Integrand size = 24, antiderivative size = 231 \[ \int \frac {x \text {sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {x \log \left (1+\frac {(a+b) e^{2 c+2 d x}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d}-\frac {x \log \left (1+\frac {(a+b) e^{2 c+2 d x}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d}+\frac {\operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 c+2 d x}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )}{4 \sqrt {-a} \sqrt {b} d^2}-\frac {\operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 c+2 d x}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{4 \sqrt {-a} \sqrt {b} d^2} \]
1/2*x*ln(1+(a+b)*exp(2*d*x+2*c)/(a-b-2*(-a)^(1/2)*b^(1/2)))/d/(-a)^(1/2)/b ^(1/2)-1/2*x*ln(1+(a+b)*exp(2*d*x+2*c)/(a-b+2*(-a)^(1/2)*b^(1/2)))/d/(-a)^ (1/2)/b^(1/2)+1/4*polylog(2,-(a+b)*exp(2*d*x+2*c)/(a-b-2*(-a)^(1/2)*b^(1/2 )))/d^2/(-a)^(1/2)/b^(1/2)-1/4*polylog(2,-(a+b)*exp(2*d*x+2*c)/(a-b+2*(-a) ^(1/2)*b^(1/2)))/d^2/(-a)^(1/2)/b^(1/2)
Time = 1.43 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.08 \[ \int \frac {x \text {sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {-4 \sqrt {-a} c \arctan \left (\frac {a-b+(a+b) e^{2 (c+d x)}}{2 \sqrt {a} \sqrt {b}}\right )+2 \sqrt {a} (c+d x) \log \left (1+\frac {(a+b) e^{2 (c+d x)}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )-2 \sqrt {a} (c+d x) \log \left (1+\frac {(a+b) e^{2 (c+d x)}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )+\sqrt {a} \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 (c+d x)}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )-\sqrt {a} \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 (c+d x)}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{4 \sqrt {-a^2} \sqrt {b} d^2} \]
(-4*Sqrt[-a]*c*ArcTan[(a - b + (a + b)*E^(2*(c + d*x)))/(2*Sqrt[a]*Sqrt[b] )] + 2*Sqrt[a]*(c + d*x)*Log[1 + ((a + b)*E^(2*(c + d*x)))/(a - 2*Sqrt[-a] *Sqrt[b] - b)] - 2*Sqrt[a]*(c + d*x)*Log[1 + ((a + b)*E^(2*(c + d*x)))/(a + 2*Sqrt[-a]*Sqrt[b] - b)] + Sqrt[a]*PolyLog[2, -(((a + b)*E^(2*(c + d*x)) )/(a - 2*Sqrt[-a]*Sqrt[b] - b))] - Sqrt[a]*PolyLog[2, -(((a + b)*E^(2*(c + d*x)))/(a + 2*Sqrt[-a]*Sqrt[b] - b))])/(4*Sqrt[-a^2]*Sqrt[b]*d^2)
Time = 1.01 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6166, 3042, 3801, 2694, 27, 2620, 2715, 2838}
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.
