Integrand size = 19, antiderivative size = 147 \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{2 d e}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{2 d \sqrt {e} \sqrt {c^2 d+e}} \]
1/2*(-a-b*arcsech(c*x))/e/(e*x^2+d)+1/2*b*arctanh((-c^2*x^2+1)^(1/2))*(1/( c*x+1))^(1/2)*(c*x+1)^(1/2)/d/e-1/2*b*arctanh(e^(1/2)*(-c^2*x^2+1)^(1/2)/( c^2*d+e)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/d/e^(1/2)/(c^2*d+e)^(1/2)
Result contains complex when optimal does not.
Time = 1.10 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.35 \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {\frac {2 a}{d+e x^2}+\frac {2 b \text {sech}^{-1}(c x)}{d+e x^2}+\frac {2 b \log (x)}{d}-\frac {2 b \log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )}{d}+\frac {b \sqrt {e} \log \left (\frac {4 \left (\frac {i d e+c^2 d^{3/2} \sqrt {e} x}{\sqrt {c^2 d+e} \left (\sqrt {d}+i \sqrt {e} x\right )}+\frac {d e \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{-i \sqrt {d} \sqrt {e}+e x}\right )}{b}\right )}{d \sqrt {c^2 d+e}}+\frac {b \sqrt {e} \log \left (\frac {4 \left (\frac {d e+i c^2 d^{3/2} \sqrt {e} x}{\sqrt {c^2 d+e} \left (i \sqrt {d}+\sqrt {e} x\right )}+\frac {d e \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{i \sqrt {d} \sqrt {e}+e x}\right )}{b}\right )}{d \sqrt {c^2 d+e}}}{4 e} \]
-1/4*((2*a)/(d + e*x^2) + (2*b*ArcSech[c*x])/(d + e*x^2) + (2*b*Log[x])/d - (2*b*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]) /d + (b*Sqrt[e]*Log[(4*((I*d*e + c^2*d^(3/2)*Sqrt[e]*x)/(Sqrt[c^2*d + e]*( Sqrt[d] + I*Sqrt[e]*x)) + (d*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/((-I)* Sqrt[d]*Sqrt[e] + e*x)))/b])/(d*Sqrt[c^2*d + e]) + (b*Sqrt[e]*Log[(4*((d*e + I*c^2*d^(3/2)*Sqrt[e]*x)/(Sqrt[c^2*d + e]*(I*Sqrt[d] + Sqrt[e]*x)) + (d *e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(I*Sqrt[d]*Sqrt[e] + e*x)))/b])/(d *Sqrt[c^2*d + e]))/e
Time = 0.46 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.86, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6853, 2036, 354, 97, 73, 221}
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.
\(\displaystyle \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 6853 |
\(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \left (e x^2+d\right )}dx}{2 e}-\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 2036 |
\(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {1}{x \sqrt {1-c^2 x^2} \left (e x^2+d\right )}dx}{2 e}-\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 354 |
\(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \left (e x^2+d\right )}dx^2}{4 e}-\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 97 |
\(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\int \frac {1}{x^2 \sqrt {1-c^2 x^2}}dx^2}{d}-\frac {e \int \frac {1}{\sqrt {1-c^2 x^2} \left (e x^2+d\right )}dx^2}{d}\right )}{4 e}-\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {2 e \int \frac {1}{-\frac {e x^4}{c^2}+d+\frac {e}{c^2}}d\sqrt {1-c^2 x^2}}{c^2 d}-\frac {2 \int \frac {1}{\frac {1}{c^2}-\frac {x^4}{c^2}}d\sqrt {1-c^2 x^2}}{c^2 d}\right )}{4 e}-\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{d \sqrt {c^2 d+e}}-\frac {2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d}\right )}{4 e}\) |
-1/2*(a + b*ArcSech[c*x])/(e*(d + e*x^2)) - (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*((-2*ArcTanh[Sqrt[1 - c^2*x^2]])/d + (2*Sqrt[e]*ArcTanh[(Sqrt[e]*S qrt[1 - c^2*x^2])/Sqrt[c^2*d + e]])/(d*Sqrt[c^2*d + e])))/(4*e)
3.2.17.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Simp[b/(b*c - a*d) Int[(e + f*x)^p/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && !IntegerQ[p]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p _.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Int[u*(a1*a2 + b1*b2 *x^n)^p*(c + d*x^n)^q, x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && E qQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && Gt Q[a2, 0]))
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSech[c*x])/(2*e*(p + 1))), x] + Simp[b*(Sqrt[1 + c*x]/(2*e*(p + 1)))*Sqrt[1/(1 + c*x)] Int[(d + e*x ^2)^(p + 1)/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, e , p}, x] && NeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(461\) vs. \(2(124)=248\).
