Integrand size = 23, antiderivative size = 123 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=-\frac {(3 a-2 b) \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 (a+b)^{7/2} d}+\frac {(a-b) \coth (c+d x)}{(a+b)^3 d}-\frac {\coth ^3(c+d x)}{3 (a+b)^2 d}-\frac {a b \tanh (c+d x)}{2 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )} \]
(a-b)*coth(d*x+c)/(a+b)^3/d-1/3*coth(d*x+c)^3/(a+b)^2/d-1/2*(3*a-2*b)*arct anh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))*b^(1/2)/(a+b)^(7/2)/d-1/2*a*b*tanh(d* x+c)/(a+b)^3/d/(a+b-b*tanh(d*x+c)^2)
Result contains complex when optimal does not.
Time = 7.42 (sec) , antiderivative size = 620, normalized size of antiderivative = 5.04 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=-\frac {(a+2 b+a \cosh (2 c+2 d x))^2 \coth (c) \text {csch}^2(c+d x) \text {sech}^4(c+d x)}{12 (a+b)^2 d \left (a+b \text {sech}^2(c+d x)\right )^2}+\frac {(3 a-2 b) (a+2 b+a \cosh (2 c+2 d x))^2 \text {sech}^4(c+d x) \left (\frac {i b \arctan \left (\text {sech}(d x) \left (-\frac {i \cosh (2 c)}{2 \sqrt {a+b} \sqrt {b \cosh (4 c)-b \sinh (4 c)}}+\frac {i \sinh (2 c)}{2 \sqrt {a+b} \sqrt {b \cosh (4 c)-b \sinh (4 c)}}\right ) (-a \sinh (d x)-2 b \sinh (d x)+a \sinh (2 c+d x))\right ) \cosh (2 c)}{8 \sqrt {a+b} d \sqrt {b \cosh (4 c)-b \sinh (4 c)}}-\frac {i b \arctan \left (\text {sech}(d x) \left (-\frac {i \cosh (2 c)}{2 \sqrt {a+b} \sqrt {b \cosh (4 c)-b \sinh (4 c)}}+\frac {i \sinh (2 c)}{2 \sqrt {a+b} \sqrt {b \cosh (4 c)-b \sinh (4 c)}}\right ) (-a \sinh (d x)-2 b \sinh (d x)+a \sinh (2 c+d x))\right ) \sinh (2 c)}{8 \sqrt {a+b} d \sqrt {b \cosh (4 c)-b \sinh (4 c)}}\right )}{(a+b)^3 \left (a+b \text {sech}^2(c+d x)\right )^2}+\frac {(a+2 b+a \cosh (2 c+2 d x))^2 \text {csch}(c) \text {csch}^3(c+d x) \text {sech}^4(c+d x) \sinh (d x)}{12 (a+b)^2 d \left (a+b \text {sech}^2(c+d x)\right )^2}+\frac {(a+2 b+a \cosh (2 c+2 d x))^2 \text {csch}(c) \text {csch}(c+d x) \text {sech}^4(c+d x) (-a \sinh (d x)+2 b \sinh (d x))}{6 (a+b)^3 d \left (a+b \text {sech}^2(c+d x)\right )^2}+\frac {(a+2 b+a \cosh (2 c+2 d x)) \text {sech}(2 c) \text {sech}^4(c+d x) \left (a b \sinh (2 c)+2 b^2 \sinh (2 c)-a b \sinh (2 d x)\right )}{8 (a+b)^3 d \left (a+b \text {sech}^2(c+d x)\right )^2} \]
-1/12*((a + 2*b + a*Cosh[2*c + 2*d*x])^2*Coth[c]*Csch[c + d*x]^2*Sech[c + d*x]^4)/((a + b)^2*d*(a + b*Sech[c + d*x]^2)^2) + ((3*a - 2*b)*(a + 2*b + a*Cosh[2*c + 2*d*x])^2*Sech[c + d*x]^4*(((I/8)*b*ArcTan[Sech[d*x]*(((-1/2* I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[ 2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b *Sinh[d*x] + a*Sinh[2*c + d*x])]*Cosh[2*c])/(Sqrt[a + b]*d*Sqrt[b*Cosh[4*c ] - b*Sinh[4*c]]) - ((I/8)*b*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[ a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*S qrt[b*Cosh[4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[ 2*c + d*x])]*Sinh[2*c])/(Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]])))/ ((a + b)^3*(a + b*Sech[c + d*x]^2)^2) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^2 *Csch[c]*Csch[c + d*x]^3*Sech[c + d*x]^4*Sinh[d*x])/(12*(a + b)^2*d*(a + b *Sech[c + d*x]^2)^2) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^2*Csch[c]*Csch[c + d*x]*Sech[c + d*x]^4*(-(a*Sinh[d*x]) + 2*b*Sinh[d*x]))/(6*(a + b)^3*d*(a + b*Sech[c + d*x]^2)^2) + ((a + 2*b + a*Cosh[2*c + 2*d*x])*Sech[2*c]*Sech[ c + d*x]^4*(a*b*Sinh[2*c] + 2*b^2*Sinh[2*c] - a*b*Sinh[2*d*x]))/(8*(a + b) ^3*d*(a + b*Sech[c + d*x]^2)^2)
Time = 0.41 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 4620, 361, 1584, 2009}
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.
