Integrand size = 26, antiderivative size = 351 \[ \int \frac {x^2 \text {sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {x^2 \log \left (1+\frac {(a+b) e^{2 c+2 d x}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d}-\frac {x^2 \log \left (1+\frac {(a+b) e^{2 c+2 d x}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d}+\frac {x \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 c+2 d x}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 c+2 d x}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d^2}-\frac {\operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 c+2 d x}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )}{4 \sqrt {-a} \sqrt {b} d^3}+\frac {\operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 c+2 d x}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{4 \sqrt {-a} \sqrt {b} d^3} \]
1/2*x^2*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^2*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*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)-1/2*x*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(3,-(a+b)*exp(2 *d*x+2*c)/(a-b-2*(-a)^(1/2)*b^(1/2)))/d^3/(-a)^(1/2)/b^(1/2)+1/4*polylog(3 ,-(a+b)*exp(2*d*x+2*c)/(a-b+2*(-a)^(1/2)*b^(1/2)))/d^3/(-a)^(1/2)/b^(1/2)
Time = 0.74 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.76 \[ \int \frac {x^2 \text {sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {2 d^2 x^2 \log \left (1+\frac {(a+b) e^{2 (c+d x)}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )-2 d^2 x^2 \log \left (1+\frac {(a+b) e^{2 (c+d x)}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )+2 d x \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 (c+d x)}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )-2 d x \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 (c+d x)}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )-\operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 (c+d x)}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )+\operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 (c+d x)}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{4 \sqrt {-a} \sqrt {b} d^3} \]
(2*d^2*x^2*Log[1 + ((a + b)*E^(2*(c + d*x)))/(a - 2*Sqrt[-a]*Sqrt[b] - b)] - 2*d^2*x^2*Log[1 + ((a + b)*E^(2*(c + d*x)))/(a + 2*Sqrt[-a]*Sqrt[b] - b )] + 2*d*x*PolyLog[2, -(((a + b)*E^(2*(c + d*x)))/(a - 2*Sqrt[-a]*Sqrt[b] - b))] - 2*d*x*PolyLog[2, -(((a + b)*E^(2*(c + d*x)))/(a + 2*Sqrt[-a]*Sqrt [b] - b))] - PolyLog[3, -(((a + b)*E^(2*(c + d*x)))/(a - 2*Sqrt[-a]*Sqrt[b ] - b))] + PolyLog[3, -(((a + b)*E^(2*(c + d*x)))/(a + 2*Sqrt[-a]*Sqrt[b] - b))])/(4*Sqrt[-a]*Sqrt[b]*d^3)
Time = 1.50 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {6166, 3042, 3801, 2694, 27, 2620, 3011, 2720, 7143}
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.
\(\displaystyle \int \frac {x^2 \text {sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx\) |
\(\Big \downarrow \) 6166 |
\(\displaystyle 2 \int \frac {x^2}{a-b+(a+b) \cosh (2 c+2 d x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 \int \frac {x^2}{a-b+(a+b) \sin \left (2 i c+2 i d x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3801 |
\(\displaystyle 4 \int \frac {e^{2 c+2 d x} x^2}{a+(a+b) e^{4 (c+d x)}+2 (a-b) e^{2 c+2 d x}+b}dx\) |
\(\Big \downarrow \) 2694 |
\(\displaystyle 4 \left (\frac {(a+b) \int \frac {e^{2 c+2 d x} x^2}{2 \left (a+(a+b) e^{2 c+2 d x}-b-2 \sqrt {-a} \sqrt {b}\right )}dx}{2 \sqrt {-a} \sqrt {b}}-\frac {(a+b) \int \frac {e^{2 c+2 d x} x^2}{2 \left (a+(a+b) e^{2 c+2 d x}-b+2 \sqrt {-a} \sqrt {b}\right )}dx}{2 \sqrt {-a} \sqrt {b}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 4 \left (\frac {(a+b) \int \frac {e^{2 c+2 d x} x^2}{a+(a+b) e^{2 c+2 d x}-b-2 \sqrt {-a} \sqrt {b}}dx}{4 \sqrt {-a} \sqrt {b}}-\frac {(a+b) \int \frac {e^{2 c+2 d x} x^2}{a+(a+b) e^{2 c+2 d x}-b+2 \sqrt {-a} \sqrt {b}}dx}{4 \sqrt {-a} \sqrt {b}}\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle 4 \left (\frac {(a+b) \left (\frac {x^2 \log \left (\frac {(a+b) e^{2 c+2 d x}}{-2 \sqrt {-a} \sqrt {b}+a-b}+1\right )}{2 d (a+b)}-\frac {\int x \log \left (\frac {e^{2 c+2 d