Integrand size = 23, antiderivative size = 112 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {15 \sqrt {b} \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{7/2} d}-\frac {\coth (c+d x)}{a^3 d}-\frac {b \tanh (c+d x)}{4 a^2 d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {7 b \tanh (c+d x)}{8 a^3 d \left (a+b \tanh ^2(c+d x)\right )} \] Output:
-15/8*b^(1/2)*arctan(b^(1/2)*tanh(d*x+c)/a^(1/2))/a^(7/2)/d-coth(d*x+c)/a^ 3/d-1/4*b*tanh(d*x+c)/a^2/d/(a+b*tanh(d*x+c)^2)^2-7/8*b*tanh(d*x+c)/a^3/d/ (a+b*tanh(d*x+c)^2)
Time = 1.18 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.97 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {-15 \sqrt {b} \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )-8 \sqrt {a} \coth (c+d x)-\frac {\sqrt {a} b (9 a-7 b+(9 a+7 b) \cosh (2 (c+d x))) \sinh (2 (c+d x))}{(a-b+(a+b) \cosh (2 (c+d x)))^2}}{8 a^{7/2} d} \] Input:
Integrate[Csch[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^3,x]
Output:
(-15*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] - 8*Sqrt[a]*Coth[c + d*x] - (Sqrt[a]*b*(9*a - 7*b + (9*a + 7*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)])^2)/(8*a^(7/2)*d)
Time = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 25, 4146, 253, 253, 264, 218}
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.
| \(\displaystyle \int \frac {\text {csch}^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\sin (i c+i d x)^2 \left (a-b \tan (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\sin (i c+i d x)^2 \left (a-b \tan (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4146 |
| \(\displaystyle \frac {\int \frac {\coth ^2(c+d x)}{\left (b \tanh ^2(c+d x)+a\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\frac {5 \int \frac {\coth ^2(c+d x)}{\left (b \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{4 a}+\frac {\coth (c+d x)}{4 a \left (a+b \tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \int \frac {\coth ^2(c+d x)}{b \tanh ^2(c+d x)+a}d\tanh (c+d x)}{2 a}+\frac {\coth (c+d x)}{2 a \left (a+b \tanh ^2(c+d x)\right )}\right )}{4 a}+\frac {\coth (c+d x)}{4 a \left (a+b \tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (-\frac {b \int \frac {1}{b \tanh ^2(c+d x)+a}d\tanh (c+d x)}{a}-\frac {\coth (c+d x)}{a}\right )}{2 a}+\frac {\coth (c+d x)}{2 a \left (a+b \tanh ^2(c+d x)\right )}\right )}{4 a}+\frac {\coth (c+d x)}{4 a \left (a+b \tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\coth (c+d x)}{a}\right )}{2 a}+\frac {\coth (c+d x)}{2 a \left (a+b \tanh ^2(c+d x)\right )}\right )}{4 a}+\frac {\coth (c+d x)}{4 a \left (a+b \tanh ^2(c+d x)\right )^2}}{d}\) |
Input:
Int[Csch[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^3,x]
Output:
(Coth[c + d*x]/(4*a*(a + b*Tanh[c + d*x]^2)^2) + (5*((3*(-((Sqrt[b]*ArcTan [(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/a^(3/2)) - Coth[c + d*x]/a))/(2*a) + Co th[c + d*x]/(2*a*(a + b*Tanh[c + d*x]^2))))/(4*a))/d
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[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[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_ )])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[c*(ff^(m + 1)/f) Subst[Int[x^m*((a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)^(m/ 2 + 1)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x ] && IntegerQ[m/2]
Leaf count of result is larger than twice the leaf count of optimal. \(305\) vs. \(2(98)=196\).