Time = 5.52 (sec) , antiderivative size = 462, normalized size of antiderivative = 3.14
method | result | size |
parts | \(-\frac {a}{2 e \left (e \,x^{2}+d \right )}+\frac {b \left (-\frac {c^{4} \operatorname {arcsech}\left (c x \right )}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {c^{3} \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (-2 \sqrt {\frac {c^{2} d +e}{e}}\, \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} d +\ln \left (-\frac {2 \left (\sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e -\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d +\ln \left (\frac {2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 \sqrt {-c^{2} d e}\, c x +2 e}{c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d -2 \sqrt {\frac {c^{2} d +e}{e}}\, \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) e +\ln \left (-\frac {2 \left (\sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e -\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) e +\ln \left (\frac {2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 \sqrt {-c^{2} d e}\, c x +2 e}{c e x +\sqrt {-c^{2} d e}}\right ) e \right )}{4 \sqrt {-c^{2} x^{2}+1}\, d \left (e +\sqrt {-c^{2} d e}\right ) \left (-e +\sqrt {-c^{2} d e}\right ) \sqrt {\frac {c^{2} d +e}{e}}}\right )}{c^{2}}\) | \(462\) |
derivativedivides | \(\frac {-\frac {a \,c^{4}}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (2 \sqrt {\frac {c^{2} d +e}{e}}\, \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} d -\ln \left (\frac {2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 \sqrt {-c^{2} d e}\, c x +2 e}{c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d -\ln \left (-\frac {2 \left (\sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e -\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d +2 \sqrt {\frac {c^{2} d +e}{e}}\, \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) e -\ln \left (\frac {2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 \sqrt {-c^{2} d e}\, c x +2 e}{c e x +\sqrt {-c^{2} d e}}\right ) e -\ln \left (-\frac {2 \left (\sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e -\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) e \right )}{4 c \sqrt {-c^{2} x^{2}+1}\, d \left (e +\sqrt {-c^{2} d e}\right ) \left (-e +\sqrt {-c^{2} d e}\right ) \sqrt {\frac {c^{2} d +e}{e}}}\right )}{c^{2}}\) | \(477\) |
default | \(\frac {-\frac {a \,c^{4}}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (2 \sqrt {\frac {c^{2} d +e}{e}}\, \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} d -\ln \left (\frac {2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 \sqrt {-c^{2} d e}\, c x +2 e}{c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d -\ln \left (-\frac {2 \left (\sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e -\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d +2 \sqrt {\frac {c^{2} d +e}{e}}\, \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) e -\ln \left (\frac {2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 \sqrt {-c^{2} d e}\, c x +2 e}{c e x +\sqrt {-c^{2} d e}}\right ) e -\ln \left (-\frac {2 \left (\sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e -\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) e \right )}{4 c \sqrt {-c^{2} x^{2}+1}\, d \left (e +\sqrt {-c^{2} d e}\right ) \left (-e +\sqrt {-c^{2} d e}\right ) \sqrt {\frac {c^{2} d +e}{e}}}\right )}{c^{2}}\) | \(477\) |
-1/2*a/e/(e*x^2+d)+b/c^2*(-1/2*c^4/e/(c^2*e*x^2+c^2*d)*arcsech(c*x)+1/4*c^ 3*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*(-2*((c^2*d+e)/e)^(1/2)*arcta nh(1/(-c^2*x^2+1)^(1/2))*c^2*d+ln(-2*(((c^2*d+e)/e)^(1/2)*(-c^2*x^2+1)^(1/ 2)*e-(-c^2*d*e)^(1/2)*c*x+e)/(-c*e*x+(-c^2*d*e)^(1/2)))*c^2*d+ln(2*(((c^2* d+e)/e)^(1/2)*(-c^2*x^2+1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(c*e*x+(-c^2*d* e)^(1/2)))*c^2*d-2*((c^2*d+e)/e)^(1/2)*arctanh(1/(-c^2*x^2+1)^(1/2))*e+ln( -2*(((c^2*d+e)/e)^(1/2)*(-c^2*x^2+1)^(1/2)*e-(-c^2*d*e)^(1/2)*c*x+e)/(-c*e *x+(-c^2*d*e)^(1/2)))*e+ln(2*(((c^2*d+e)/e)^(1/2)*(-c^2*x^2+1)^(1/2)*e+(-c ^2*d*e)^(1/2)*c*x+e)/(c*e*x+(-c^2*d*e)^(1/2)))*e)/(-c^2*x^2+1)^(1/2)/d/(e+ (-c^2*d*e)^(1/2))/(-e+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (89) = 178\).
