Integrand size = 23, antiderivative size = 126 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {15 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 (a+b)^{7/2} d}-\frac {\coth (c+d x)}{(a+b)^3 d}+\frac {b \tanh (c+d x)}{4 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {7 b \tanh (c+d x)}{8 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )} \] Output:
15/8*b^(1/2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))/(a+b)^(7/2)/d-coth(d *x+c)/(a+b)^3/d+1/4*b*tanh(d*x+c)/(a+b)^2/d/(a+b-b*tanh(d*x+c)^2)^2+7/8*b* tanh(d*x+c)/(a+b)^3/d/(a+b-b*tanh(d*x+c)^2)
Leaf count is larger than twice the leaf count of optimal. \(741\) vs. \(2(126)=252\).
Time = 6.00 (sec) , antiderivative size = 741, normalized size of antiderivative = 5.88 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^6(c+d x) \left (\frac {120 b \text {arctanh}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (a+2 b+a \cosh (2 (c+d x)))^2 (\cosh (2 c)-\sinh (2 c))}{\sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}+\frac {\text {csch}(c) \text {csch}(c+d x) \text {sech}(2 c) \left (\left (-32 a^4-64 a^3 b+22 a^2 b^2+80 a b^3+16 b^4\right ) \sinh (d x)+2 a \left (16 a^3+23 a^2 b-27 a b^2-4 b^3\right ) \sinh (3 d x)-48 a^4 \sinh (2 c-d x)-128 a^3 b \sinh (2 c-d x)-106 a^2 b^2 \sinh (2 c-d x)+80 a b^3 \sinh (2 c-d x)+16 b^4 \sinh (2 c-d x)+48 a^4 \sinh (2 c+d x)+146 a^3 b \sinh (2 c+d x)+182 a^2 b^2 \sinh (2 c+d x)+80 a b^3 \sinh (2 c+d x)+16 b^4 \sinh (2 c+d x)-32 a^4 \sinh (4 c+d x)-82 a^3 b \sinh (4 c+d x)-54 a^2 b^2 \sinh (4 c+d x)-80 a b^3 \sinh (4 c+d x)-16 b^4 \sinh (4 c+d x)-8 a^4 \sinh (2 c+3 d x)+18 a^3 b \sinh (2 c+3 d x)+54 a^2 b^2 \sinh (2 c+3 d x)+8 a b^3 \sinh (2 c+3 d x)+32 a^4 \sinh (4 c+3 d x)+73 a^3 b \sinh (4 c+3 d x)+24 a^2 b^2 \sinh (4 c+3 d x)+8 a b^3 \sinh (4 c+3 d x)-8 a^4 \sinh (6 c+3 d x)-9 a^3 b \sinh (6 c+3 d x)-24 a^2 b^2 \sinh (6 c+3 d x)-8 a b^3 \sinh (6 c+3 d x)+8 a^4 \sinh (2 c+5 d x)-9 a^3 b \sinh (2 c+5 d x)-2 a^2 b^2 \sinh (2 c+5 d x)+9 a^3 b \sinh (4 c+5 d x)+2 a^2 b^2 \sinh (4 c+5 d x)+8 a^4 \sinh (6 c+5 d x)\right )}{a^2}\right )}{512 (a+b)^3 d \left (a+b \text {sech}^2(c+d x)\right )^3} \] Input:
Integrate[Csch[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3,x]
Output:
((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^6*((120*b*ArcTanh[(Sech[d*x ]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sq rt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])^ 2*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + ( Csch[c]*Csch[c + d*x]*Sech[2*c]*((-32*a^4 - 64*a^3*b + 22*a^2*b^2 + 80*a*b ^3 + 16*b^4)*Sinh[d*x] + 2*a*(16*a^3 + 23*a^2*b - 27*a*b^2 - 4*b^3)*Sinh[3 *d*x] - 48*a^4*Sinh[2*c - d*x] - 128*a^3*b*Sinh[2*c - d*x] - 106*a^2*b^2*S inh[2*c - d*x] + 80*a*b^3*Sinh[2*c - d*x] + 16*b^4*Sinh[2*c - d*x] + 48*a^ 4*Sinh[2*c + d*x] + 146*a^3*b*Sinh[2*c + d*x] + 182*a^2*b^2*Sinh[2*c + d*x ] + 80*a*b^3*Sinh[2*c + d*x] + 16*b^4*Sinh[2*c + d*x] - 32*a^4*Sinh[4*c + d*x] - 82*a^3*b*Sinh[4*c + d*x] - 54*a^2*b^2*Sinh[4*c + d*x] - 80*a*b^3*Si nh[4*c + d*x] - 16*b^4*Sinh[4*c + d*x] - 8*a^4*Sinh[2*c + 3*d*x] + 18*a^3* b*Sinh[2*c + 3*d*x] + 54*a^2*b^2*Sinh[2*c + 3*d*x] + 8*a*b^3*Sinh[2*c + 3* d*x] + 32*a^4*Sinh[4*c + 3*d*x] + 73*a^3*b*Sinh[4*c + 3*d*x] + 24*a^2*b^2* Sinh[4*c + 3*d*x] + 8*a*b^3*Sinh[4*c + 3*d*x] - 8*a^4*Sinh[6*c + 3*d*x] - 9*a^3*b*Sinh[6*c + 3*d*x] - 24*a^2*b^2*Sinh[6*c + 3*d*x] - 8*a*b^3*Sinh[6* c + 3*d*x] + 8*a^4*Sinh[2*c + 5*d*x] - 9*a^3*b*Sinh[2*c + 5*d*x] - 2*a^2*b ^2*Sinh[2*c + 5*d*x] + 9*a^3*b*Sinh[4*c + 5*d*x] + 2*a^2*b^2*Sinh[4*c + 5* d*x] + 8*a^4*Sinh[6*c + 5*d*x]))/a^2))/(512*(a + b)^3*d*(a + b*Sech[c + d* x]^2)^3)
Time = 0.31 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 25, 4620, 253, 253, 264, 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.