Time = 10.32 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.73
| method | result | size |
| derivativedivides | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{3}}-\frac {1}{2 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 b \left (\frac {-\frac {9 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8}+\left (-\frac {27 a}{8}-\frac {7 b}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {27 a}{8}-\frac {7 b}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-\frac {9 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{8}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a \right )^{2}}+\frac {15 a \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{8}\right )}{a^{3}}}{d}\) | \(306\) |
| default | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{3}}-\frac {1}{2 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 b \left (\frac {-\frac {9 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8}+\left (-\frac {27 a}{8}-\frac {7 b}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {27 a}{8}-\frac {7 b}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-\frac {9 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{8}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a \right )^{2}}+\frac {15 a \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{8}\right )}{a^{3}}}{d}\) | \(306\) |
| risch | \(-\frac {8 \,{\mathrm e}^{8 d x +8 c} a^{4}+23 \,{\mathrm e}^{8 d x +8 c} a^{3} b +45 \,{\mathrm e}^{8 d x +8 c} a^{2} b^{2}+45 \,{\mathrm e}^{8 d x +8 c} a \,b^{3}+15 \,{\mathrm e}^{8 d x +8 c} b^{4}+32 \,{\mathrm e}^{6 d x +6 c} a^{4}+46 b \,a^{3} {\mathrm e}^{6 d x +6 c}-90 \,{\mathrm e}^{6 d x +6 c} a \,b^{3}-60 \,{\mathrm e}^{6 d x +6 c} b^{4}+48 \,{\mathrm e}^{4 d x +4 c} a^{4}+64 \,{\mathrm e}^{4 d x +4 c} a^{3} b +10 \,{\mathrm e}^{4 d x +4 c} a^{2} b^{2}+100 \,{\mathrm e}^{4 d x +4 c} a \,b^{3}+90 \,{\mathrm e}^{4 d x +4 c} b^{4}+32 \,{\mathrm e}^{2 d x +2 c} a^{4}+82 \,{\mathrm e}^{2 d x +2 c} a^{3} b -110 \,{\mathrm e}^{2 d x +2 c} a \,b^{3}-60 \,{\mathrm e}^{2 d x +2 c} b^{4}+8 a^{4}+41 a^{3} b +73 a^{2} b^{2}+55 a \,b^{3}+15 b^{4}}{4 \left ({\mathrm e}^{2 d x +2 c}-1\right ) \left (a^{2}+2 a b +b^{2}\right ) \left ({\mathrm e}^{4 d x +4 c} a +b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 \,{\mathrm e}^{2 d x +2 c} b +a +b \right )^{2} a^{3} d}+\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{16 a^{4} d}-\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{16 a^{4} d}\) | \(473\) |
Input:
int(csch(d*x+c)^2/(a+tanh(d*x+c)^2*b)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/2/a^3*tanh(1/2*d*x+1/2*c)-1/2/a^3/tanh(1/2*d*x+1/2*c)+2*b/a^3*((-9 /8*a*tanh(1/2*d*x+1/2*c)^7+(-27/8*a-7/2*b)*tanh(1/2*d*x+1/2*c)^5+(-27/8*a- 7/2*b)*tanh(1/2*d*x+1/2*c)^3-9/8*tanh(1/2*d*x+1/2*c)*a)/(tanh(1/2*d*x+1/2* c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*b*tanh(1/2*d*x+1/2*c)^2+a)^2+15/8*a*(1/ 2*(a+((a+b)*b)^(1/2)+b)/a/((a+b)*b)^(1/2)/((2*((a+b)*b)^(1/2)+a+2*b)*a)^(1 /2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*((a+b)*b)^(1/2)+a+2*b)*a)^(1/2))-1/2* (-a+((a+b)*b)^(1/2)-b)/a/((a+b)*b)^(1/2)/((2*((a+b)*b)^(1/2)-a-2*b)*a)^(1/ 2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*((a+b)*b)^(1/2)-a-2*b)*a)^(1/2)))))
Leaf count of result is larger than twice the leaf count of optimal. 3995 vs. \(2 (98) = 196\).
Time = 0.24 (sec) , antiderivative size = 8312, normalized size of antiderivative = 74.21 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^2/(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {\operatorname {csch}^{2}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3}}\, dx \] Input:
integrate(csch(d*x+c)**2/(a+b*tanh(d*x+c)**2)**3,x)
Output:
Integral(csch(c + d*x)**2/(a + b*tanh(c + d*x)**2)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (98) = 196\).