x} (a+b)}{a-b-2 \sqrt {-a} \sqrt {b}}+1\right )dx}{d (a+b)}\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {(a+b) \left (\frac {x^2 \log \left (\frac {(a+b) e^{2 c+2 d x}}{2 \sqrt {-a} \sqrt {b}+a-b}+1\right )}{2 d (a+b)}-\frac {\int x \log \left (\frac {e^{2 c+2 d x} (a+b)}{a-b+2 \sqrt {-a} \sqrt {b}}+1\right )dx}{d (a+b)}\right )}{4 \sqrt {-a} \sqrt {b}}\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle 4 \left (\frac {(a+b) \left (\frac {x^2 \log \left (\frac {(a+b) e^{2 c+2 d x}}{-2 \sqrt {-a} \sqrt {b}+a-b}+1\right )}{2 d (a+b)}-\frac {\frac {\int \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 c+2 d x}}{a-b-2 \sqrt {-a} \sqrt {b}}\right )dx}{2 d}-\frac {x \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 c+2 d x}}{a-b-2 \sqrt {-a} \sqrt {b}}\right )}{2 d}}{d (a+b)}\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {(a+b) \left (\frac {x^2 \log \left (\frac {(a+b) e^{2 c+2 d x}}{2 \sqrt {-a} \sqrt {b}+a-b}+1\right )}{2 d (a+b)}-\frac {\frac {\int \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 c+2 d x}}{a-b+2 \sqrt {-a} \sqrt {b}}\right )dx}{2 d}-\frac {x \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 c+2 d x}}{a-b+2 \sqrt {-a} \sqrt {b}}\right )}{2 d}}{d (a+b)}\right )}{4 \sqrt {-a} \sqrt {b}}\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle 4 \left (\frac {(a+b) \left (\frac {x^2 \log \left (\frac {(a+b) e^{2 c+2 d x}}{-2 \sqrt {-a} \sqrt {b}+a-b}+1\right )}{2 d (a+b)}-\frac {\frac {\int e^{-2 c-2 d x} \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 c+2 d x}}{a-b-2 \sqrt {-a} \sqrt {b}}\right )de^{2 c+2 d x}}{4 d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 c+2 d x}}{a-b-2 \sqrt {-a} \sqrt {b}}\right )}{2 d}}{d (a+b)}\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {(a+b) \left (\frac {x^2 \log \left (\frac {(a+b) e^{2 c+2 d x}}{2 \sqrt {-a} \sqrt {b}+a-b}+1\right )}{2 d (a+b)}-\frac {\frac {\int e^{-2 c-2 d x} \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 c+2 d x}}{a-b+2 \sqrt {-a} \sqrt {b}}\right )de^{2 c+2 d x}}{4 d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 c+2 d x}}{a-b+2 \sqrt {-a} \sqrt {b}}\right )}{2 d}}{d (a+b)}\right )}{4 \sqrt {-a} \sqrt {b}}\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle 4 \left (\frac {(a+b) \left (\frac {x^2 \log \left (\frac {(a+b) e^{2 c+2 d x}}{-2 \sqrt {-a} \sqrt {b}+a-b}+1\right )}{2 d (a+b)}-\frac {\frac {\operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 c+2 d x}}{a-b-2 \sqrt {-a} \sqrt {b}}\right )}{4 d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 c+2 d x}}{a-b-2 \sqrt {-a} \sqrt {b}}\right )}{2 d}}{d (a+b)}\right )}{4 \sqrt {-a} \sqrt {b}}-\frac {(a+b) \left (\frac {x^2 \log \left (\frac {(a+b) e^{2 c+2 d x}}{2 \sqrt {-a} \sqrt {b}+a-b}+1\right )}{2 d (a+b)}-\frac {\frac {\operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 c+2 d x}}{a-b+2 \sqrt {-a} \sqrt {b}}\right )}{4 d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 c+2 d x}}{a-b+2 \sqrt {-a} \sqrt {b}}\right )}{2 d}}{d (a+b)}\right )}{4 \sqrt {-a} \sqrt {b}}\right )\) |
4*(((a + b)*((x^2*Log[1 + ((a + b)*E^(2*c + 2*d*x))/(a - 2*Sqrt[-a]*Sqrt[b ] - b)])/(2*(a + b)*d) - (-1/2*(x*PolyLog[2, -(((a + b)*E^(2*c + 2*d*x))/( a - 2*Sqrt[-a]*Sqrt[b] - b))])/d + PolyLog[3, -(((a + b)*E^(2*c + 2*d*x))/ (a - 2*Sqrt[-a]*Sqrt[b] - b))]/(4*d^2))/((a + b)*d)))/(4*Sqrt[-a]*Sqrt[b]) - ((a + b)*((x^2*Log[1 + ((a + b)*E^(2*c + 2*d*x))/(a + 2*Sqrt[-a]*Sqrt[b ] - b)])/(2*(a + b)*d) - (-1/2*(x*PolyLog[2, -(((a + b)*E^(2*c + 2*d*x))/( a + 2*Sqrt[-a]*Sqrt[b] - b))])/d + PolyLog[3, -(((a + b)*E^(2*c + 2*d*x))/ (a + 2*Sqrt[-a]*Sqrt[b] - b))]/(4*d^2))/((a + b)*d)))/(4*Sqrt[-a]*Sqrt[b]) )
3.2.45.3.1 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[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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] + Simp[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]
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]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
Leaf count of result is larger than twice the leaf count of optimal. \(1185\) vs. \(2(285)=570\).
Time = 3.21 (sec) , antiderivative size = 1186, normalized size of antiderivative = 3.38
1/d^3*c^2/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*exp(2*d*x+2*c)+2*a-2*b)/(a*b)^(1 /2))-1/2/d^3/(-a*b)^(1/2)*ln(1-(a+b)*exp(2*d*x+2*c)/(2*(-a*b)^(1/2)-a+b))* c^2+1/d^2/(-a*b)^(1/2)*c^2*x+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^2+1/2/d^2/(-a*b)^(1/2)*polylog(2,(a+b)*exp(2*d*x+2 *c)/(2*(-a*b)^(1/2)-a+b))*x-1/d^3/(-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^2+1/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))*x+2/d^2/(-2*(-a*b)^(1/2)-a+b)*c ^2*x+4/3/d^3/(-2*(-a*b)^(1/2)-a+b)*c^3-1/2/d^3/(-2*(-a*b)^(1/2)-a+b)*polyl og(3,(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))+2/3/d^3/(-a*b)^(1/2)*c^3- 1/4/d^3/(-a*b)^(1/2)*polylog(3,(a+b)*exp(2*d*x+2*c)/(2*(-a*b)^(1/2)-a+b))+ 1/2/d^2/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*a*polylog(2,(a+b)*exp(2*d*x+2*c )/(-2*(-a*b)^(1/2)-a+b))*x-1/2/d/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*b*ln(1 -(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))*x^2-1/2/d^2/(-a*b)^(1/2)/(-2* (-a*b)^(1/2)-a+b)*b*polylog(2,(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))* x+1/d^2/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*a*c^2*x-1/2/d^3/(-a*b)^(1/2)/(- 2*(-a*b)^(1/2)-a+b)*a*ln(1-(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))*c^2 +1/2/d^3/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*b*ln(1-(a+b)*exp(2*d*x+2*c)/(- 2*(-a*b)^(1/2)-a+b))*c^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+1/3/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*b*x^3 -1/3/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*a*x^3+2/3/d^3/(-a*b)^(1/2)/(-2*...
Leaf count of result is larger than twice the leaf count of optimal. 2110 vs. \(2 (283) = 566\).
Time = 0.33 (sec) , antiderivative size = 2110, normalized size of antiderivative = 6.01 \[ \int \frac {x^2 \text {sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\text {Too large to display} \]
1/2*(2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*d*x*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) + 2*(a + b)*sqrt(-a *b/(a^2 + 2*a*b + b^2))*d*x*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) - 2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) *d*x*dilog(-(((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c) + 2*((a + b)*c osh(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) - 2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*d*x*dilog((((a - b)*co sh(d*x + c) + (a - b)*sinh(d*x + c) + 2*((a + b)*cosh(d*x + c) + (a + b)*s inh(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))*c^2*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^2*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^2*log(2*sqrt((2*(a + b)*sqrt(-...
\[ \int \frac {x^2 \text {sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int \frac {x^{2} \operatorname {sech}^{2}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {x^2 \text {sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int { \frac {x^{2} \operatorname {sech}\left (d x + c\right )^{2}}{b \tanh \left (d x + c\right )^{2} + a} \,d x } \]
\[ \int \frac {x^2 \text {sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int { \frac {x^{2} \operatorname {sech}\left (d x + c\right )^{2}}{b \tanh \left (d x + c\right )^{2} + a} \,d x } \]
Timed out. \[ \int \frac {x^2 \text {sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int \frac {x^2}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )} \,d x \]