| \(\displaystyle \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\sin (i c+i d x)^2 \left (a+b \sec (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\left (b \sec (i c+i d x)^2+a\right )^3 \sin (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 4620 |
| \(\displaystyle \frac {\int \frac {\coth ^2(c+d x)}{\left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\frac {5 \int \frac {\coth ^2(c+d x)}{\left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 (a+b)}+\frac {\coth (c+d x)}{4 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \int \frac {\coth ^2(c+d x)}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{2 (a+b)}+\frac {\coth (c+d x)}{2 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{4 (a+b)}+\frac {\coth (c+d x)}{4 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (\frac {b \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a+b}-\frac {\coth (c+d x)}{a+b}\right )}{2 (a+b)}+\frac {\coth (c+d x)}{2 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{4 (a+b)}+\frac {\coth (c+d x)}{4 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}-\frac {\coth (c+d x)}{a+b}\right )}{2 (a+b)}+\frac {\coth (c+d x)}{2 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{4 (a+b)}+\frac {\coth (c+d x)}{4 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
Input:
Int[Csch[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3,x]
Output:
(Coth[c + d*x]/(4*(a + b)*(a + b - b*Tanh[c + d*x]^2)^2) + (5*((3*((Sqrt[b ]*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(3/2) - Coth[c + d *x]/(a + b)))/(2*(a + b)) + Coth[c + d*x]/(2*(a + b)*(a + b - b*Tanh[c + d *x]^2))))/(4*(a + b)))/d
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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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. \(818\) vs. \(2(112)=224\).
Time = 0.24 (sec) , antiderivative size = 819, normalized size of antiderivative = 6.50
\[\text {Expression too large to display}\]
Input:
int(csch(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x)
Output:
-1/2/d*tanh(1/2*d*x+1/2*c)/(a^3+3*a^2*b+3*a*b^2+b^3)-1/2/d/(a+b)^3/tanh(1/ 2*d*x+1/2*c)+9/4/d*b/(a+b)^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*tanh(1/2*d* x+1/2*c)^7*a+9/4/d*b^2/(a+b)^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*tanh(1/2* d*x+1/2*c)^7+27/4/d*b/(a+b)^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*tanh(1/2*d *x+1/2*c)^5*a-1/4/d*b^2/(a+b)^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*tanh(1/2 *d*x+1/2*c)^5+27/4/d*b/(a+b)^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*tanh(1/2* d*x+1/2*c)^3*a-1/4/d*b^2/(a+b)^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*tanh(1/ 2*d*x+1/2*c)^3+9/4/d*b/(a+b)^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*tanh(1/2* d*x+1/2*c)*a+9/4/d*b^2/(a+b)^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*tanh(1/2* d*x+1/2*c)+15/16/d*b^(1/2)/(a+b)^(7/2)*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))-15/16/d*b^(1/2)/(a+b)^(7/2)*l n(-(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...
Leaf count of result is larger than twice the leaf count of optimal. 3499 vs. \(2 (118) = 236\).
Time = 0.42 (sec) , antiderivative size = 7275, normalized size of antiderivative = 57.74 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {\operatorname {csch}^{2}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \] Input:
integrate(csch(d*x+c)**2/(a+b*sech(d*x+c)**2)**3,x)
Output:
Integral(csch(c + d*x)**2/(a + b*sech(c + d*x)**2)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (118) = 236\).
