Integrand size = 23, antiderivative size = 104 \[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {(a+b) (a+5 b) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {\left (3 a^2+6 a b+5 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{6 d}-\frac {b (6 a+5 b) \text {sech}(c+d x)}{3 d}+\frac {b^2 \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{3 d} \]
1/2*(a+b)*(a+5*b)*arctanh(cosh(d*x+c))/d-1/6*(3*a^2+6*a*b+5*b^2)*coth(d*x+ c)*csch(d*x+c)/d-1/3*b*(6*a+5*b)*sech(d*x+c)/d+1/3*b^2*csch(d*x+c)^2*sech( d*x+c)^3/d
Time = 5.08 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.38 \[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=-\frac {\left (b+a \cosh ^2(c+d x)\right )^2 \left (8 b^2+48 b (a+b) \cosh ^2(c+d x)+3 (a+b) \cosh ^3(c+d x) \left ((a+b) \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-4 (a+5 b) \left (\log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )+(a+b) \text {sech}^2\left (\frac {1}{2} (c+d x)\right )\right )\right ) \text {sech}^3(c+d x)}{6 d (a+2 b+a \cosh (2 (c+d x)))^2} \]
-1/6*((b + a*Cosh[c + d*x]^2)^2*(8*b^2 + 48*b*(a + b)*Cosh[c + d*x]^2 + 3* (a + b)*Cosh[c + d*x]^3*((a + b)*Csch[(c + d*x)/2]^2 - 4*(a + 5*b)*(Log[Co sh[(c + d*x)/2]] - Log[Sinh[(c + d*x)/2]]) + (a + b)*Sech[(c + d*x)/2]^2)) *Sech[c + d*x]^3)/(d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)
Time = 0.33 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 26, 4621, 365, 361, 25, 359, 219}
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.
| \(\displaystyle \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \left (a+b \sec (i c+i d x)^2\right )^2}{\sin (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\left (b \sec (i c+i d x)^2+a\right )^2}{\sin (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 4621 |
| \(\displaystyle \frac {\int \frac {\left (a \cosh ^2(c+d x)+b\right )^2 \text {sech}^4(c+d x)}{\left (1-\cosh ^2(c+d x)\right )^2}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {\left (3 a^2 \cosh ^2(c+d x)+b (6 a+5 b)\right ) \text {sech}^2(c+d x)}{\left (1-\cosh ^2(c+d x)\right )^2}d\cosh (c+d x)-\frac {b^2 \text {sech}^3(c+d x)}{3 \left (1-\cosh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 361 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {\left (3 a^2+6 a b+5 b^2\right ) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}-\frac {1}{2} \int -\frac {\left (\left (3 a^2+6 b a+5 b^2\right ) \cosh ^2(c+d x)+2 b (6 a+5 b)\right ) \text {sech}^2(c+d x)}{1-\cosh ^2(c+d x)}d\cosh (c+d x)\right )-\frac {b^2 \text {sech}^3(c+d x)}{3 \left (1-\cosh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \int \frac {\left (\left (3 a^2+6 b a+5 b^2\right ) \cosh ^2(c+d x)+2 b (6 a+5 b)\right ) \text {sech}^2(c+d x)}{1-\cosh ^2(c+d x)}d\cosh (c+d x)+\frac {\left (3 a^2+6 a b+5 b^2\right ) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}\right )-\frac {b^2 \text {sech}^3(c+d x)}{3 \left (1-\cosh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 (a+b) (a+5 b) \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)-2 b (6 a+5 b) \text {sech}(c+d x)\right )+\frac {\left (3 a^2+6 a b+5 b^2\right ) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}\right )-\frac {b^2 \text {sech}^3(c+d x)}{3 \left (1-\cosh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {\left (3 a^2+6 a b+5 b^2\right ) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}+\frac {1}{2} (3 (a+b) (a+5 b) \text {arctanh}(\cosh (c+d x))-2 b (6 a+5 b) \text {sech}(c+d x))\right )-\frac {b^2 \text {sech}^3(c+d x)}{3 \left (1-\cosh ^2(c+d x)\right )}}{d}\) |
(-1/3*(b^2*Sech[c + d*x]^3)/(1 - Cosh[c + d*x]^2) + (((3*a^2 + 6*a*b + 5*b ^2)*Cosh[c + d*x])/(2*(1 - Cosh[c + d*x]^2)) + (3*(a + b)*(a + 5*b)*ArcTan h[Cosh[c + d*x]] - 2*b*(6*a + 5*b)*Sech[c + d*x])/2)/3)/d
3.1.15.3.1 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[((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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
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[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x _Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*Simp[2*b*c^2*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*d^2*(m + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff*x)^n)^p/(ff*x)^(n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/ 2] && IntegerQ[n] && IntegerQ[p]
