Integrand size = 20, antiderivative size = 149 \[ \int x^2 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx=\frac {4 x^2 \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}-\frac {\text {arctanh}(\cosh (2 a+2 b x))}{b^3}-\frac {2 x \text {csch}(2 a+2 b x)}{b^2}-\frac {2 x^2 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{b}+\frac {2 x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{b^2}-\frac {2 x \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )}{b^2}-\frac {\operatorname {PolyLog}\left (3,-e^{2 a+2 b x}\right )}{b^3}+\frac {\operatorname {PolyLog}\left (3,e^{2 a+2 b x}\right )}{b^3} \]
4*x^2*arctanh(exp(2*b*x+2*a))/b-arctanh(cosh(2*b*x+2*a))/b^3-2*x*csch(2*b* x+2*a)/b^2-2*x^2*coth(2*b*x+2*a)*csch(2*b*x+2*a)/b+2*x*polylog(2,-exp(2*b* x+2*a))/b^2-2*x*polylog(2,exp(2*b*x+2*a))/b^2-polylog(3,-exp(2*b*x+2*a))/b ^3+polylog(3,exp(2*b*x+2*a))/b^3
Time = 3.06 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.29 \[ \int x^2 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx=-\frac {4 \text {arctanh}\left (e^{2 (a+b x)}\right )+b^2 x^2 \text {csch}^2(a+b x)+4 b^2 x^2 \log \left (1-e^{2 (a+b x)}\right )-4 b^2 x^2 \log \left (1+e^{2 (a+b x)}\right )-4 b x \operatorname {PolyLog}\left (2,-e^{2 (a+b x)}\right )+4 b x \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )+2 \operatorname {PolyLog}\left (3,-e^{2 (a+b x)}\right )-2 \operatorname {PolyLog}\left (3,e^{2 (a+b x)}\right )+2 b x \text {csch}(a) \text {sech}(a)+b^2 x^2 \text {sech}^2(a+b x)-2 b x \text {csch}(a) \text {csch}(a+b x) \sinh (b x)-2 b x \text {sech}(a) \text {sech}(a+b x) \sinh (b x)}{2 b^3} \]
-1/2*(4*ArcTanh[E^(2*(a + b*x))] + b^2*x^2*Csch[a + b*x]^2 + 4*b^2*x^2*Log [1 - E^(2*(a + b*x))] - 4*b^2*x^2*Log[1 + E^(2*(a + b*x))] - 4*b*x*PolyLog [2, -E^(2*(a + b*x))] + 4*b*x*PolyLog[2, E^(2*(a + b*x))] + 2*PolyLog[3, - E^(2*(a + b*x))] - 2*PolyLog[3, E^(2*(a + b*x))] + 2*b*x*Csch[a]*Sech[a] + b^2*x^2*Sech[a + b*x]^2 - 2*b*x*Csch[a]*Csch[a + b*x]*Sinh[b*x] - 2*b*x*S ech[a]*Sech[a + b*x]*Sinh[b*x])/b^3
Result contains complex when optimal does not.
Time = 0.76 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.32, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5984, 3042, 26, 4674, 26, 3042, 26, 4257, 4670, 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.