| \(\displaystyle \int \frac {\text {csch}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (i c+i d x)^4 \left (a+b \sec (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 4620 |
| \(\displaystyle \frac {\int \frac {\coth ^4(c+d x) \left (1-\tanh ^2(c+d x)\right )}{\left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 361 |
| \(\displaystyle \frac {\frac {1}{2} b \int \frac {\coth ^4(c+d x) \left (-\frac {a \tanh ^4(c+d x)}{(a+b)^3}-\frac {2 a \tanh ^2(c+d x)}{b (a+b)^2}+\frac {2}{b (a+b)}\right )}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)-\frac {a b \tanh (c+d x)}{2 (a+b)^3 \left (a-b \tanh ^2(c+d x)+b\right )}}{d}\) |
| \(\Big \downarrow \) 1584 |
| \(\displaystyle \frac {\frac {1}{2} b \int \left (\frac {2 \coth ^4(c+d x)}{b (a+b)^2}-\frac {2 (a-b) \coth ^2(c+d x)}{b (a+b)^3}+\frac {2 b-3 a}{(a+b)^3 \left (-b \tanh ^2(c+d x)+a+b\right )}\right )d\tanh (c+d x)-\frac {a b \tanh (c+d x)}{2 (a+b)^3 \left (a-b \tanh ^2(c+d x)+b\right )}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{2} b \left (-\frac {(3 a-2 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {b} (a+b)^{7/2}}-\frac {2 \coth ^3(c+d x)}{3 b (a+b)^2}+\frac {2 (a-b) \coth (c+d x)}{b (a+b)^3}\right )-\frac {a b \tanh (c+d x)}{2 (a+b)^3 \left (a-b \tanh ^2(c+d x)+b\right )}}{d}\) |
((b*(-(((3*a - 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(Sqrt[b] *(a + b)^(7/2))) + (2*(a - b)*Coth[c + d*x])/(b*(a + b)^3) - (2*Coth[c + d *x]^3)/(3*b*(a + b)^2)))/2 - (a*b*Tanh[c + d*x])/(2*(a + b)^3*(a + b - b*T anh[c + d*x]^2)))/d
3.1.40.3.1 Defintions of rubi rules used
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[x^m*(a + b*x^2)^(p + 1)*E xpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)] - ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ILtQ[m/ 2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* (a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1)/f Subst[Int[x^m*(ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p/(1 + f f^2*x^2)^(m/2 + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]
Leaf count of result is larger than twice the leaf count of optimal. \(326\) vs. \(2(109)=218\).
Time = 14.55 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.66
| method | result | size |
| derivativedivides | \(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{3}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b}{3}-3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a +5 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right )}+\frac {2 b \left (\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{2}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (3 a -2 b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (-\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}-\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2}\right )}{\left (a +b \right )^{3}}-\frac {1}{24 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-3 a +5 b}{8 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(327\) |
| default | \(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{3}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b}{3}-3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a +5 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right )}+\frac {2 b \left (\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{2}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (3 a -2 b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (-\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}-\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2}\right )}{\left (a +b \right )^{3}}-\frac {1}{24 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-3 a +5 b}{8 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(327\) |
| risch | \(-\frac {9 a b \,{\mathrm e}^{8 d x +8 c}-6 b^{2} {\mathrm e}^{8 d x +8 c}+12 a^{2} {\mathrm e}^{6 d x +6 c}+18 a b \,{\mathrm e}^{6 d x +6 c}+66 b^{2} {\mathrm e}^{6 d x +6 c}+20 a^{2} {\mathrm e}^{4 d x +4 c}+44 a b \,{\mathrm e}^{4 d x +4 c}-66 \,{\mathrm e}^{4 d x +4 c} b^{2}+4 a^{2} {\mathrm e}^{2 d x +2 c}-18 a b \,{\mathrm e}^{2 d x +2 c}+38 \,{\mathrm e}^{2 d x +2 c} b^{2}-4 a^{2}+11 a b}{3 d \left (a +b \right )^{3} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3} \left (a \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a +4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}+\frac {3 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}+a +2 b}{a}\right ) a}{4 \left (a +b \right )^{4} d}-\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}+a +2 b}{a}\right ) b}{2 \left (a +b \right )^{4} d}-\frac {3 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}-a -2 b}{a}\right ) a}{4 \left (a +b \right )^{4} d}+\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}-a -2 b}{a}\right ) b}{2 \left (a +b \right )^{4} d}\) | \(418\) |
1/d*(-1/8/(a^2+2*a*b+b^2)/(a+b)*(1/3*tanh(1/2*d*x+1/2*c)^3*a+1/3*tanh(1/2* d*x+1/2*c)^3*b-3*tanh(1/2*d*x+1/2*c)*a+5*b*tanh(1/2*d*x+1/2*c))+2*b/(a+b)^ 3*((-1/2*tanh(1/2*d*x+1/2*c)^3*a-1/2*tanh(1/2*d*x+1/2*c)*a)/(tanh(1/2*d*x+ 1/2*c)^4*a+tanh(1/2*d*x+1/2*c)^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d* x+1/2*c)^2*b+a+b)+1/2*(3*a-2*b)*(-1/4/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*t anh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))+1/4/b^(1/2 )/(a+b)^(1/2)*ln(-(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)* b^(1/2)-(a+b)^(1/2))))-1/24/(a+b)^2/tanh(1/2*d*x+1/2*c)^3-1/8/(a+b)^3*(-3* a+5*b)/tanh(1/2*d*x+1/2*c))
Leaf count of result is larger than twice the leaf count of optimal. 2933 vs. \(2 (112) = 224\).
Time = 0.34 (sec) , antiderivative size = 6143, normalized size of antiderivative = 49.94 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
\[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {\operatorname {csch}^{4}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (112) = 224\).
Time = 0.34 (sec) , antiderivative size = 430, normalized size of antiderivative = 3.50 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {{\left (3 \, a b - 2 \, b^{2}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{4 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {{\left (a + b\right )} b} d} + \frac {4 \, a^{2} - 11 \, a b - 2 \, {\left (2 \, a^{2} - 9 \, a b + 19 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, {\left (10 \, a^{2} + 22 \, a b - 33 \, b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 6 \, {\left (2 \, a^{2} + 3 \, a b + 11 \, b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )} - 3 \, {\left (3 \, a b - 2 \, b^{2}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{3 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} - {\left (a^{4} - a^{3} b - 9 \, a^{2} b^{2} - 11 \, a b^{3} - 4 \, b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, {\left (a^{4} + 9 \, a^{3} b + 21 \, a^{2} b^{2} + 19 \, a b^{3} + 6 \, b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 2 \, {\left (a^{4} + 9 \, a^{3} b + 21 \, a^{2} b^{2} + 19 \, a b^{3} + 6 \, b^{4}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{4} - a^{3} b - 9 \, a^{2} b^{2} - 11 \, a b^{3} - 4 \, b^{4}\right )} e^{\left (-8 \, d x - 8 \, c\right )} - {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} e^{\left (-10 \, d x - 10 \, c\right )}\right )} d} \]
1/4*(3*a*b - 2*b^2)*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b)) /(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^3 + 3*a^2*b + 3*a *b^2 + b^3)*sqrt((a + b)*b)*d) + 1/3*(4*a^2 - 11*a*b - 2*(2*a^2 - 9*a*b + 19*b^2)*e^(-2*d*x - 2*c) - 2*(10*a^2 + 22*a*b - 33*b^2)*e^(-4*d*x - 4*c) - 6*(2*a^2 + 3*a*b + 11*b^2)*e^(-6*d*x - 6*c) - 3*(3*a*b - 2*b^2)*e^(-8*d*x - 8*c))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3 - (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*e^(-2*d*x - 2*c) - 2*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a* b^3 + 6*b^4)*e^(-4*d*x - 4*c) + 2*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*e^(-6*d*x - 6*c) + (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*e^ (-8*d*x - 8*c) - (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*e^(-10*d*x - 10*c))*d )
\[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{4}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^4}{{\mathrm {sinh}\left (c+d\,x\right )}^4\,{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^2} \,d x \]