Integrand size = 23, antiderivative size = 187 \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=-\frac {(a+6 b) x}{2 a^4}+\frac {\sqrt {b} \left (15 a^2+40 a b+24 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^4 (a+b)^{3/2} d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {3 b \tanh (c+d x)}{4 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {b (11 a+12 b) \tanh (c+d x)}{8 a^3 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )} \]
-1/2*(a+6*b)*x/a^4+1/8*(15*a^2+40*a*b+24*b^2)*arctanh(b^(1/2)*tanh(d*x+c)/ (a+b)^(1/2))*b^(1/2)/a^4/(a+b)^(3/2)/d+1/2*cosh(d*x+c)*sinh(d*x+c)/a/d/(a+ b-b*tanh(d*x+c)^2)^2+3/4*b*tanh(d*x+c)/a^2/d/(a+b-b*tanh(d*x+c)^2)^2+1/8*b *(11*a+12*b)*tanh(d*x+c)/a^3/(a+b)/d/(a+b-b*tanh(d*x+c)^2)
Leaf count is larger than twice the leaf count of optimal. \(2544\) vs. \(2(187)=374\).
Time = 22.29 (sec) , antiderivative size = 2544, normalized size of antiderivative = 13.60 \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Result too large to show} \]
(-5*(a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*(((3*a^2 + 8*a*b + 8 *b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) - (a*Sqr t[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(8192*b^(5/2)*d* (a + b*Sech[c + d*x]^2)^3) - ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d *x]^6*((-3*a*(a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) + (Sqrt[b]*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3 + a*(3*a^2 + 4*a *b + 4*b^2)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a* Cosh[2*(c + d*x)])^2)))/(2048*b^(5/2)*d*(a + b*Sech[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((-2*(3*a^5 - 10*a^4*b + 80* a^3*b^2 + 480*a^2*b^3 + 640*a*b^4 + 256*b^5)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sq rt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b *(Cosh[c] - Sinh[c])^4]) + (Sech[2*c]*(256*b^2*(a + b)^2*(3*a^2 + 8*a*b + 8*b^2)*d*x*Cosh[2*c] + 512*a*b^2*(a + b)^2*(a + 2*b)*d*x*Cosh[2*d*x] + 128 *a^4*b^2*d*x*Cosh[2*(c + 2*d*x)] + 256*a^3*b^3*d*x*Cosh[2*(c + 2*d*x)] + 1 28*a^2*b^4*d*x*Cosh[2*(c + 2*d*x)] + 512*a^4*b^2*d*x*Cosh[4*c + 2*d*x] + 2 048*a^3*b^3*d*x*Cosh[4*c + 2*d*x] + 2560*a^2*b^4*d*x*Cosh[4*c + 2*d*x] + 1 024*a*b^5*d*x*Cosh[4*c + 2*d*x] + 128*a^4*b^2*d*x*Cosh[6*c + 4*d*x] + 256* a^3*b^3*d*x*Cosh[6*c + 4*d*x] + 128*a^2*b^4*d*x*Cosh[6*c + 4*d*x] - 9*a...
Time = 0.45 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.18, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 25, 4620, 373, 402, 27, 402, 25, 397, 219, 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.
| \(\displaystyle \int \frac {\sinh ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin (i c+i d x)^2}{\left (a+b \sec (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sin (i c+i d x)^2}{\left (b \sec (i c+i d x)^2+a\right )^3}dx\) |
| \(\Big \downarrow \) 4620 |
| \(\displaystyle \frac {\int \frac {\tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right )^2 \left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 373 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\int \frac {5 b \tanh ^2(c+d x)+a+b}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{2 a}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {-\frac {\int -\frac {2 (a+b) \left (9 b \tanh ^2(c+d x)+2 a+3 b\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 a (a+b)}-\frac {3 b \tanh (c+d x)}{2 a \left (a-b \tanh ^2(c+d x)+b\right )^2}}{2 a}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\int \frac {9 b \tanh ^2(c+d x)+2 a+3 b}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{2 a}-\frac {3 b \tanh (c+d x)}{2 a \left (a-b \tanh ^2(c+d x)+b\right )^2}}{2 a}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {-\frac {\int -\frac {4 a^2+17 b a+12 b^2+b (11 a+12 b) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b (11 a+12 b) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{2 a}-\frac {3 b \tanh (c+d x)}{2 a \left (a-b \tanh ^2(c+d x)+b\right )^2}}{2 a}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\int \frac {4 a^2+17 b a+12 b^2+b (11 a+12 b) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b (11 a+12 b) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{2 a}-\frac {3 b \tanh (c+d x)}{2 a \left (a-b \tanh ^2(c+d x)+b\right )^2}}{2 a}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\frac {4 (a+b) (a+6 b) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a}-\frac {b \left (15 a^2+40 a b+24 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{2 a (a+b)}-\frac {b (11 a+12 b) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{2 a}-\frac {3 b \tanh (c+d x)}{2 a \left (a-b \tanh ^2(c+d x)+b\right )^2}}{2 a}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\frac {4 (a+b) (a+6 b) \text {arctanh}(\tanh (c+d x))}{a}-\frac {b \left (15 a^2+40 a b+24 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{2 a (a+b)}-\frac {b (11 a+12 b) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{2 a}-\frac {3 b \tanh (c+d x)}{2 a \left (a-b \tanh ^2(c+d x)+b\right )^2}}{2 a}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\frac {4 (a+b) (a+6 b) \text {arctanh}(\tanh (c+d x))}{a}-\frac {\sqrt {b} \left (15 a^2+40 a b+24 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{2 a (a+b)}-\frac {b (11 a+12 b) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{2 a}-\frac {3 b \tanh (c+d x)}{2 a \left (a-b \tanh ^2(c+d x)+b\right )^2}}{2 a}}{d}\) |
(Tanh[c + d*x]/(2*a*(1 - Tanh[c + d*x]^2)*(a + b - b*Tanh[c + d*x]^2)^2) - ((-3*b*Tanh[c + d*x])/(2*a*(a + b - b*Tanh[c + d*x]^2)^2) + (((4*(a + b)* (a + 6*b)*ArcTanh[Tanh[c + d*x]])/a - (Sqrt[b]*(15*a^2 + 40*a*b + 24*b^2)* ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + b]))/(2*a*(a + b )) - (b*(11*a + 12*b)*Tanh[c + d*x])/(2*a*(a + b)*(a + b - b*Tanh[c + d*x] ^2)))/(2*a))/(2*a))/d
3.1.43.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*(b*c - a*d)*(p + 1))), x] - Simp[e^2/(2*(b*c - a*d)*(p + 1)) Int[(e *x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(m - 1) + d*(m + 2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[m, 3] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
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. \(1319\) vs. \(2(169)=338\).
