Integrand size = 12, antiderivative size = 203 \[ \int e^{2 \text {sech}^{-1}(a x)} x^4 \, dx=\frac {5 \sqrt {\frac {1-a x}{1+a x}} (1+a x)^2}{4 a^5}+\frac {(1-a x) (1+a x)^4}{5 a^5}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^4 \left (5-6 \sqrt {\frac {1-a x}{1+a x}}\right )}{10 a^5}+\frac {(1+a x) \left (4-\sqrt {\frac {1-a x}{1+a x}}\right )}{4 a^5}-\frac {(1+a x)^3 \left (4+45 \sqrt {\frac {1-a x}{1+a x}}\right )}{30 a^5}-\frac {\arctan \left (\sqrt {\frac {1-a x}{1+a x}}\right )}{2 a^5} \]
1/5*(-a*x+1)*(a*x+1)^4/a^5-1/2*arctan(((-a*x+1)/(a*x+1))^(1/2))/a^5+1/4*(a *x+1)*(4-((-a*x+1)/(a*x+1))^(1/2))/a^5+5/4*(a*x+1)^2*((-a*x+1)/(a*x+1))^(1 /2)/a^5+1/10*(a*x+1)^4*(5-6*((-a*x+1)/(a*x+1))^(1/2))*((-a*x+1)/(a*x+1))^( 1/2)/a^5-1/30*(a*x+1)^3*(4+45*((-a*x+1)/(a*x+1))^(1/2))/a^5
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.52 \[ \int e^{2 \text {sech}^{-1}(a x)} x^4 \, dx=\frac {40 a^3 x^3-12 a^5 x^5-15 a \sqrt {\frac {1-a x}{1+a x}} \left (x+a x^2-2 a^2 x^3-2 a^3 x^4\right )+15 i \log \left (-2 i a x+2 \sqrt {\frac {1-a x}{1+a x}} (1+a x)\right )}{60 a^5} \]
(40*a^3*x^3 - 12*a^5*x^5 - 15*a*Sqrt[(1 - a*x)/(1 + a*x)]*(x + a*x^2 - 2*a ^2*x^3 - 2*a^3*x^4) + (15*I)*Log[(-2*I)*a*x + 2*Sqrt[(1 - a*x)/(1 + a*x)]* (1 + a*x)])/(60*a^5)
Time = 1.01 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.31, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6891, 7268, 2335, 27, 2342, 2335, 27, 2345, 27, 2345, 454, 216}
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.
| \(\displaystyle \int x^4 e^{2 \text {sech}^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6891 |
| \(\displaystyle \int x^4 \left (\frac {\sqrt {\frac {1-a x}{a x+1}}}{a x}+\sqrt {\frac {1-a x}{a x+1}}+\frac {1}{a x}\right )^2dx\) |
| \(\Big \downarrow \) 7268 |
| \(\displaystyle -\frac {4 \int \frac {\sqrt {\frac {1-a x}{a x+1}} \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^6}{\left (\frac {1-a x}{a x+1}+1\right )^6}d\sqrt {\frac {1-a x}{a x+1}}}{a^5}\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle -\frac {4 \left (-\frac {1}{10} \int -\frac {2 \left (5 \left (\frac {1-a x}{a x+1}\right )^{7/2}+15 \left (\frac {1-a x}{a x+1}\right )^{5/2}-65 \left (\frac {1-a x}{a x+1}\right )^{3/2}+21 \sqrt {\frac {1-a x}{a x+1}}+\frac {20 (1-a x)}{a x+1}-\frac {40 (1-a x)^2}{(a x+1)^2}+\frac {20 (1-a x)^3}{(a x+1)^3}\right )}{\left (\frac {1-a x}{a x+1}+1\right )^5}d\sqrt {\frac {1-a x}{a x+1}}-\frac {8 (1-a x)}{5 (a x+1) \left (\frac {1-a x}{a x+1}+1\right )^5}\right )}{a^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4 \left (\frac {1}{5} \int \frac {5 \left (\frac {1-a x}{a x+1}\right )^{7/2}+15 \left (\frac {1-a x}{a x+1}\right )^{5/2}-65 \left (\frac {1-a x}{a x+1}\right )^{3/2}+21 \sqrt {\frac {1-a x}{a x+1}}+\frac {20 (1-a x)}{a x+1}-\frac {40 (1-a x)^2}{(a x+1)^2}+\frac {20 (1-a x)^3}{(a x+1)^3}}{\left (\frac {1-a x}{a x+1}+1\right )^5}d\sqrt {\frac {1-a x}{a x+1}}-\frac {8 (1-a x)}{5 (a x+1) \left (\frac {1-a x}{a x+1}+1\right )^5}\right )}{a^5}\) |