| \(\displaystyle \int x^2 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle 8 \int x^2 \text {csch}^3(2 a+2 b x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 8 \int -i x^2 \csc (2 i a+2 i b x)^3dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -8 i \int x^2 \csc (2 i a+2 i b x)^3dx\) |
| \(\Big \downarrow \) 4674 |
| \(\displaystyle -8 i \left (-\frac {\int -i \text {csch}(2 a+2 b x)dx}{4 b^2}+\frac {1}{2} \int -i x^2 \text {csch}(2 a+2 b x)dx-\frac {i x \text {csch}(2 a+2 b x)}{4 b^2}-\frac {i x^2 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -8 i \left (\frac {i \int \text {csch}(2 a+2 b x)dx}{4 b^2}-\frac {1}{2} i \int x^2 \text {csch}(2 a+2 b x)dx-\frac {i x \text {csch}(2 a+2 b x)}{4 b^2}-\frac {i x^2 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -8 i \left (\frac {i \int i \csc (2 i a+2 i b x)dx}{4 b^2}-\frac {1}{2} i \int i x^2 \csc (2 i a+2 i b x)dx-\frac {i x \text {csch}(2 a+2 b x)}{4 b^2}-\frac {i x^2 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -8 i \left (-\frac {\int \csc (2 i a+2 i b x)dx}{4 b^2}+\frac {1}{2} \int x^2 \csc (2 i a+2 i b x)dx-\frac {i x \text {csch}(2 a+2 b x)}{4 b^2}-\frac {i x^2 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -8 i \left (\frac {1}{2} \int x^2 \csc (2 i a+2 i b x)dx-\frac {i \text {arctanh}(\cosh (2 a+2 b x))}{8 b^3}-\frac {i x \text {csch}(2 a+2 b x)}{4 b^2}-\frac {i x^2 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -8 i \left (\frac {1}{2} \left (\frac {i \int x \log \left (1-e^{2 a+2 b x}\right )dx}{b}-\frac {i \int x \log \left (1+e^{2 a+2 b x}\right )dx}{b}+\frac {i x^2 \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}\right )-\frac {i \text {arctanh}(\cosh (2 a+2 b x))}{8 b^3}-\frac {i x \text {csch}(2 a+2 b x)}{4 b^2}-\frac {i x^2 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -8 i \left (\frac {1}{2} \left (-\frac {i \left (\frac {\int \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )dx}{2 b}-\frac {x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{2 b}\right )}{b}+\frac {i \left (\frac {\int \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )dx}{2 b}-\frac {x \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )}{2 b}\right )}{b}+\frac {i x^2 \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}\right )-\frac {i \text {arctanh}(\cosh (2 a+2 b x))}{8 b^3}-\frac {i x \text {csch}(2 a+2 b x)}{4 b^2}-\frac {i x^2 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -8 i \left (\frac {1}{2} \left (-\frac {i \left (\frac {\int e^{-2 a-2 b x} \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )de^{2 a+2 b x}}{4 b^2}-\frac {x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{2 b}\right )}{b}+\frac {i \left (\frac {\int e^{-2 a-2 b x} \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )de^{2 a+2 b x}}{4 b^2}-\frac {x \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )}{2 b}\right )}{b}+\frac {i x^2 \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}\right )-\frac {i \text {arctanh}(\cosh (2 a+2 b x))}{8 b^3}-\frac {i x \text {csch}(2 a+2 b x)}{4 b^2}-\frac {i x^2 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -8 i \left (-\frac {i \text {arctanh}(\cosh (2 a+2 b x))}{8 b^3}+\frac {1}{2} \left (\frac {i x^2 \text {arctanh}\left (e^{2 a+2 b x}\right )}{b}-\frac {i \left (\frac {\operatorname {PolyLog}\left (3,-e^{2 a+2 b x}\right )}{4 b^2}-\frac {x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{2 b}\right )}{b}+\frac {i \left (\frac {\operatorname {PolyLog}\left (3,e^{2 a+2 b x}\right )}{4 b^2}-\frac {x \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )}{2 b}\right )}{b}\right )-\frac {i x \text {csch}(2 a+2 b x)}{4 b^2}-\frac {i x^2 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{4 b}\right )\) |
(-8*I)*(((-1/8*I)*ArcTanh[Cosh[2*a + 2*b*x]])/b^3 - ((I/4)*x*Csch[2*a + 2* b*x])/b^2 - ((I/4)*x^2*Coth[2*a + 2*b*x]*Csch[2*a + 2*b*x])/b + ((I*x^2*Ar cTanh[E^(2*a + 2*b*x)])/b - (I*(-1/2*(x*PolyLog[2, -E^(2*a + 2*b*x)])/b + PolyLog[3, -E^(2*a + 2*b*x)]/(4*b^2)))/b + (I*(-1/2*(x*PolyLog[2, E^(2*a + 2*b*x)])/b + PolyLog[3, E^(2*a + 2*b*x)]/(4*b^2)))/b)/2)
3.6.23.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbo l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ (n - 1)) Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c , d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csch[2*a + 2*b*x ]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
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. \(298\) vs. \(2(144)=288\).
Time = 21.85 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.01
| method | result | size |
| risch | \(-\frac {4 x \,{\mathrm e}^{2 b x +2 a} \left ({\mathrm e}^{4 b x +4 a} b x +{\mathrm e}^{4 b x +4 a}+b x -1\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2} \left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}-\frac {2 a^{2} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{3}}-\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b}-\frac {4 x \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {2 x^{2} \ln \left (1+{\mathrm e}^{2 b x +2 a}\right )}{b}+\frac {2 x \operatorname {polylog}\left (2, -{\mathrm e}^{2 b x +2 a}\right )}{b^{2}}-\frac {2 \ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{b}-\frac {4 x \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {4 \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {\operatorname {polylog}\left (3, -{\mathrm e}^{2 b x +2 a}\right )}{b^{3}}+\frac {4 \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{3}}+\frac {\ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{3}}-\frac {\ln \left (1+{\mathrm e}^{2 b x +2 a}\right )}{b^{3}}+\frac {\ln \left ({\mathrm e}^{b x +a}+1\right )}{b^{3}}\) | \(299\) |
-4*x*exp(2*b*x+2*a)*(exp(4*b*x+4*a)*b*x+exp(4*b*x+4*a)+b*x-1)/b^2/(exp(2*b *x+2*a)-1)^2/(1+exp(2*b*x+2*a))^2-2/b^3*a^2*ln(exp(b*x+a)-1)-2/b*ln(1-exp( b*x+a))*x^2-4*x*polylog(2,exp(b*x+a))/b^2+2*x^2*ln(1+exp(2*b*x+2*a))/b+2*x *polylog(2,-exp(2*b*x+2*a))/b^2-2/b*ln(exp(b*x+a)+1)*x^2-4*x*polylog(2,-ex p(b*x+a))/b^2+4*polylog(3,exp(b*x+a))/b^3-polylog(3,-exp(2*b*x+2*a))/b^3+4 *polylog(3,-exp(b*x+a))/b^3+2/b^3*ln(1-exp(b*x+a))*a^2+1/b^3*ln(exp(b*x+a) -1)-1/b^3*ln(1+exp(2*b*x+2*a))+1/b^3*ln(exp(b*x+a)+1)
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 4779, normalized size of antiderivative = 32.07 \[ \int x^2 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx=\text {Too large to display} \]
-(4*(b^2*x^2 + b*x)*cosh(b*x + a)^6 + 80*(b^2*x^2 + b*x)*cosh(b*x + a)^3*s inh(b*x + a)^3 + 60*(b^2*x^2 + b*x)*cosh(b*x + a)^2*sinh(b*x + a)^4 + 24*( b^2*x^2 + b*x)*cosh(b*x + a)*sinh(b*x + a)^5 + 4*(b^2*x^2 + b*x)*sinh(b*x + a)^6 + 4*(b^2*x^2 - b*x)*cosh(b*x + a)^2 + 4*(15*(b^2*x^2 + b*x)*cosh(b* x + a)^4 + b^2*x^2 - b*x)*sinh(b*x + a)^2 + 4*(b*x*cosh(b*x + a)^8 + 56*b* x*cosh(b*x + a)^3*sinh(b*x + a)^5 + 28*b*x*cosh(b*x + a)^2*sinh(b*x + a)^6 + 8*b*x*cosh(b*x + a)*sinh(b*x + a)^7 + b*x*sinh(b*x + a)^8 - 2*b*x*cosh( b*x + a)^4 + 2*(35*b*x*cosh(b*x + a)^4 - b*x)*sinh(b*x + a)^4 + 8*(7*b*x*c osh(b*x + a)^5 - b*x*cosh(b*x + a))*sinh(b*x + a)^3 + 4*(7*b*x*cosh(b*x + a)^6 - 3*b*x*cosh(b*x + a)^2)*sinh(b*x + a)^2 + b*x + 8*(b*x*cosh(b*x + a) ^7 - b*x*cosh(b*x + a)^3)*sinh(b*x + a))*dilog(cosh(b*x + a) + sinh(b*x + a)) - 4*(b*x*cosh(b*x + a)^8 + 56*b*x*cosh(b*x + a)^3*sinh(b*x + a)^5 + 28 *b*x*cosh(b*x + a)^2*sinh(b*x + a)^6 + 8*b*x*cosh(b*x + a)*sinh(b*x + a)^7 + b*x*sinh(b*x + a)^8 - 2*b*x*cosh(b*x + a)^4 + 2*(35*b*x*cosh(b*x + a)^4 - b*x)*sinh(b*x + a)^4 + 8*(7*b*x*cosh(b*x + a)^5 - b*x*cosh(b*x + a))*si nh(b*x + a)^3 + 4*(7*b*x*cosh(b*x + a)^6 - 3*b*x*cosh(b*x + a)^2)*sinh(b*x + a)^2 + b*x + 8*(b*x*cosh(b*x + a)^7 - b*x*cosh(b*x + a)^3)*sinh(b*x + a ))*dilog(I*cosh(b*x + a) + I*sinh(b*x + a)) - 4*(b*x*cosh(b*x + a)^8 + 56* b*x*cosh(b*x + a)^3*sinh(b*x + a)^5 + 28*b*x*cosh(b*x + a)^2*sinh(b*x + a) ^6 + 8*b*x*cosh(b*x + a)*sinh(b*x + a)^7 + b*x*sinh(b*x + a)^8 - 2*b*x*...
\[ \int x^2 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx=\int x^{2} \operatorname {csch}^{3}{\left (a + b x \right )} \operatorname {sech}^{3}{\left (a + b x \right )}\, dx \]
Time = 0.22 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.83 \[ \int x^2 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx=-\frac {4 \, {\left ({\left (b x^{2} e^{\left (6 \, a\right )} + x e^{\left (6 \, a\right )}\right )} e^{\left (6 \, b x\right )} + {\left (b x^{2} e^{\left (2 \, a\right )} - x e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}\right )}}{b^{2} e^{\left (8 \, b x + 8 \, a\right )} - 2 \, b^{2} e^{\left (4 \, b x + 4 \, a\right )} + b^{2}} + \frac {2 \, b^{2} x^{2} \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 \, b x + 2 \, a\right )})}{b^{3}} - \frac {2 \, {\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})\right )}}{b^{3}} - \frac {2 \, {\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})\right )}}{b^{3}} - \frac {\log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{b^{3}} + \frac {\log \left (e^{\left (b x + a\right )} + 1\right )}{b^{3}} + \frac {\log \left (e^{\left (b x + a\right )} - 1\right )}{b^{3}} \]
-4*((b*x^2*e^(6*a) + x*e^(6*a))*e^(6*b*x) + (b*x^2*e^(2*a) - x*e^(2*a))*e^ (2*b*x))/(b^2*e^(8*b*x + 8*a) - 2*b^2*e^(4*b*x + 4*a) + b^2) + (2*b^2*x^2* log(e^(2*b*x + 2*a) + 1) + 2*b*x*dilog(-e^(2*b*x + 2*a)) - polylog(3, -e^( 2*b*x + 2*a)))/b^3 - 2*(b^2*x^2*log(e^(b*x + a) + 1) + 2*b*x*dilog(-e^(b*x + a)) - 2*polylog(3, -e^(b*x + a)))/b^3 - 2*(b^2*x^2*log(-e^(b*x + a) + 1 ) + 2*b*x*dilog(e^(b*x + a)) - 2*polylog(3, e^(b*x + a)))/b^3 - log(e^(2*b *x + 2*a) + 1)/b^3 + log(e^(b*x + a) + 1)/b^3 + log(e^(b*x + a) - 1)/b^3
\[ \int x^2 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx=\int { x^{2} \operatorname {csch}\left (b x + a\right )^{3} \operatorname {sech}\left (b x + a\right )^{3} \,d x } \]
Timed out. \[ \int x^2 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx=\int \frac {x^2}{{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \]