Integrand size = 23, antiderivative size = 144 \[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {(a+b)^2 (a+7 b) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {(a+b)^2 (a+7 b) \text {sech}(c+d x)}{2 d}-\frac {b \left (6 a^2+15 a b+7 b^2\right ) \text {sech}^3(c+d x)}{6 d}-\frac {b^2 (5 a+7 b) \text {sech}^5(c+d x)}{10 d}-\frac {(a+b) \left (b+a \cosh ^2(c+d x)\right )^2 \text {csch}^2(c+d x) \text {sech}^5(c+d x)}{2 d} \]
1/2*(a+b)^2*(a+7*b)*arctanh(cosh(d*x+c))/d-1/2*(a+b)^2*(a+7*b)*sech(d*x+c) /d-1/6*b*(6*a^2+15*a*b+7*b^2)*sech(d*x+c)^3/d-1/10*b^2*(5*a+7*b)*sech(d*x+ c)^5/d-1/2*(a+b)*(b+a*cosh(d*x+c)^2)^2*csch(d*x+c)^2*sech(d*x+c)^5/d
Time = 9.97 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.56 \[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=-\frac {\left (b+a \cosh ^2(c+d x)\right )^3 \text {sech}^6(c+d x) \left (\left (150 a^3+270 a^2 b-30 a b^2-206 b^3+10 \left (9 a^3+45 a^2 b+75 a b^2+35 b^3\right ) \cosh (4 (c+d x))+15 (a+b)^2 (a+7 b) \cosh (6 (c+d x))\right ) \coth (c+d x) \text {csch}(c+d x)-480 (a+b)^2 (a+7 b) \cosh ^6(c+d x) \left (\log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )+\frac {3}{4} \left (75 a^3+195 a^2 b+165 a b^2+77 b^3\right ) \text {csch}^3(c+d x) \sinh (4 (c+d x))\right )}{120 d (a+2 b+a \cosh (2 (c+d x)))^3} \]
-1/120*((b + a*Cosh[c + d*x]^2)^3*Sech[c + d*x]^6*((150*a^3 + 270*a^2*b - 30*a*b^2 - 206*b^3 + 10*(9*a^3 + 45*a^2*b + 75*a*b^2 + 35*b^3)*Cosh[4*(c + d*x)] + 15*(a + b)^2*(a + 7*b)*Cosh[6*(c + d*x)])*Coth[c + d*x]*Csch[c + d*x] - 480*(a + b)^2*(a + 7*b)*Cosh[c + d*x]^6*(Log[Cosh[(c + d*x)/2]] - L og[Sinh[(c + d*x)/2]]) + (3*(75*a^3 + 195*a^2*b + 165*a*b^2 + 77*b^3)*Csch [c + d*x]^3*Sinh[4*(c + d*x)])/4))/(d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)
Time = 0.37 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 26, 4621, 370, 437, 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.
| \(\displaystyle \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \left (a+b \sec (i c+i d x)^2\right )^3}{\sin (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\left (b \sec (i c+i d x)^2+a\right )^3}{\sin (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 4621 |
| \(\displaystyle \frac {\int \frac {\left (a \cosh ^2(c+d x)+b\right )^3 \text {sech}^6(c+d x)}{\left (1-\cosh ^2(c+d x)\right )^2}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 370 |
| \(\displaystyle \frac {\frac {1}{2} \int \frac {\left (a \cosh ^2(c+d x)+b\right ) \left (a (a+3 b) \cosh ^2(c+d x)+b (5 a+7 b)\right ) \text {sech}^6(c+d x)}{1-\cosh ^2(c+d x)}d\cosh (c+d x)+\frac {(a+b) \text {sech}^5(c+d x) \left (a \cosh ^2(c+d x)+b\right )^2}{2 \left (1-\cosh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 437 |
| \(\displaystyle \frac {\frac {1}{2} \int \left (b^2 (5 a+7 b) \text {sech}^6(c+d x)+b \left (6 a^2+15 b a+7 b^2\right ) \text {sech}^4(c+d x)+(a+b)^2 (a+7 b) \text {sech}^2(c+d x)-\frac {(a+b)^2 (a+7 b)}{\cosh ^2(c+d x)-1}\right )d\cosh (c+d x)+\frac {(a+b) \text {sech}^5(c+d x) \left (a \cosh ^2(c+d x)+b\right )^2}{2 \left (1-\cosh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\frac {1}{3} b \left (6 a^2+15 a b+7 b^2\right ) \text {sech}^3(c+d x)+(a+b)^2 (a+7 b) \text {arctanh}(\cosh (c+d x))-\frac {1}{5} b^2 (5 a+7 b) \text {sech}^5(c+d x)-(a+b)^2 (a+7 b) \text {sech}(c+d x)\right )+\frac {(a+b) \text {sech}^5(c+d x) \left (a \cosh ^2(c+d x)+b\right )^2}{2 \left (1-\cosh ^2(c+d x)\right )}}{d}\) |
(((a + b)*(b + a*Cosh[c + d*x]^2)^2*Sech[c + d*x]^5)/(2*(1 - Cosh[c + d*x] ^2)) + ((a + b)^2*(a + 7*b)*ArcTanh[Cosh[c + d*x]] - (a + b)^2*(a + 7*b)*S ech[c + d*x] - (b*(6*a^2 + 15*a*b + 7*b^2)*Sech[c + d*x]^3)/3 - (b^2*(5*a + 7*b)*Sech[c + d*x]^5)/5)/2)/d
3.1.23.3.1 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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(a*b*e*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(e*x) ^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*Simp[c*(b*c*2*(p + 1) + (b*c - a *d)*(m + 1)) + d*(b*c*2*(p + 1) + (b*c - a*d)*(m + 2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*( a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f , g, m}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]
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 = 0.40 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.33
\[\frac {a^{3} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+3 a^{2} b \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2} \cosh \left (d x +c \right )}-\frac {3}{2 \cosh \left (d x +c \right )}+3 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+3 a \,b^{2} \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2} \cosh \left (d x +c \right )^{3}}-\frac {5}{6 \cosh \left (d x +c \right )^{3}}-\frac {5}{2 \cosh \left (d x +c \right )}+5 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b^{3} \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2} \cosh \left (d x +c \right )^{5}}-\frac {7}{10 \cosh \left (d x +c \right )^{5}}-\frac {7}{6 \cosh \left (d x +c \right )^{3}}-\frac {7}{2 \cosh \left (d x +c \right )}+7 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )}{d}\]
1/d*(a^3*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+3*a^2*b*(-1/2/ sinh(d*x+c)^2/cosh(d*x+c)-3/2/cosh(d*x+c)+3*arctanh(exp(d*x+c)))+3*a*b^2*( -1/2/sinh(d*x+c)^2/cosh(d*x+c)^3-5/6/cosh(d*x+c)^3-5/2/cosh(d*x+c)+5*arcta nh(exp(d*x+c)))+b^3*(-1/2/sinh(d*x+c)^2/cosh(d*x+c)^5-7/10/cosh(d*x+c)^5-7 /6/cosh(d*x+c)^3-7/2/cosh(d*x+c)+7*arctanh(exp(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 6717 vs. \(2 (134) = 268\).
Time = 0.32 (sec) , antiderivative size = 6717, normalized size of antiderivative = 46.65 \[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\text {Too large to display} \]
\[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3} \operatorname {csch}^{3}{\left (c + d x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (134) = 268\).
Time = 0.20 (sec) , antiderivative size = 556, normalized size of antiderivative = 3.86 \[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {1}{30} \, b^{3} {\left (\frac {105 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {105 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {2 \, {\left (105 \, e^{\left (-d x - c\right )} + 350 \, e^{\left (-3 \, d x - 3 \, c\right )} + 231 \, e^{\left (-5 \, d x - 5 \, c\right )} - 412 \, e^{\left (-7 \, d x - 7 \, c\right )} + 231 \, e^{\left (-9 \, d x - 9 \, c\right )} + 350 \, e^{\left (-11 \, d x - 11 \, c\right )} + 105 \, e^{\left (-13 \, d x - 13 \, c\right )}\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} - 5 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 3 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}}\right )} + \frac {1}{2} \, a 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 )} + \frac {3}{2} \, a^{2} 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^{3} {\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 )} \]
1/30*b^3*(105*log(e^(-d*x - c) + 1)/d - 105*log(e^(-d*x - c) - 1)/d - 2*(1 05*e^(-d*x - c) + 350*e^(-3*d*x - 3*c) + 231*e^(-5*d*x - 5*c) - 412*e^(-7* d*x - 7*c) + 231*e^(-9*d*x - 9*c) + 350*e^(-11*d*x - 11*c) + 105*e^(-13*d* x - 13*c))/(d*(3*e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) - 5*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 3*e^(-12*d*x - 12*c) + e^(-14* d*x - 14*c) + 1))) + 1/2*a*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 - 1 0*c) + 1))) + 3/2*a^2*b*(3*log(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^3*(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. 341 vs. \(2 (134) = 268\).
Time = 0.31 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.37 \[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {15 \, {\left (a^{3} + 9 \, a^{2} b + 15 \, a b^{2} + 7 \, b^{3}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - 15 \, {\left (a^{3} + 9 \, a^{2} b + 15 \, a b^{2} + 7 \, b^{3}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {60 \, {\left (a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 3 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 3 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4} - \frac {8 \, {\left (45 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} + 90 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} + 45 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} + 60 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 40 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 48 \, b^{3}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5}}}{60 \, d} \]
1/60*(15*(a^3 + 9*a^2*b + 15*a*b^2 + 7*b^3)*log(e^(d*x + c) + e^(-d*x - c) + 2) - 15*(a^3 + 9*a^2*b + 15*a*b^2 + 7*b^3)*log(e^(d*x + c) + e^(-d*x - c) - 2) - 60*(a^3*(e^(d*x + c) + e^(-d*x - c)) + 3*a^2*b*(e^(d*x + c) + e^ (-d*x - c)) + 3*a*b^2*(e^(d*x + c) + e^(-d*x - c)) + b^3*(e^(d*x + c) + e^ (-d*x - c)))/((e^(d*x + c) + e^(-d*x - c))^2 - 4) - 8*(45*a^2*b*(e^(d*x + c) + e^(-d*x - c))^4 + 90*a*b^2*(e^(d*x + c) + e^(-d*x - c))^4 + 45*b^3*(e ^(d*x + c) + e^(-d*x - c))^4 + 60*a*b^2*(e^(d*x + c) + e^(-d*x - c))^2 + 4 0*b^3*(e^(d*x + c) + e^(-d*x - c))^2 + 48*b^3)/(e^(d*x + c) + e^(-d*x - c) )^5)/d
Time = 2.30 (sec) , antiderivative size = 536, normalized size of antiderivative = 3.72 \[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^3\,\sqrt {-d^2}+7\,b^3\,\sqrt {-d^2}+15\,a\,b^2\,\sqrt {-d^2}+9\,a^2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {a^6+18\,a^5\,b+111\,a^4\,b^2+284\,a^3\,b^3+351\,a^2\,b^4+210\,a\,b^5+49\,b^6}}\right )\,\sqrt {a^6+18\,a^5\,b+111\,a^4\,b^2+284\,a^3\,b^3+351\,a^2\,b^4+210\,a\,b^5+49\,b^6}}{\sqrt {-d^2}}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,{\mathrm {e}}^{c+d\,x}\,\left (2\,b^3+3\,a\,b^2\right )}{3\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {6\,{\mathrm {e}}^{c+d\,x}\,\left (a^2\,b+2\,a\,b^2+b^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {64\,b^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {8\,{\mathrm {e}}^{c+d\,x}\,\left (15\,a\,b^2-2\,b^3\right )}{15\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {32\,b^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \]
(atan((exp(d*x)*exp(c)*(a^3*(-d^2)^(1/2) + 7*b^3*(-d^2)^(1/2) + 15*a*b^2*( -d^2)^(1/2) + 9*a^2*b*(-d^2)^(1/2)))/(d*(210*a*b^5 + 18*a^5*b + a^6 + 49*b ^6 + 351*a^2*b^4 + 284*a^3*b^3 + 111*a^4*b^2)^(1/2)))*(210*a*b^5 + 18*a^5* b + a^6 + 49*b^6 + 351*a^2*b^4 + 284*a^3*b^3 + 111*a^4*b^2)^(1/2))/(-d^2)^ (1/2) - (2*exp(c + d*x)*(3*a*b^2 + 3*a^2*b + a^3 + b^3))/(d*(exp(4*c + 4*d *x) - 2*exp(2*c + 2*d*x) + 1)) - (8*exp(c + d*x)*(3*a*b^2 + 2*b^3))/(3*d*( 2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - (6*exp(c + d*x)*(2*a*b^2 + a ^2*b + b^3))/(d*(exp(2*c + 2*d*x) + 1)) + (64*b^3*exp(c + d*x))/(5*d*(4*ex p(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x ) + 1)) + (8*exp(c + d*x)*(15*a*b^2 - 2*b^3))/(15*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) - (32*b^3*exp(c + d*x))/(5*d*( 5*exp(2*c + 2*d*x) + 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) + 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) + 1)) - (exp(c + d*x)*(3*a*b^2 + 3*a^2*b + a^3 + b^3))/(d*(exp(2*c + 2*d*x) - 1))