2 + 78*a^6*(a - 1)^2*(a + 1)^2 + 16*a^6*(a - 1)^3*(a + 1)^3 - 39*a^8*(a - 1)^2*(a + 1)^2 + 33*a^
2*(a - 1)*(a + 1) - 132*a^4*(a - 1)*(a + 1) + 198*a^6*(a - 1)*(a + 1) - 132*a^8*(a - 1)*(a + 1) + 33*a^10*(a -
1)*(a + 1) - 10))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))*(15*a^4 - 6*a^2 - 20*a^6 + 15*a^8 - 6*a^10 + a^12 +
1))) + (32*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))*(17*(a - 1)^(1/2)*(a + 1)^(1/2) - 106*a^2*(a - 1)^(1/2)*(a +
1)^(1/2) + 275*a^4*(a - 1)^(1/2)*(a + 1)^(1/2) - 48*a^2*(a - 1)^(3/2)*(a + 1)^(3/2) - 380*a^6*(a - 1)^(1/2)*(a
+ 1)^(1/2) + 196*a^4*(a - 1)^(3/2)*(a + 1)^(3/2) + 295*a^8*(a - 1)^(1/2)*(a + 1)^(1/2) - 300*a^6*(a - 1)^(3/2
)*(a + 1)^(3/2) - 122*a^10*(a - 1)^(1/2)*(a + 1)^(1/2) + 47*a^4*(a - 1)^(5/2)*(a + 1)^(5/2) + 204*a^8*(a - 1)^
(3/2)*(a + 1)^(3/2) + 21*a^12*(a - 1)^(1/2)*(a + 1)^(1/2) - 94*a^6*(a - 1)^(5/2)*(a + 1)^(5/2) - 52*a^10*(a -
1)^(3/2)*(a + 1)^(3/2) + 47*a^8*(a - 1)^(5/2)*(a + 1)^(5/2) - 16*a^6*(a - 1)^(7/2)*(a + 1)^(7/2)))/(((a + 1)^(
1/2) - (a + b*x + 1)^(1/2))*(15*a^4 - 6*a^2 - 20*a^6 + 15*a^8 - 6*a^10 + a^12 + 1))) + (32*((a - 1)^(1/2) - (a
+ b*x - 1)^(1/2))*(36*a^2 - 216*a^4 + 540*a^6 - 720*a^8 + 540*a^10 - 216*a^12 + 36*a^14 - 6*(a - 1)*(a + 1) +
15*a^2*(a - 1)^2*(a + 1)^2 - 64*a^4*(a - 1)^2*(a + 1)^2 - 9*a^4*(a - 1)^3*(a + 1)^3 + 262*a^6*(a - 1)^2*(a +
1)^2 + 18*a^6*(a - 1)^3*(a + 1)^3 - 392*a^8*(a - 1)^2*(a + 1)^2 - 73*a^8*(a - 1)^3*(a + 1)^3 + 179*a^10*(a - 1
)^2*(a + 1)^2 + 40*a^2*(a - 1)*(a + 1) - 242*a^4*(a - 1)*(a + 1) + 688*a^6*(a - 1)*(a + 1) - 922*a^8*(a - 1)*(
a + 1) + 584*a^10*(a - 1)*(a + 1) - 142*a^12*(a - 1)*(a + 1)))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))*(15*a^4
- 6*a^2 - 20*a^6 + 15*a^8 - 6*a^10 + a^12 + 1))) - (32*(2*a*(a - 1)^2*(a + 1)^2 - 2*a*(a - 1)*(a + 1) - 8*a^3*
(a - 1)^2*(a + 1)^2 - 2*a^3*(a - 1)^3*(a + 1)^3 + 12*a^5*(a - 1)^2*(a + 1)^2 + 6*a^5*(a - 1)^3*(a + 1)^3 - 8*a
^7*(a - 1)^2*(a + 1)^2 - 4*a^7*(a - 1)^3*(a + 1)^3 + 2*a^9*(a - 1)^2*(a + 1)^2 + 10*a^3*(a - 1)*(a + 1) - 20*a
^5*(a - 1)*(a + 1) + 20*a^7*(a - 1)*(a + 1) - 10*a^9*(a - 1)*(a + 1) + 2*a^11*(a - 1)*(a + 1)))/(5*a^2 - 10*a^
4 + 10*a^6 - 5*a^8 + a^10 - 1) + (32*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))*((a - 1)^(3/2)*(a + 1)^(3/2) + 64*a
^2*(a - 1)^(1/2)*(a + 1)^(1/2) - 384*a^4*(a - 1)^(1/2)*(a + 1)^(1/2) - 10*a^2*(a - 1)^(3/2)*(a + 1)^(3/2) + 96
0*a^6*(a - 1)^(1/2)*(a + 1)^(1/2) - 157*a^4*(a - 1)^(3/2)*(a + 1)^(3/2) - 1280*a^8*(a - 1)^(1/2)*(a + 1)^(1/2)
+ 708*a^6*(a - 1)^(3/2)*(a + 1)^(3/2) + 960*a^10*(a - 1)^(1/2)*(a + 1)^(1/2) + 4*a^4*(a - 1)^(5/2)*(a + 1)^(5
/2) - 1097*a^8*(a - 1)^(3/2)*(a + 1)^(3/2) - 384*a^12*(a - 1)^(1/2)*(a + 1)^(1/2) + 180*a^6*(a - 1)^(5/2)*(a +
1)^(5/2) + 742*a^10*(a - 1)^(3/2)*(a + 1)^(3/2) + 64*a^14*(a - 1)^(1/2)*(a + 1)^(1/2) - a^4*(a - 1)^(7/2)*(a
+ 1)^(7/2) - 372*a^8*(a - 1)^(5/2)*(a + 1)^(5/2) - 187*a^12*(a - 1)^(3/2)*(a + 1)^(3/2) + 2*a^6*(a - 1)^(7/2)*
(a + 1)^(7/2) + 188*a^10*(a - 1)^(5/2)*(a + 1)^(5/2) - 65*a^8*(a - 1)^(7/2)*(a + 1)^(7/2)))/(((a + 1)^(1/2) -
(a + b*x + 1)^(1/2))*(15*a^4 - 6*a^2 - 20*a^6 + 15*a^8 - 6*a^10 + a^12 + 1)))*2i - a*((32*(2*a*(a - 1)^2*(a +
1)^2 - 2*a*(a - 1)*(a + 1) - 8*a^3*(a - 1)^2*(a + 1)^2 - 2*a^3*(a - 1)^3*(a + 1)^3 + 12*a^5*(a - 1)^2*(a + 1)^
2 + 6*a^5*(a - 1)^3*(a + 1)^3 - 8*a^7*(a - 1)^2*(a + 1)^2 - 4*a^7*(a - 1)^3*(a + 1)^3 + 2*a^9*(a - 1)^2*(a + 1
)^2 + 10*a^3*(a - 1)*(a + 1) - 20*a^5*(a - 1)*(a + 1) + 20*a^7*(a - 1)*(a + 1) - 10*a^9*(a - 1)*(a + 1) + 2*a^
11*(a - 1)*(a + 1)))/(5*a^2 - 10*a^4 + 10*a^6 - 5*a^8 + a^10 - 1) + 2*a*((32*(a*(a - 1)^(3/2)*(a + 1)^(3/2) -
2*a*(a - 1)^(1/2)*(a + 1)^(1/2) + 22*a^3*(a - 1...
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