Time = 70.24 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.21
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+2 a b \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2} \cosh \left (d x +c \right )}-\frac {3}{2 \cosh \left (d x +c \right )}+3 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2} \cosh \left (d x +c \right )^{3}}-\frac {5}{6 \cosh \left (d x +c \right )^{3}}-\frac {5}{2 \cosh \left (d x +c \right )}+5 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(126\) |
| default | \(\frac {a^{2} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+2 a b \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2} \cosh \left (d x +c \right )}-\frac {3}{2 \cosh \left (d x +c \right )}+3 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2} \cosh \left (d x +c \right )^{3}}-\frac {5}{6 \cosh \left (d x +c \right )^{3}}-\frac {5}{2 \cosh \left (d x +c \right )}+5 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(126\) |
| risch | \(-\frac {{\mathrm e}^{d x +c} \left (3 a^{2} {\mathrm e}^{8 d x +8 c}+18 a b \,{\mathrm e}^{8 d x +8 c}+15 b^{2} {\mathrm e}^{8 d x +8 c}+12 a^{2} {\mathrm e}^{6 d x +6 c}+24 a b \,{\mathrm e}^{6 d x +6 c}+20 b^{2} {\mathrm e}^{6 d x +6 c}+18 a^{2} {\mathrm e}^{4 d x +4 c}+12 a b \,{\mathrm e}^{4 d x +4 c}-22 \,{\mathrm e}^{4 d x +4 c} b^{2}+12 a^{2} {\mathrm e}^{2 d x +2 c}+24 a b \,{\mathrm e}^{2 d x +2 c}+20 \,{\mathrm e}^{2 d x +2 c} b^{2}+3 a^{2}+18 a b +15 b^{2}\right )}{3 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{3} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d}+\frac {3 \ln \left ({\mathrm e}^{d x +c}+1\right ) a b}{d}+\frac {5 \ln \left ({\mathrm e}^{d x +c}+1\right ) b^{2}}{2 d}-\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d}-\frac {3 \ln \left ({\mathrm e}^{d x +c}-1\right ) a b}{d}-\frac {5 \ln \left ({\mathrm e}^{d x +c}-1\right ) b^{2}}{2 d}\) | \(318\) |
1/d*(a^2*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+2*a*b*(-1/2/si nh(d*x+c)^2/cosh(d*x+c)-3/2/cosh(d*x+c)+3*arctanh(exp(d*x+c)))+b^2*(-1/2/s inh(d*x+c)^2/cosh(d*x+c)^3-5/6/cosh(d*x+c)^3-5/2/cosh(d*x+c)+5*arctanh(exp (d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 2930 vs. \(2 (96) = 192\).
Time = 0.28 (sec) , antiderivative size = 2930, normalized size of antiderivative = 28.17 \[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\text {Too large to display} \]
-1/6*(6*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^9 + 54*(a^2 + 6*a*b + 5*b^2)*c osh(d*x + c)*sinh(d*x + c)^8 + 6*(a^2 + 6*a*b + 5*b^2)*sinh(d*x + c)^9 + 8 *(3*a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^7 + 8*(27*(a^2 + 6*a*b + 5*b^2)*cos h(d*x + c)^2 + 3*a^2 + 6*a*b + 5*b^2)*sinh(d*x + c)^7 + 56*(9*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^3 + (3*a^2 + 6*a*b + 5*b^2)*cosh(d*x + c))*sinh(d* x + c)^6 + 4*(9*a^2 + 6*a*b - 11*b^2)*cosh(d*x + c)^5 + 4*(189*(a^2 + 6*a* b + 5*b^2)*cosh(d*x + c)^4 + 42*(3*a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^2 + 9*a^2 + 6*a*b - 11*b^2)*sinh(d*x + c)^5 + 4*(189*(a^2 + 6*a*b + 5*b^2)*cos h(d*x + c)^5 + 70*(3*a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^3 + 5*(9*a^2 + 6*a *b - 11*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 + 8*(3*a^2 + 6*a*b + 5*b^2)*co sh(d*x + c)^3 + 8*(63*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^6 + 35*(3*a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^4 + 5*(9*a^2 + 6*a*b - 11*b^2)*cosh(d*x + c)^ 2 + 3*a^2 + 6*a*b + 5*b^2)*sinh(d*x + c)^3 + 8*(27*(a^2 + 6*a*b + 5*b^2)*c osh(d*x + c)^7 + 21*(3*a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^5 + 5*(9*a^2 + 6 *a*b - 11*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 + 6*a*b + 5*b^2)*cosh(d*x + c))* sinh(d*x + c)^2 + 6*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c) - 3*((a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^10 + 10*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)*sinh(d* x + c)^9 + (a^2 + 6*a*b + 5*b^2)*sinh(d*x + c)^10 + (a^2 + 6*a*b + 5*b^2)* cosh(d*x + c)^8 + (45*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^2 + a^2 + 6*a*b + 5*b^2)*sinh(d*x + c)^8 + 8*(15*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^3 ...
\[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2} \operatorname {csch}^{3}{\left (c + d x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (96) = 192\).