| \(\Big \downarrow \) 2342 |
| \(\displaystyle -\frac {4 \left (\frac {1}{5} \int \frac {\sqrt {\frac {1-a x}{a x+1}} \left (\frac {5 (1-a x)^3}{(a x+1)^3}+\frac {15 (1-a x)^2}{(a x+1)^2}-\frac {65 (1-a x)}{a x+1}+20 \left (\frac {1-a x}{a x+1}\right )^{5/2}-40 \left (\frac {1-a x}{a x+1}\right )^{3/2}+20 \sqrt {\frac {1-a x}{a x+1}}+21\right )}{\left (\frac {1-a x}{a x+1}+1\right )^5}d\sqrt {\frac {1-a x}{a x+1}}-\frac {8 (1-a x)}{5 (a x+1) \left (\frac {1-a x}{a x+1}+1\right )^5}\right )}{a^5}\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle -\frac {4 \left (\frac {1}{5} \left (-\frac {1}{8} \int -\frac {8 \left (5 \left (\frac {1-a x}{a x+1}\right )^{5/2}+10 \left (\frac {1-a x}{a x+1}\right )^{3/2}-3 \sqrt {\frac {1-a x}{a x+1}}-\frac {60 (1-a x)}{a x+1}+\frac {20 (1-a x)^2}{(a x+1)^2}+10\right )}{\left (\frac {1-a x}{a x+1}+1\right )^4}d\sqrt {\frac {1-a x}{a x+1}}-\frac {2 \sqrt {\frac {1-a x}{a x+1}} \left (5-6 \sqrt {\frac {1-a x}{a x+1}}\right )}{\left (\frac {1-a x}{a x+1}+1\right )^4}\right )-\frac {8 (1-a x)}{5 (a x+1) \left (\frac {1-a x}{a x+1}+1\right )^5}\right )}{a^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4 \left (\frac {1}{5} \left (\int \frac {5 \left (\frac {1-a x}{a x+1}\right )^{5/2}+10 \left (\frac {1-a x}{a x+1}\right )^{3/2}-3 \sqrt {\frac {1-a x}{a x+1}}-\frac {60 (1-a x)}{a x+1}+\frac {20 (1-a x)^2}{(a x+1)^2}+10}{\left (\frac {1-a x}{a x+1}+1\right )^4}d\sqrt {\frac {1-a x}{a x+1}}-\frac {2 \sqrt {\frac {1-a x}{a x+1}} \left (5-6 \sqrt {\frac {1-a x}{a x+1}}\right )}{\left (\frac {1-a x}{a x+1}+1\right )^4}\right )-\frac {8 (1-a x)}{5 (a x+1) \left (\frac {1-a x}{a x+1}+1\right )^5}\right )}{a^5}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle -\frac {4 \left (\frac {1}{5} \left (-\frac {1}{6} \int \frac {30 \left (-\left (\frac {1-a x}{a x+1}\right )^{3/2}-\sqrt {\frac {1-a x}{a x+1}}-\frac {4 (1-a x)}{a x+1}+1\right )}{\left (\frac {1-a x}{a x+1}+1\right )^3}d\sqrt {\frac {1-a x}{a x+1}}-\frac {2 \sqrt {\frac {1-a x}{a x+1}} \left (5-6 \sqrt {\frac {1-a x}{a x+1}}\right )}{\left (\frac {1-a x}{a x+1}+1\right )^4}+\frac {45 \sqrt {\frac {1-a x}{a x+1}}+4}{3 \left (\frac {1-a x}{a x+1}+1\right )^3}\right )-\frac {8 (1-a x)}{5 (a x+1) \left (\frac {1-a x}{a x+1}+1\right )^5}\right )}{a^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4 \left (\frac {1}{5} \left (-5 \int \frac {-\left (\frac {1-a x}{a x+1}\right )^{3/2}-\sqrt {\frac {1-a x}{a x+1}}-\frac {4 (1-a x)}{a x+1}+1}{\left (\frac {1-a x}{a x+1}+1\right )^3}d\sqrt {\frac {1-a x}{a x+1}}-\frac {2 \sqrt {\frac {1-a x}{a x+1}} \left (5-6 \sqrt {\frac {1-a x}{a x+1}}\right )}{\left (\frac {1-a x}{a x+1}+1\right )^4}+\frac {45 \sqrt {\frac {1-a x}{a x+1}}+4}{3 \left (\frac {1-a x}{a x+1}+1\right )^3}\right )-\frac {8 (1-a x)}{5 (a x+1) \left (\frac {1-a x}{a x+1}+1\right )^5}\right )}{a^5}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle -\frac {4 \left (\frac {1}{5} \left (-5 \left (\frac {5 \sqrt {\frac {1-a x}{a x+1}}}{4 \left (\frac {1-a x}{a x+1}+1\right )^2}-\frac {1}{4} \int \frac {4 \sqrt {\frac {1-a x}{a x+1}}+1}{\left (\frac {1-a x}{a x+1}+1\right )^2}d\sqrt {\frac {1-a x}{a x+1}}\right )-\frac {2 \sqrt {\frac {1-a x}{a x+1}} \left (5-6 \sqrt {\frac {1-a x}{a x+1}}\right )}{\left (\frac {1-a x}{a x+1}+1\right )^4}+\frac {45 \sqrt {\frac {1-a x}{a x+1}}+4}{3 \left (\frac {1-a x}{a x+1}+1\right )^3}\right )-\frac {8 (1-a x)}{5 (a x+1) \left (\frac {1-a x}{a x+1}+1\right )^5}\right )}{a^5}\) |
| \(\Big \downarrow \) 454 |
| \(\displaystyle -\frac {4 \left (\frac {1}{5} \left (-5 \left (\frac {1}{4} \left (\frac {4-\sqrt {\frac {1-a x}{a x+1}}}{2 \left (\frac {1-a x}{a x+1}+1\right )}-\frac {1}{2} \int \frac {1}{\frac {1-a x}{a x+1}+1}d\sqrt {\frac {1-a x}{a x+1}}\right )+\frac {5 \sqrt {\frac {1-a x}{a x+1}}}{4 \left (\frac {1-a x}{a x+1}+1\right )^2}\right )-\frac {2 \sqrt {\frac {1-a x}{a x+1}} \left (5-6 \sqrt {\frac {1-a x}{a x+1}}\right )}{\left (\frac {1-a x}{a x+1}+1\right )^4}+\frac {45 \sqrt {\frac {1-a x}{a x+1}}+4}{3 \left (\frac {1-a x}{a x+1}+1\right )^3}\right )-\frac {8 (1-a x)}{5 (a x+1) \left (\frac {1-a x}{a x+1}+1\right )^5}\right )}{a^5}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {4 \left (\frac {1}{5} \left (-5 \left (\frac {1}{4} \left (\frac {4-\sqrt {\frac {1-a x}{a x+1}}}{2 \left (\frac {1-a x}{a x+1}+1\right )}-\frac {1}{2} \arctan \left (\sqrt {\frac {1-a x}{a x+1}}\right )\right )+\frac {5 \sqrt {\frac {1-a x}{a x+1}}}{4 \left (\frac {1-a x}{a x+1}+1\right )^2}\right )-\frac {2 \sqrt {\frac {1-a x}{a x+1}} \left (5-6 \sqrt {\frac {1-a x}{a x+1}}\right )}{\left (\frac {1-a x}{a x+1}+1\right )^4}+\frac {45 \sqrt {\frac {1-a x}{a x+1}}+4}{3 \left (\frac {1-a x}{a x+1}+1\right )^3}\right )-\frac {8 (1-a x)}{5 (a x+1) \left (\frac {1-a x}{a x+1}+1\right )^5}\right )}{a^5}\) |
(-4*((-8*(1 - a*x))/(5*(1 + a*x)*(1 + (1 - a*x)/(1 + a*x))^5) + ((-2*Sqrt[ (1 - a*x)/(1 + a*x)]*(5 - 6*Sqrt[(1 - a*x)/(1 + a*x)]))/(1 + (1 - a*x)/(1 + a*x))^4 + (4 + 45*Sqrt[(1 - a*x)/(1 + a*x)])/(3*(1 + (1 - a*x)/(1 + a*x) )^3) - 5*((5*Sqrt[(1 - a*x)/(1 + a*x)])/(4*(1 + (1 - a*x)/(1 + a*x))^2) + ((4 - Sqrt[(1 - a*x)/(1 + a*x)])/(2*(1 + (1 - a*x)/(1 + a*x))) - ArcTan[Sq rt[(1 - a*x)/(1 + a*x)]]/2)/4))/5))/a^5
3.1.65.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*d - b*c*x)/(2*a*b*(p + 1)))*(a + b*x^2)^(p + 1), x] + Simp[c*((2*p + 3)/(2*a *(p + 1))) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && L tQ[p, -1] && NeQ[p, -3/2]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq , a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(c*x)^m*(a + b*x^2)^(p + 1)*((a*g - b*f*x)/(2*a*b*(p + 1))), x] + Simp[c/(2*a*b*(p + 1)) Int[(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSu m[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[x*PolynomialQuotient [Pq, x, x]*(a + b*x^2)^p, x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && EqQ[ Coeff[Pq, x, 0], 0] && !MatchQ[Pq, x^(m_.)