Time = 0.31 (sec) , antiderivative size = 602, normalized size of antiderivative = 4.10 \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\left [-\frac {2 \, a c^{2} d^{2} + 2 \, a d e - \sqrt {c^{2} d e + e^{2}} {\left (b e x^{2} + b d\right )} \log \left (\frac {c^{4} d^{2} + 4 \, c^{2} d e - {\left (c^{4} d e + 2 \, c^{2} e^{2}\right )} x^{2} + 4 \, {\left (c^{3} d e + c e^{2}\right )} x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 4 \, e^{2} + 2 \, {\left (c^{2} e x^{2} - c^{2} d - {\left (c^{3} d + 2 \, c e\right )} x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 2 \, e\right )} \sqrt {c^{2} d e + e^{2}}}{e x^{2} + d}\right ) + 2 \, {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 2 \, {\left (b c^{2} d^{2} + b d e\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{4 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}, -\frac {a c^{2} d^{2} + a d e + \sqrt {-c^{2} d e - e^{2}} {\left (b e x^{2} + b d\right )} \arctan \left (\frac {\sqrt {-c^{2} d e - e^{2}} c d x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - \sqrt {-c^{2} d e - e^{2}} {\left (e x^{2} + d\right )}}{{\left (c^{2} d e + e^{2}\right )} x^{2}}\right ) + {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + {\left (b c^{2} d^{2} + b d e\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{2 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}\right ] \]
[-1/4*(2*a*c^2*d^2 + 2*a*d*e - sqrt(c^2*d*e + e^2)*(b*e*x^2 + b*d)*log((c^ 4*d^2 + 4*c^2*d*e - (c^4*d*e + 2*c^2*e^2)*x^2 + 4*(c^3*d*e + c*e^2)*x*sqrt (-(c^2*x^2 - 1)/(c^2*x^2)) + 4*e^2 + 2*(c^2*e*x^2 - c^2*d - (c^3*d + 2*c*e )*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 2*e)*sqrt(c^2*d*e + e^2))/(e*x^2 + d) ) + 2*(b*c^2*d^2 + b*d*e + (b*c^2*d*e + b*e^2)*x^2)*log((c*x*sqrt(-(c^2*x^ 2 - 1)/(c^2*x^2)) - 1)/x) + 2*(b*c^2*d^2 + b*d*e)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)))/(c^2*d^3*e + d^2*e^2 + (c^2*d^2*e^2 + d*e^3)* x^2), -1/2*(a*c^2*d^2 + a*d*e + sqrt(-c^2*d*e - e^2)*(b*e*x^2 + b*d)*arcta n((sqrt(-c^2*d*e - e^2)*c*d*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - sqrt(-c^2*d *e - e^2)*(e*x^2 + d))/((c^2*d*e + e^2)*x^2)) + (b*c^2*d^2 + b*d*e + (b*c^ 2*d*e + b*e^2)*x^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x) + (b*c ^2*d^2 + b*d*e)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)))/(c^2* d^3*e + d^2*e^2 + (c^2*d^2*e^2 + d*e^3)*x^2)]
\[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]
\[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
1/2*(2*c^2*integrate(1/2*x^3/(c^2*d^2*x^2 + (c^2*d*e*x^2 - d*e)*x^2 + (c^2 *d^2*x^2 + (c^2*d*e*x^2 - d*e)*x^2 - d^2)*sqrt(c*x + 1)*sqrt(-c*x + 1) - d ^2), x) + (x^2*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1) - x^2*log(c) - x^2*lo g(x))/(d*e*x^2 + d^2) - 2*integrate(1/2*x/(c^2*d^2*x^2 + (c^2*d*e*x^2 - d* e)*x^2 - d^2), x))*b - 1/2*a/(e^2*x^2 + d*e)
\[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]