Time = 0.20 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.40 \[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {1}{6} \, b^{2} {\left (\frac {15 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {15 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {2 \, {\left (15 \, e^{\left (-d x - c\right )} + 20 \, e^{\left (-3 \, d x - 3 \, c\right )} - 22 \, e^{\left (-5 \, d x - 5 \, c\right )} + 20 \, e^{\left (-7 \, d x - 7 \, c\right )} + 15 \, e^{\left (-9 \, d x - 9 \, c\right )}\right )}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-4 \, d x - 4 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + a b {\left (\frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (3 \, e^{\left (-d x - c\right )} - 2 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )}\right )}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} - e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac {1}{2} \, a^{2} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \]
1/6*b^2*(15*log(e^(-d*x - c) + 1)/d - 15*log(e^(-d*x - c) - 1)/d - 2*(15*e ^(-d*x - c) + 20*e^(-3*d*x - 3*c) - 22*e^(-5*d*x - 5*c) + 20*e^(-7*d*x - 7 *c) + 15*e^(-9*d*x - 9*c))/(d*(e^(-2*d*x - 2*c) - 2*e^(-4*d*x - 4*c) - 2*e ^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1))) + a*b*(3*lo g(e^(-d*x - c) + 1)/d - 3*log(e^(-d*x - c) - 1)/d + 2*(3*e^(-d*x - c) - 2* e^(-3*d*x - 3*c) + 3*e^(-5*d*x - 5*c))/(d*(e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) - e^(-6*d*x - 6*c) - 1))) + 1/2*a^2*(log(e^(-d*x - c) + 1)/d - log(e^ (-d*x - c) - 1)/d + 2*(e^(-d*x - c) + e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c) - 1)))
Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (96) = 192\).
Time = 0.30 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.19 \[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {3 \, {\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - 3 \, {\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {12 \, {\left (a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 2 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4} - \frac {16 \, {\left (3 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 3 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 2 \, b^{2}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3}}}{12 \, d} \]
1/12*(3*(a^2 + 6*a*b + 5*b^2)*log(e^(d*x + c) + e^(-d*x - c) + 2) - 3*(a^2 + 6*a*b + 5*b^2)*log(e^(d*x + c) + e^(-d*x - c) - 2) - 12*(a^2*(e^(d*x + c) + e^(-d*x - c)) + 2*a*b*(e^(d*x + c) + e^(-d*x - c)) + b^2*(e^(d*x + c) + e^(-d*x - c)))/((e^(d*x + c) + e^(-d*x - c))^2 - 4) - 16*(3*a*b*(e^(d*x + c) + e^(-d*x - c))^2 + 3*b^2*(e^(d*x + c) + e^(-d*x - c))^2 + 2*b^2)/(e ^(d*x + c) + e^(-d*x - c))^3)/d
Time = 2.19 (sec) , antiderivative size = 316, normalized size of antiderivative = 3.04 \[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^2\,\sqrt {-d^2}+5\,b^2\,\sqrt {-d^2}+6\,a\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {a^4+12\,a^3\,b+46\,a^2\,b^2+60\,a\,b^3+25\,b^4}}\right )\,\sqrt {a^4+12\,a^3\,b+46\,a^2\,b^2+60\,a\,b^3+25\,b^4}}{\sqrt {-d^2}}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^2+2\,a\,b+b^2\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}+\frac {8\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (a^2+2\,a\,b+b^2\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {4\,{\mathrm {e}}^{c+d\,x}\,\left (b^2+a\,b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]
(atan((exp(d*x)*exp(c)*(a^2*(-d^2)^(1/2) + 5*b^2*(-d^2)^(1/2) + 6*a*b*(-d^ 2)^(1/2)))/(d*(60*a*b^3 + 12*a^3*b + a^4 + 25*b^4 + 46*a^2*b^2)^(1/2)))*(6 0*a*b^3 + 12*a^3*b + a^4 + 25*b^4 + 46*a^2*b^2)^(1/2))/(-d^2)^(1/2) - (exp (c + d*x)*(2*a*b + a^2 + b^2))/(d*(exp(2*c + 2*d*x) - 1)) + (8*b^2*exp(c + d*x))/(3*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) - (2*exp(c + d*x)*(2*a*b + a^2 + b^2))/(d*(exp(4*c + 4*d*x) - 2*exp(2* c + 2*d*x) + 1)) - (4*exp(c + d*x)*(a*b + b^2))/(d*(exp(2*c + 2*d*x) + 1)) - (8*b^2*exp(c + d*x))/(3*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1))