*(u_.) /; IntegerQ[m]]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Int[E^(ArcSech[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(1/u + Sqrt[(1 - u)/(1 + u)] + (1/u)*Sqrt[(1 - u)/(1 + u)])^n, x] /; FreeQ[m, x] && Integer Q[n]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfQuotientOfLinears [u, x]}, Simp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/ls t[[2]])], x] /; !FalseQ[lst]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.67
| method | result | size |
| default | \(\frac {-\frac {1}{5} a^{2} x^{5}+\frac {1}{3} x^{3}}{a^{2}}+\frac {\sqrt {\frac {a x +1}{a x}}\, x \sqrt {-\frac {a x -1}{a x}}\, \left (2 \,\operatorname {csgn}\left (a \right ) a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-\sqrt {-a^{2} x^{2}+1}\, x \,\operatorname {csgn}\left (a \right ) a +\arctan \left (\frac {\operatorname {csgn}\left (a \right ) a x}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) \operatorname {csgn}\left (a \right )}{4 a^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {x^{3}}{3 a^{2}}\) | \(136\) |
1/a^2*(-1/5*a^2*x^5+1/3*x^3)+1/4/a^4*((a*x+1)/a/x)^(1/2)*x*(-(a*x-1)/a/x)^ (1/2)*(2*csgn(a)*a^3*x^3*(-a^2*x^2+1)^(1/2)-(-a^2*x^2+1)^(1/2)*x*csgn(a)*a +arctan(csgn(a)*a*x/(-a^2*x^2+1)^(1/2)))*csgn(a)/(-a^2*x^2+1)^(1/2)+1/3*x^ 3/a^2
Time = 0.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.51 \[ \int e^{2 \text {sech}^{-1}(a x)} x^4 \, dx=-\frac {12 \, a^{5} x^{5} - 40 \, a^{3} x^{3} - 15 \, {\left (2 \, a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 15 \, \arctan \left (\sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}\right )}{60 \, a^{5}} \]
-1/60*(12*a^5*x^5 - 40*a^3*x^3 - 15*(2*a^4*x^4 - a^2*x^2)*sqrt((a*x + 1)/( a*x))*sqrt(-(a*x - 1)/(a*x)) + 15*arctan(sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x))))/a^5
\[ \int e^{2 \text {sech}^{-1}(a x)} x^4 \, dx=\frac {\int 2 x^{2}\, dx + \int \left (- a^{2} x^{4}\right )\, dx + \int 2 a x^{3} \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}\, dx}{a^{2}} \]
(Integral(2*x**2, x) + Integral(-a**2*x**4, x) + Integral(2*a*x**3*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x)), x))/a**2
\[ \int e^{2 \text {sech}^{-1}(a x)} x^4 \, dx=\int { x^{4} {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}^{2} \,d x } \]
Exception generated. \[ \int e^{2 \text {sech}^{-1}(a x)} x^4 \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{-1,[0,4,0,6,0,0]%%%}+%%%{1,[0,2,4,4,0,0]%%%}+%%%{1,[0,2,0, 4,0,0]%%%
Time = 25.51 (sec) , antiderivative size = 808, normalized size of antiderivative = 3.98 \[ \int e^{2 \text {sech}^{-1}(a x)} x^4 \, dx=-\frac {\frac {1{}\mathrm {i}}{512\,a^5}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,3{}\mathrm {i}}{64\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4\,53{}\mathrm {i}}{256\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6\,87{}\mathrm {i}}{128\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8\,657{}\mathrm {i}}{512\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{10}\,121{}\mathrm {i}}{128\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{10}}}{\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}+\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {6\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}+\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{10}}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{12}}}-\frac {\frac {1{}\mathrm {i}}{16\,a^5}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{8\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4\,15{}\mathrm {i}}{16\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}}{\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+\frac {2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}}-\frac {x^5\,\left (\frac {a^2}{5}-\frac {2}{3\,x^2}\right )}{a^2}-\frac {\ln \left (\frac {a\,\sqrt {\frac {1}{a\,x}+1}-\frac {1}{x}+a\,\sqrt {\frac {1}{a\,x}-1}\,1{}\mathrm {i}}{2\,a-2\,a\,\sqrt {\frac {1}{a\,x}+1}+\frac {1}{x}}\right )\,3{}\mathrm {i}}{4\,a^5}-\frac {\ln \left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )\,1{}\mathrm {i}}{4\,a^5}+\frac {\ln \left (\frac {2\,a\,\sqrt {\frac {a+\frac {1}{x}}{a}}-\frac {2}{x}+a\,\sqrt {-\frac {a-\frac {1}{x}}{a}}\,2{}\mathrm {i}}{2\,a+\frac {1}{x}-2\,a\,\sqrt {\frac {a+\frac {1}{x}}{a}}}\right )\,1{}\mathrm {i}}{a^5}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{128\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4\,1{}\mathrm {i}}{512\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4} \]
(log((a*(-(a - 1/x)/a)^(1/2)*2i - 2/x + 2*a*((a + 1/x)/a)^(1/2))/(2*a + 1/ x - 2*a*((a + 1/x)/a)^(1/2)))*1i)/a^5 - (1i/(512*a^5) - (((1/(a*x) - 1)^(1 /2) - 1i)^2*3i)/(64*a^5*((1/(a*x) + 1)^(1/2) - 1)^2) - (((1/(a*x) - 1)^(1/ 2) - 1i)^4*53i)/(256*a^5*((1/(a*x) + 1)^(1/2) - 1)^4) + (((1/(a*x) - 1)^(1 /2) - 1i)^6*87i)/(128*a^5*((1/(a*x) + 1)^(1/2) - 1)^6) + (((1/(a*x) - 1)^( 1/2) - 1i)^8*657i)/(512*a^5*((1/(a*x) + 1)^(1/2) - 1)^8) + (((1/(a*x) - 1) ^(1/2) - 1i)^10*121i)/(128*a^5*((1/(a*x) + 1)^(1/2) - 1)^10))/(((1/(a*x) - 1)^(1/2) - 1i)^4/((1/(a*x) + 1)^(1/2) - 1)^4 + (4*((1/(a*x) - 1)^(1/2) - 1i)^6)/((1/(a*x) + 1)^(1/2) - 1)^6 + (6*((1/(a*x) - 1)^(1/2) - 1i)^8)/((1/ (a*x) + 1)^(1/2) - 1)^8 + (4*((1/(a*x) - 1)^(1/2) - 1i)^10)/((1/(a*x) + 1) ^(1/2) - 1)^10 + ((1/(a*x) - 1)^(1/2) - 1i)^12/((1/(a*x) + 1)^(1/2) - 1)^1 2) - (log(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1))*1i)/(4*a^5 ) - (1i/(16*a^5) + (((1/(a*x) - 1)^(1/2) - 1i)^2*1i)/(8*a^5*((1/(a*x) + 1) ^(1/2) - 1)^2) - (((1/(a*x) - 1)^(1/2) - 1i)^4*15i)/(16*a^5*((1/(a*x) + 1) ^(1/2) - 1)^4))/(((1/(a*x) - 1)^(1/2) - 1i)^2/((1/(a*x) + 1)^(1/2) - 1)^2 + (2*((1/(a*x) - 1)^(1/2) - 1i)^4)/((1/(a*x) + 1)^(1/2) - 1)^4 + ((1/(a*x) - 1)^(1/2) - 1i)^6/((1/(a*x) + 1)^(1/2) - 1)^6) - (log((a*(1/(a*x) - 1)^( 1/2)*1i + a*(1/(a*x) + 1)^(1/2) - 1/x)/(2*a - 2*a*(1/(a*x) + 1)^(1/2) + 1/ x))*3i)/(4*a^5) - (((1/(a*x) - 1)^(1/2) - 1i)^2*1i)/(128*a^5*((1/(a*x) + 1 )^(1/2) - 1)^2) - (((1/(a*x) - 1)^(1/2) - 1i)^4*1i)/(512*a^5*((1/(a*x) ...