Time = 0.19 (sec) , antiderivative size = 533, normalized size of antiderivative = 4.23 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=-\frac {15 \, b \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {8 \, a^{4} - 9 \, a^{3} b - 2 \, a^{2} b^{2} + 2 \, {\left (16 \, a^{4} + 23 \, a^{3} b - 27 \, a^{2} b^{2} - 4 \, a b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (24 \, a^{4} + 64 \, a^{3} b + 53 \, a^{2} b^{2} - 40 \, a b^{3} - 8 \, b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 2 \, {\left (16 \, a^{4} + 41 \, a^{3} b + 27 \, a^{2} b^{2} + 40 \, a b^{3} + 8 \, b^{4}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (8 \, a^{4} + 9 \, a^{3} b + 24 \, a^{2} b^{2} + 8 \, a b^{3}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{4 \, {\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3} + {\left (3 \, a^{7} + 17 \, a^{6} b + 33 \, a^{5} b^{2} + 27 \, a^{4} b^{3} + 8 \, a^{3} b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (a^{7} + 7 \, a^{6} b + 23 \, a^{5} b^{2} + 37 \, a^{4} b^{3} + 28 \, a^{3} b^{4} + 8 \, a^{2} b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 2 \, {\left (a^{7} + 7 \, a^{6} b + 23 \, a^{5} b^{2} + 37 \, a^{4} b^{3} + 28 \, a^{3} b^{4} + 8 \, a^{2} b^{5}\right )} e^{\left (-6 \, d x - 6 \, c\right )} - {\left (3 \, a^{7} + 17 \, a^{6} b + 33 \, a^{5} b^{2} + 27 \, a^{4} b^{3} + 8 \, a^{3} b^{4}\right )} e^{\left (-8 \, d x - 8 \, c\right )} - {\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} e^{\left (-10 \, d x - 10 \, c\right )}\right )} d} \] Input:
integrate(csch(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")
Output:
-15/16*b*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) - 1/4*(8*a^4 - 9*a^3*b - 2*a^2*b^2 + 2*(16*a^4 + 23*a^ 3*b - 27*a^2*b^2 - 4*a*b^3)*e^(-2*d*x - 2*c) + 2*(24*a^4 + 64*a^3*b + 53*a ^2*b^2 - 40*a*b^3 - 8*b^4)*e^(-4*d*x - 4*c) + 2*(16*a^4 + 41*a^3*b + 27*a^ 2*b^2 + 40*a*b^3 + 8*b^4)*e^(-6*d*x - 6*c) + (8*a^4 + 9*a^3*b + 24*a^2*b^2 + 8*a*b^3)*e^(-8*d*x - 8*c))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3 + (3*a ^7 + 17*a^6*b + 33*a^5*b^2 + 27*a^4*b^3 + 8*a^3*b^4)*e^(-2*d*x - 2*c) + 2* (a^7 + 7*a^6*b + 23*a^5*b^2 + 37*a^4*b^3 + 28*a^3*b^4 + 8*a^2*b^5)*e^(-4*d *x - 4*c) - 2*(a^7 + 7*a^6*b + 23*a^5*b^2 + 37*a^4*b^3 + 28*a^3*b^4 + 8*a^ 2*b^5)*e^(-6*d*x - 6*c) - (3*a^7 + 17*a^6*b + 33*a^5*b^2 + 27*a^4*b^3 + 8* a^3*b^4)*e^(-8*d*x - 8*c) - (a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*e^(-10*d *x - 10*c))*d)
Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (118) = 236\).
Time = 0.36 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.75 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {\frac {15 \, b \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {-a b - b^{2}}} - \frac {2 \, {\left (9 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} + 24 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 8 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 27 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 78 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 88 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 16 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 27 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 56 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a^{3} b + 2 \, a^{2} b^{2}\right )}}{{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{2}} - \frac {16}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}}}{8 \, d} \] Input:
integrate(csch(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")
Output:
1/8*(15*b*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt(-a*b - b^2)) - 2*(9*a^3*b*e^(6*d*x + 6*c) + 24*a^2*b^2*e^(6*d*x + 6*c) + 8*a*b^3*e^(6*d*x + 6*c) + 27*a^3*b*e^(4*d* x + 4*c) + 78*a^2*b^2*e^(4*d*x + 4*c) + 88*a*b^3*e^(4*d*x + 4*c) + 16*b^4* e^(4*d*x + 4*c) + 27*a^3*b*e^(2*d*x + 2*c) + 56*a^2*b^2*e^(2*d*x + 2*c) + 8*a*b^3*e^(2*d*x + 2*c) + 9*a^3*b + 2*a^2*b^2)/((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*(a*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)^2) - 16/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*(e^(2*d*x + 2*c) - 1)))/d
Timed out. \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \] Input:
int(1/(sinh(c + d*x)^2*(a + b/cosh(c + d*x)^2)^3),x)
Output:
int(cosh(c + d*x)^6/(sinh(c + d*x)^2*(b + a*cosh(c + d*x)^2)^3), x)
Time = 0.45 (sec) , antiderivative size = 3705, normalized size of antiderivative = 29.40 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:
int(csch(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x)
Output:
(45*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**4 + 120*e**(10*c + 10*d*x)*sqrt(b )*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)* sqrt(a))*a**3*b + 45*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqr t(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**4 + 120*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e* *(c + d*x)*sqrt(a))*a**3*b - 45*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a + b)*log (2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**4 - 120*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x )*a + a + 2*b)*a**3*b + 135*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log( - sq rt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**4 + 720*e** (8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**3*b + 960*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))* a**2*b**2 + 135*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sq rt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**4 + 720*e**(8*c + 8*d*x)*s qrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x )*sqrt(a))*a**3*b + 960*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sq rt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**2*b**2 - 135*e**(8 *c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2...