Time = 0.29 (sec) , antiderivative size = 478, normalized size of antiderivative = 4.27 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {8 \, a^{4} + 41 \, a^{3} b + 73 \, a^{2} b^{2} + 55 \, a b^{3} + 15 \, b^{4} + 2 \, {\left (16 \, a^{4} + 41 \, a^{3} b - 55 \, a b^{3} - 30 \, b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (24 \, a^{4} + 32 \, a^{3} b + 5 \, a^{2} b^{2} + 50 \, a b^{3} + 45 \, b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 2 \, {\left (16 \, a^{4} + 23 \, a^{3} b - 45 \, a b^{3} - 30 \, b^{4}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (8 \, a^{4} + 23 \, a^{3} b + 45 \, a^{2} b^{2} + 45 \, a b^{3} + 15 \, b^{4}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{4 \, {\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4} + {\left (3 \, a^{7} + 4 \, a^{6} b - 6 \, a^{5} b^{2} - 12 \, a^{4} b^{3} - 5 \, a^{3} b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (a^{7} + 2 \, a^{5} b^{2} + 8 \, a^{4} b^{3} + 5 \, a^{3} b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 2 \, {\left (a^{7} + 2 \, a^{5} b^{2} + 8 \, a^{4} b^{3} + 5 \, a^{3} b^{4}\right )} e^{\left (-6 \, d x - 6 \, c\right )} - {\left (3 \, a^{7} + 4 \, a^{6} b - 6 \, a^{5} b^{2} - 12 \, a^{4} b^{3} - 5 \, a^{3} b^{4}\right )} e^{\left (-8 \, d x - 8 \, c\right )} - {\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} e^{\left (-10 \, d x - 10 \, c\right )}\right )} d} + \frac {15 \, b \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{3} d} \] Input:
integrate(csch(d*x+c)^2/(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")
Output:
-1/4*(8*a^4 + 41*a^3*b + 73*a^2*b^2 + 55*a*b^3 + 15*b^4 + 2*(16*a^4 + 41*a ^3*b - 55*a*b^3 - 30*b^4)*e^(-2*d*x - 2*c) + 2*(24*a^4 + 32*a^3*b + 5*a^2* b^2 + 50*a*b^3 + 45*b^4)*e^(-4*d*x - 4*c) + 2*(16*a^4 + 23*a^3*b - 45*a*b^ 3 - 30*b^4)*e^(-6*d*x - 6*c) + (8*a^4 + 23*a^3*b + 45*a^2*b^2 + 45*a*b^3 + 15*b^4)*e^(-8*d*x - 8*c))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b ^4 + (3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*e^(-2*d*x - 2* c) + 2*(a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*e^(-4*d*x - 4*c) - 2*(a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*e^(-6*d*x - 6*c) - (3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*e^(-8*d*x - 8*c) - (a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*e^(-10*d*x - 10*c))*d) + 15/8*b*arctan(1/ 2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/(sqrt(a*b)*a^3*d)
Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (98) = 196\).
Time = 0.60 (sec) , antiderivative size = 351, normalized size of antiderivative = 3.13 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {\frac {15 \, b \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} a^{3}} - \frac {2 \, {\left (9 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 13 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 7 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 27 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 13 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 21 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 27 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 25 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 23 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 21 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a^{3} b + 25 \, a^{2} b^{2} + 23 \, a b^{3} + 7 \, b^{4}\right )}}{{\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}^{2}} + \frac {16}{a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}}}{8 \, d} \] Input:
integrate(csch(d*x+c)^2/(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")
Output:
-1/8*(15*b*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/sqrt (a*b))/(sqrt(a*b)*a^3) - 2*(9*a^3*b*e^(6*d*x + 6*c) + 3*a^2*b^2*e^(6*d*x + 6*c) - 13*a*b^3*e^(6*d*x + 6*c) - 7*b^4*e^(6*d*x + 6*c) + 27*a^3*b*e^(4*d *x + 4*c) + 3*a^2*b^2*e^(4*d*x + 4*c) + 13*a*b^3*e^(4*d*x + 4*c) + 21*b^4* e^(4*d*x + 4*c) + 27*a^3*b*e^(2*d*x + 2*c) + 25*a^2*b^2*e^(2*d*x + 2*c) - 23*a*b^3*e^(2*d*x + 2*c) - 21*b^4*e^(2*d*x + 2*c) + 9*a^3*b + 25*a^2*b^2 + 23*a*b^3 + 7*b^4)/((a^5 + 2*a^4*b + a^3*b^2)*(a*e^(4*d*x + 4*c) + b*e^(4* d*x + 4*c) + 2*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + a + b)^2) + 16/(a ^3*(e^(2*d*x + 2*c) - 1)))/d
Timed out. \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \] Input:
int(1/(sinh(c + d*x)^2*(a + b*tanh(c + d*x)^2)^3),x)
Output:
int(1/(sinh(c + d*x)^2*(a + b*tanh(c + d*x)^2)^3), x)
Time = 0.44 (sec) , antiderivative size = 3001, normalized size of antiderivative = 26.79 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:
int(csch(d*x+c)^2/(a+b*tanh(d*x+c)^2)^3,x)
Output:
( - 45*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**4 - 60*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a)*atan((e**( c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**3*b + 90*e**(10*c + 10*d*x)*sq rt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**2*b**2 + 180*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a*b**3 + 75*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a)*atan((e* *(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*b**4 - 135*e**(8*c + 8*d*x)*sqr t(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**4 + 180 *e**(8*c + 8*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b) )/sqrt(a))*a**3*b + 390*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x )*sqrt(a + b) - sqrt(b))/sqrt(a))*a**2*b**2 - 300*e**(8*c + 8*d*x)*sqrt(b) *sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a*b**3 - 375*e **(8*c + 8*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/ sqrt(a))*b**4 - 90*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqr t(a + b) - sqrt(b))/sqrt(a))*a**4 + 240*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*a tan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**3*b - 420*e**(6*c + 6 *d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a **2*b**2 + 750*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*b**4 + 90*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*atan(( e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**4 - 240*e**(4*c + 4*d*x...