| \(\displaystyle \int \frac {x \text {sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 6166 |
| \(\displaystyle 2 \int \frac {x}{a-b+(a+b) \cosh (2 c+2 d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 \int \frac {x}{a-b+(a+b) \sin \left (2 i c+2 i d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3801 |
| \(\displaystyle 4 \int \frac {e^{2 c+2 d x} x}{a+(a+b) e^{4 (c+d x)}+2 (a-b) e^{2 c+2 d x}+b}dx\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle 4 \left (\frac {(a+b) \int \frac {e^{2 c+2 d x} x}{2 \left (a+(a+b) e^{2 c+2 d x}-b-2 \sqrt {-a} \sqrt {b}\right )}dx}{2 \sqrt {-a} \sqrt {b}}-\frac {(a+b) \int \frac {e^{2 c+2 d x} x}{2 \left (a+(a+b) e^{2 c+2 d x}-b+2 \sqrt {-a} \sqrt {b}\right )}dx}{2 \sqrt {-a} \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \left (\frac {(a+b) \int \frac {e^{2 c+2 d x} x}{a+(a+b) e^{2 c+2 d x}-b-2 \sqrt {-a} \sqrt {b}}dx}{4 \sqrt {-a} \sqrt {b}}-\frac {(a+b) \int \frac {e^{2 c+2 d x} x}{a+(a+b) e^{2 c+2 d x}-b+2 \sqrt {-a} \sqrt {b}}dx}{4 \sqrt {-a} \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle 4 \left (\frac {(a+b) \left (\frac {x \log \left (\frac {(a+b) e^{2 c+2 d x}}{-2 \sqrt {-a} \sqrt {b}+a-b}+1\right )}{2 d (a+b)}-\frac {\int \log \left (\frac {e^{2 c+2 d x} (a+b)}{a-b-2 \sqrt {-a} \sqrt {b}}+1\right )dx}{2 d (a+b)}\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {(a+b) \left (\frac {x \log \left (\frac {(a+b) e^{2 c+2 d x}}{2 \sqrt {-a} \sqrt {b}+a-b}+1\right )}{2 d (a+b)}-\frac {\int \log \left (\frac {e^{2 c+2 d x} (a+b)}{a-b+2 \sqrt {-a} \sqrt {b}}+1\right )dx}{2 d (a+b)}\right )}{4 \sqrt {-a} \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle 4 \left (\frac {(a+b) \left (\frac {x \log \left (\frac {(a+b) e^{2 c+2 d x}}{-2 \sqrt {-a} \sqrt {b}+a-b}+1\right )}{2 d (a+b)}-\frac {\int e^{-2 c-2 d x} \log \left (\frac {e^{2 c+2 d x} (a+b)}{a-b-2 \sqrt {-a} \sqrt {b}}+1\right )de^{2 c+2 d x}}{4 d^2 (a+b)}\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {(a+b) \left (\frac {x \log \left (\frac {(a+b) e^{2 c+2 d x}}{2 \sqrt {-a} \sqrt {b}+a-b}+1\right )}{2 d (a+b)}-\frac {\int e^{-2 c-2 d x} \log \left (\frac {e^{2 c+2 d x} (a+b)}{a-b+2 \sqrt {-a} \sqrt {b}}+1\right )de^{2 c+2 d x}}{4 d^2 (a+b)}\right )}{4 \sqrt {-a} \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle 4 \left (\frac {(a+b) \left (\frac {\operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 c+2 d x}}{a-b-2 \sqrt {-a} \sqrt {b}}\right )}{4 d^2 (a+b)}+\frac {x \log \left (\frac {(a+b) e^{2 c+2 d x}}{-2 \sqrt {-a} \sqrt {b}+a-b}+1\right )}{2 d (a+b)}\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {(a+b) \left (\frac {\operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 c+2 d x}}{a-b+2 \sqrt {-a} \sqrt {b}}\right )}{4 d^2 (a+b)}+\frac {x \log \left (\frac {(a+b) e^{2 c+2 d x}}{2 \sqrt {-a} \sqrt {b}+a-b}+1\right )}{2 d (a+b)}\right )}{4 \sqrt {-a} \sqrt {b}}\right )\) |
4*(((a + b)*((x*Log[1 + ((a + b)*E^(2*c + 2*d*x))/(a - 2*Sqrt[-a]*Sqrt[b] - b)])/(2*(a + b)*d) + PolyLog[2, -(((a + b)*E^(2*c + 2*d*x))/(a - 2*Sqrt[ -a]*Sqrt[b] - b))]/(4*(a + b)*d^2)))/(4*Sqrt[-a]*Sqrt[b]) - ((a + b)*((x*L og[1 + ((a + b)*E^(2*c + 2*d*x))/(a + 2*Sqrt[-a]*Sqrt[b] - b)])/(2*(a + b) *d) + PolyLog[2, -(((a + b)*E^(2*c + 2*d*x))/(a + 2*Sqrt[-a]*Sqrt[b] - b)) ]/(4*(a + b)*d^2)))/(4*Sqrt[-a]*Sqrt[b]))
3.2.44.3.1 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[(((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]
Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) *(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q) Int[(f + g*x) ^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Comple x[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Simp[2 Int[((c + d*x)^m*(E^((-I)*e + f*fz*x)/(b + (2*a*E^((-I)*e + f*fz*x))/E^(I*Pi*(k - 1/2)) - (b*E^(2*((-I) *e + f*fz*x)))/E^(2*I*k*Pi))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c , d, e, f, fz}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[(((f_.) + (g_.)*(x_))^(m_.)*Sech[(d_.) + (e_.)*(x_)]^2)/((b_) + (c_.)*T anh[(d_.) + (e_.)*(x_)]^2), x_Symbol] :> Simp[2 Int[(f + g*x)^m/(b - c + (b + c)*Cosh[2*d + 2*e*x]), x], x] /; FreeQ[{b, c, d, e, f, g}, x] && IGtQ[ m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(952\) vs. \(2(187)=374\).
Time = 3.20 (sec) , antiderivative size = 953, normalized size of antiderivative = 4.13
| method | result | size |
| risch | \(-\frac {c^{2}}{d^{2} \left (-2 \sqrt {-a b}-a +b \right )}+\frac {\operatorname {polylog}\left (2, \frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right )}{2 d^{2} \left (-2 \sqrt {-a b}-a +b \right )}-\frac {c^{2}}{2 d^{2} \sqrt {-a b}}+\frac {\operatorname {polylog}\left (2, \frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{2 \sqrt {-a b}-a +b}\right )}{4 d^{2} \sqrt {-a b}}-\frac {a \,x^{2}}{2 \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}+\frac {b \,x^{2}}{2 \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}+\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) c}{d^{2} \left (-2 \sqrt {-a b}-a +b \right )}+\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{2 \sqrt {-a b}-a +b}\right ) x}{2 d \sqrt {-a b}}-\frac {c x}{d \sqrt {-a b}}-\frac {c \arctan \left (\frac {2 \left (a +b \right ) {\mathrm e}^{2 d x +2 c}+2 a -2 b}{4 \sqrt {a b}}\right )}{d^{2} \sqrt {a b}}+\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) x}{d \left (-2 \sqrt {-a b}-a +b \right )}-\frac {2 c x}{d \left (-2 \sqrt {-a b}-a +b \right )}+\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{2 \sqrt {-a b}-a +b}\right ) c}{2 d^{2} \sqrt {-a b}}-\frac {x^{2}}{-2 \sqrt {-a b}-a +b}-\frac {x^{2}}{2 \sqrt {-a b}}-\frac {a \,c^{2}}{2 d^{2} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}+\frac {c^{2} b}{2 d^{2} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}+\frac {\operatorname {polylog}\left (2, \frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) a}{4 d^{2} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}-\frac {\operatorname {polylog}\left (2, \frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) b}{4 d^{2} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}+\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) a c}{2 d^{2} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}-\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) b c}{2 d^{2} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}+\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) a x}{2 d \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}-\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) b x}{2 d \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}-\frac {a c x}{d \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}+\frac {b c x}{d \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}\) | \(953\) |
-1/d^2/(-2*(-a*b)^(1/2)-a+b)*c^2+1/2/d^2/(-2*(-a*b)^(1/2)-a+b)*polylog(2,( a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))-1/2/d^2/(-a*b)^(1/2)*c^2+1/4/d^ 2/(-a*b)^(1/2)*polylog(2,(a+b)*exp(2*d*x+2*c)/(2*(-a*b)^(1/2)-a+b))-1/2/(- a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*a*x^2+1/2/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a +b)*b*x^2+1/d^2/(-2*(-a*b)^(1/2)-a+b)*ln(1-(a+b)*exp(2*d*x+2*c)/(-2*(-a*b) ^(1/2)-a+b))*c+1/2/d/(-a*b)^(1/2)*ln(1-(a+b)*exp(2*d*x+2*c)/(2*(-a*b)^(1/2 )-a+b))*x-1/d/(-a*b)^(1/2)*c*x-1/d^2*c/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*exp (2*d*x+2*c)+2*a-2*b)/(a*b)^(1/2))+1/d/(-2*(-a*b)^(1/2)-a+b)*ln(1-(a+b)*exp (2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))*x-2/d/(-2*(-a*b)^(1/2)-a+b)*c*x+1/2/d^2 /(-a*b)^(1/2)*ln(1-(a+b)*exp(2*d*x+2*c)/(2*(-a*b)^(1/2)-a+b))*c-1/(-2*(-a* b)^(1/2)-a+b)*x^2-1/2/(-a*b)^(1/2)*x^2-1/2/d^2/(-a*b)^(1/2)/(-2*(-a*b)^(1/ 2)-a+b)*a*c^2+1/2/d^2/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*c^2*b+1/4/d^2/(-a *b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*polylog(2,(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^ (1/2)-a+b))*a-1/4/d^2/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*polylog(2,(a+b)*e xp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))*b+1/2/d^2/(-a*b)^(1/2)/(-2*(-a*b)^(1/ 2)-a+b)*ln(1-(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))*a*c-1/2/d^2/(-a*b )^(1/2)/(-2*(-a*b)^(1/2)-a+b)*ln(1-(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a +b))*b*c+1/2/d/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*ln(1-(a+b)*exp(2*d*x+2*c )/(-2*(-a*b)^(1/2)-a+b))*a*x-1/2/d/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*ln(1 -(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))*b*x-1/d/(-a*b)^(1/2)/(-2*(...
Leaf count of result is larger than twice the leaf count of optimal. 1516 vs. \(2 (185) = 370\).
Time = 0.35 (sec) , antiderivative size = 1516, normalized size of antiderivative = 6.56 \[ \int \frac {x \text {sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\text {Too large to display} \]
-1/2*((a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*c*log(2*sqrt(-(2*(a + b)*sqrt (-a*b/(a^2 + 2*a*b + b^2)) + a - b)/(a + b)) + 2*cosh(d*x + c) + 2*sinh(d* x + c)) + (a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*c*log(-2*sqrt(-(2*(a + b) *sqrt(-a*b/(a^2 + 2*a*b + b^2)) + a - b)/(a + b)) + 2*cosh(d*x + c) + 2*si nh(d*x + c)) - (a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*c*log(2*sqrt((2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) - a + b)/(a + b)) + 2*cosh(d*x + c) + 2 *sinh(d*x + c)) - (a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*c*log(-2*sqrt((2* (a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) - a + b)/(a + b)) + 2*cosh(d*x + c) + 2*sinh(d*x + c)) - (a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*dilog(-(((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c) - 2*((a + b)*cosh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-a*b/(a^2 + 2*a*b + b^2)))*sqrt(-(2*(a + b)*sqrt( -a*b/(a^2 + 2*a*b + b^2)) + a - b)/(a + b)) + a + b)/(a + b) + 1) - (a + b )*sqrt(-a*b/(a^2 + 2*a*b + b^2))*dilog((((a - b)*cosh(d*x + c) + (a - b)*s inh(d*x + c) - 2*((a + b)*cosh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-a*b /(a^2 + 2*a*b + b^2)))*sqrt(-(2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) + a - b)/(a + b)) - a - b)/(a + b) + 1) + (a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^ 2))*dilog(-(((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c) + 2*((a + b)*co sh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-a*b/(a^2 + 2*a*b + b^2)))*sqrt( (2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) - a + b)/(a + b)) + a + b)/(a + b) + 1) + (a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*dilog((((a - b)*cosh(d...
\[ \int \frac {x \text {sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int \frac {x \operatorname {sech}^{2}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {x \text {sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int { \frac {x \operatorname {sech}\left (d x + c\right )^{2}}{b \tanh \left (d x + c\right )^{2} + a} \,d x } \]
\[ \int \frac {x \text {sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int { \frac {x \operatorname {sech}\left (d x + c\right )^{2}}{b \tanh \left (d x + c\right )^{2} + a} \,d x } \]
Timed out. \[ \int \frac {x \text {sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int \frac {x}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )} \,d x \]