Time = 0.24 (sec) , antiderivative size = 1320, normalized size of antiderivative = 7.06
\[\text {Expression too large to display}\]
1/2/d/a^3/(tanh(1/2*d*x+1/2*c)-1)^2+1/2/d/a^3/(tanh(1/2*d*x+1/2*c)-1)+1/2/ d/a^3*ln(tanh(1/2*d*x+1/2*c)-1)+3/d/a^4*ln(tanh(1/2*d*x+1/2*c)-1)*b-1/2/d/ a^3/(1+tanh(1/2*d*x+1/2*c))^2+1/2/d/a^3/(1+tanh(1/2*d*x+1/2*c))-1/2/d/a^3* ln(1+tanh(1/2*d*x+1/2*c))-3/d/a^4*ln(1+tanh(1/2*d*x+1/2*c))*b+9/4/d*b/a^2/ (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)^2*tanh(1/2*d*x+1/2*c)^7+2/d*b^2/a^3/(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*tan h(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^7+27/4/d*b/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)^2/(a+b)*tanh(1/2*d*x+1/2*c)^5+23/4/d*b^2/a^2/(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)^2/(a+b)*tanh(1/2*d*x+1/2*c)^5-2/d*b^3/a^3/(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)^2/(a+b)*tanh(1/2*d*x+1/2*c)^5+27/4/d*b/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)^2/(a+b)*tanh(1/2*d*x+1/2*c)^3+23/4/d*b^2/a^2/(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)^2/(a+b)*tanh(1/2*d*x+1/2*c)^3-2/d*b^3/a^3/(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)^2/(a+b)*tanh(1/2*d*x+1/2*c)^3+9/4/d*b/a^2/(tanh...
Leaf count of result is larger than twice the leaf count of optimal. 4724 vs. \(2 (178) = 356\).
Time = 0.38 (sec) , antiderivative size = 9730, normalized size of antiderivative = 52.03 \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1373 vs. \(2 (178) = 356\).
Time = 0.37 (sec) , antiderivative size = 1373, normalized size of antiderivative = 7.34 \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
3/64*(5*a^3*b + 30*a^2*b^2 + 40*a*b^3 + 16*b^4)*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^6 + 2*a^5*b + a^4*b^2)*sqrt((a + b)*b)*d) - 3/64*(5*a^3*b + 30*a^ 2*b^2 + 40*a*b^3 + 16*b^4)*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^6 + 2*a^5* b + a^4*b^2)*sqrt((a + b)*b)*d) - 1/32*(15*a^2*b + 20*a*b^2 + 8*b^3)*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^5 + 2*a^4*b + a^3*b^2)*sqrt((a + b)*b)*d) - 1/16*(9*a^4*b + 32*a^3*b^2 + 20*a^2*b^3 + 3*(3*a^4*b + 34*a^3*b^2 + 64*a ^2*b^3 + 32*a*b^4)*e^(6*d*x + 6*c) + (27*a^4*b + 264*a^3*b^2 + 740*a^2*b^3 + 832*a*b^4 + 320*b^5)*e^(4*d*x + 4*c) + (27*a^4*b + 194*a^3*b^2 + 336*a^ 2*b^3 + 160*a*b^4)*e^(2*d*x + 2*c))/((a^8 + 2*a^7*b + a^6*b^2 + (a^8 + 2*a ^7*b + a^6*b^2)*e^(8*d*x + 8*c) + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3 )*e^(6*d*x + 6*c) + 2*(3*a^8 + 14*a^7*b + 27*a^6*b^2 + 24*a^5*b^3 + 8*a^4* b^4)*e^(4*d*x + 4*c) + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*e^(2*d*x + 2*c))*d) + 1/16*(9*a^4*b + 32*a^3*b^2 + 20*a^2*b^3 + (27*a^4*b + 194*a^3 *b^2 + 336*a^2*b^3 + 160*a*b^4)*e^(-2*d*x - 2*c) + (27*a^4*b + 264*a^3*b^2 + 740*a^2*b^3 + 832*a*b^4 + 320*b^5)*e^(-4*d*x - 4*c) + 3*(3*a^4*b + 34*a ^3*b^2 + 64*a^2*b^3 + 32*a*b^4)*e^(-6*d*x - 6*c))/((a^8 + 2*a^7*b + a^6*b^ 2 + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*e^(-2*d*x - 2*c) + 2*(3*a...
\[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int { \frac {\sinh \left (d x + c\right )^{2}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\mathrm {sinh}\left (c+d\,x\right )}^2}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \]