Integrand size = 23, antiderivative size = 96 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {\left (3 a^2+2 a b+3 b^2\right ) \arctan (\sinh (c+d x))}{8 d}+\frac {3 \left (a^2-b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {(a-b) \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \tanh (c+d x)}{4 d} \]
1/8*(3*a^2+2*a*b+3*b^2)*arctan(sinh(d*x+c))/d+3/8*(a^2-b^2)*sech(d*x+c)*ta nh(d*x+c)/d+1/4*(a-b)*sech(d*x+c)^3*(a+b*sinh(d*x+c)^2)*tanh(d*x+c)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 6.16 (sec) , antiderivative size = 303, normalized size of antiderivative = 3.16 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=-\frac {\text {csch}^3(c+d x) \left (128 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {9}{2};-\sinh ^2(c+d x)\right ) \sinh ^6(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2+128 \, _4F_3\left (\frac {3}{2},2,2,2;1,1,\frac {9}{2};-\sinh ^2(c+d x)\right ) \sinh ^6(c+d x) \left (7 a^2+12 a b \sinh ^2(c+d x)+5 b^2 \sinh ^4(c+d x)\right )+35 \left (3375 a^2+a (657 a+4643 b+607 b \cosh (2 (c+d x))) \sinh ^2(c+d x)+1947 b^2 \sinh ^4(c+d x)+485 b^2 \sinh ^6(c+d x)\right )-\frac {105 \text {arctanh}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \left (1125 a^2+2 a (297 a+875 b) \sinh ^2(c+d x)+\left (37 a^2+988 a b+649 b^2\right ) \sinh ^4(c+d x)+2 b (11 a+189 b) \sinh ^6(c+d x)+9 b^2 \sinh ^8(c+d x)\right )}{\sqrt {-\sinh ^2(c+d x)}}\right )}{6720 d} \]
-1/6720*(Csch[c + d*x]^3*(128*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6*(a + b*Sinh[c + d*x]^2)^2 + 128 *HypergeometricPFQ[{3/2, 2, 2, 2}, {1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6*(7*a^2 + 12*a*b*Sinh[c + d*x]^2 + 5*b^2*Sinh[c + d*x]^4) + 35*(337 5*a^2 + a*(657*a + 4643*b + 607*b*Cosh[2*(c + d*x)])*Sinh[c + d*x]^2 + 194 7*b^2*Sinh[c + d*x]^4 + 485*b^2*Sinh[c + d*x]^6) - (105*ArcTanh[Sqrt[-Sinh [c + d*x]^2]]*(1125*a^2 + 2*a*(297*a + 875*b)*Sinh[c + d*x]^2 + (37*a^2 + 988*a*b + 649*b^2)*Sinh[c + d*x]^4 + 2*b*(11*a + 189*b)*Sinh[c + d*x]^6 + 9*b^2*Sinh[c + d*x]^8))/Sqrt[-Sinh[c + d*x]^2]))/d
Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 3669, 315, 298, 216}
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.
| \(\displaystyle \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-b \sin (i c+i d x)^2\right )^2}{\cos (i c+i d x)^5}dx\) |
| \(\Big \downarrow \) 3669 |
| \(\displaystyle \frac {\int \frac {\left (b \sinh ^2(c+d x)+a\right )^2}{\left (\sinh ^2(c+d x)+1\right )^3}d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {\frac {1}{4} \int \frac {b (a+3 b) \sinh ^2(c+d x)+a (3 a+b)}{\left (\sinh ^2(c+d x)+1\right )^2}d\sinh (c+d x)+\frac {(a-b) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )}{4 \left (\sinh ^2(c+d x)+1\right )^2}}{d}\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} \left (3 a^2+2 a b+3 b^2\right ) \int \frac {1}{\sinh ^2(c+d x)+1}d\sinh (c+d x)+\frac {3 \left (a^2-b^2\right ) \sinh (c+d x)}{2 \left (\sinh ^2(c+d x)+1\right )}\right )+\frac {(a-b) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )}{4 \left (\sinh ^2(c+d x)+1\right )^2}}{d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} \left (3 a^2+2 a b+3 b^2\right ) \arctan (\sinh (c+d x))+\frac {3 \left (a^2-b^2\right ) \sinh (c+d x)}{2 \left (\sinh ^2(c+d x)+1\right )}\right )+\frac {(a-b) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )}{4 \left (\sinh ^2(c+d x)+1\right )^2}}{d}\) |
(((a - b)*Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2))/(4*(1 + Sinh[c + d*x]^2)^ 2) + (((3*a^2 + 2*a*b + 3*b^2)*ArcTan[Sinh[c + d*x]])/2 + (3*(a^2 - b^2)*S inh[c + d*x])/(2*(1 + Sinh[c + d*x]^2)))/4)/d
3.4.1.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) *x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f S ubst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x] /ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 57.09 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.84
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )+\frac {3 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )+2 a b \left (-\frac {\sinh \left (d x +c \right )}{3 \cosh \left (d x +c \right )^{4}}+\frac {\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )}{3}+\frac {\arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )+b^{2} \left (-\frac {\sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{4}}-\frac {\sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{4}}+\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )+\frac {3 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )}{d}\) | \(177\) |
| default | \(\frac {a^{2} \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )+\frac {3 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )+2 a b \left (-\frac {\sinh \left (d x +c \right )}{3 \cosh \left (d x +c \right )^{4}}+\frac {\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )}{3}+\frac {\arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )+b^{2} \left (-\frac {\sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{4}}-\frac {\sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{4}}+\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )+\frac {3 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )}{d}\) | \(177\) |
| risch | \(\frac {{\mathrm e}^{d x +c} \left (3 a^{2} {\mathrm e}^{6 d x +6 c}+2 \,{\mathrm e}^{6 d x +6 c} a b -5 b^{2} {\mathrm e}^{6 d x +6 c}+11 \,{\mathrm e}^{4 d x +4 c} a^{2}-14 \,{\mathrm e}^{4 d x +4 c} a b +3 b^{2} {\mathrm e}^{4 d x +4 c}-11 \,{\mathrm e}^{2 d x +2 c} a^{2}+14 \,{\mathrm e}^{2 d x +2 c} b a -3 b^{2} {\mathrm e}^{2 d x +2 c}-3 a^{2}-2 a b +5 b^{2}\right )}{4 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{4}}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{8 d}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a b}{4 d}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{2}}{8 d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{8 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a b}{4 d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) b^{2}}{8 d}\) | \(276\) |
1/d*(a^2*((1/4*sech(d*x+c)^3+3/8*sech(d*x+c))*tanh(d*x+c)+3/4*arctan(exp(d *x+c)))+2*a*b*(-1/3*sinh(d*x+c)/cosh(d*x+c)^4+1/3*(1/4*sech(d*x+c)^3+3/8*s ech(d*x+c))*tanh(d*x+c)+1/4*arctan(exp(d*x+c)))+b^2*(-sinh(d*x+c)^3/cosh(d *x+c)^4-sinh(d*x+c)/cosh(d*x+c)^4+(1/4*sech(d*x+c)^3+3/8*sech(d*x+c))*tanh (d*x+c)+3/4*arctan(exp(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 1472 vs. \(2 (90) = 180\).
Time = 0.28 (sec) , antiderivative size = 1472, normalized size of antiderivative = 15.33 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\text {Too large to display} \]
1/4*((3*a^2 + 2*a*b - 5*b^2)*cosh(d*x + c)^7 + 7*(3*a^2 + 2*a*b - 5*b^2)*c osh(d*x + c)*sinh(d*x + c)^6 + (3*a^2 + 2*a*b - 5*b^2)*sinh(d*x + c)^7 + ( 11*a^2 - 14*a*b + 3*b^2)*cosh(d*x + c)^5 + (21*(3*a^2 + 2*a*b - 5*b^2)*cos h(d*x + c)^2 + 11*a^2 - 14*a*b + 3*b^2)*sinh(d*x + c)^5 + 5*(7*(3*a^2 + 2* a*b - 5*b^2)*cosh(d*x + c)^3 + (11*a^2 - 14*a*b + 3*b^2)*cosh(d*x + c))*si nh(d*x + c)^4 - (11*a^2 - 14*a*b + 3*b^2)*cosh(d*x + c)^3 + (35*(3*a^2 + 2 *a*b - 5*b^2)*cosh(d*x + c)^4 + 10*(11*a^2 - 14*a*b + 3*b^2)*cosh(d*x + c) ^2 - 11*a^2 + 14*a*b - 3*b^2)*sinh(d*x + c)^3 + (21*(3*a^2 + 2*a*b - 5*b^2 )*cosh(d*x + c)^5 + 10*(11*a^2 - 14*a*b + 3*b^2)*cosh(d*x + c)^3 - 3*(11*a ^2 - 14*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 + ((3*a^2 + 2*a*b + 3* b^2)*cosh(d*x + c)^8 + 8*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^2 + 2*a*b + 3*b^2)*sinh(d*x + c)^8 + 4*(3*a^2 + 2*a*b + 3*b^2) *cosh(d*x + c)^6 + 4*(7*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^2 + 3*a^2 + 2*a*b + 3*b^2)*sinh(d*x + c)^6 + 8*(7*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c )^3 + 3*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 6*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^4 + 2*(35*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^4 + 30*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^2 + 9*a^2 + 6*a*b + 9*b^ 2)*sinh(d*x + c)^4 + 8*(7*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^5 + 10*(3* a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^2 + 4*(...
Timed out. \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (90) = 180\).
Time = 0.31 (sec) , antiderivative size = 347, normalized size of antiderivative = 3.61 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=-\frac {1}{4} \, b^{2} {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {5 \, e^{\left (-d x - c\right )} - 3 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} - \frac {1}{4} \, a^{2} {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {3 \, e^{\left (-d x - c\right )} + 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} - 3 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} - \frac {1}{2} \, a b {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - 7 \, e^{\left (-3 \, d x - 3 \, c\right )} + 7 \, e^{\left (-5 \, d x - 5 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} \]
-1/4*b^2*(3*arctan(e^(-d*x - c))/d + (5*e^(-d*x - c) - 3*e^(-3*d*x - 3*c) + 3*e^(-5*d*x - 5*c) - 5*e^(-7*d*x - 7*c))/(d*(4*e^(-2*d*x - 2*c) + 6*e^(- 4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) + 1))) - 1/4*a^2*(3*a rctan(e^(-d*x - c))/d - (3*e^(-d*x - c) + 11*e^(-3*d*x - 3*c) - 11*e^(-5*d *x - 5*c) - 3*e^(-7*d*x - 7*c))/(d*(4*e^(-2*d*x - 2*c) + 6*e^(-4*d*x - 4*c ) + 4*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) + 1))) - 1/2*a*b*(arctan(e^(-d*x - c))/d - (e^(-d*x - c) - 7*e^(-3*d*x - 3*c) + 7*e^(-5*d*x - 5*c) - e^(-7 *d*x - 7*c))/(d*(4*e^(-2*d*x - 2*c) + 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6 *c) + e^(-8*d*x - 8*c) + 1)))
Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (90) = 180\).
Time = 0.31 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.27 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (3 \, a^{2} + 2 \, a b + 3 \, b^{2}\right )} + \frac {4 \, {\left (3 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 2 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 5 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 20 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 8 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 12 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{2}}}{16 \, d} \]
1/16*((pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(3*a^2 + 2*a *b + 3*b^2) + 4*(3*a^2*(e^(d*x + c) - e^(-d*x - c))^3 + 2*a*b*(e^(d*x + c) - e^(-d*x - c))^3 - 5*b^2*(e^(d*x + c) - e^(-d*x - c))^3 + 20*a^2*(e^(d*x + c) - e^(-d*x - c)) - 8*a*b*(e^(d*x + c) - e^(-d*x - c)) - 12*b^2*(e^(d* x + c) - e^(-d*x - c)))/((e^(d*x + c) - e^(-d*x - c))^2 + 4)^2)/d
Time = 1.72 (sec) , antiderivative size = 327, normalized size of antiderivative = 3.41 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (3\,a^2\,\sqrt {d^2}+3\,b^2\,\sqrt {d^2}+2\,a\,b\,\sqrt {d^2}\right )}{d\,\sqrt {9\,a^4+12\,a^3\,b+22\,a^2\,b^2+12\,a\,b^3+9\,b^4}}\right )\,\sqrt {9\,a^4+12\,a^3\,b+22\,a^2\,b^2+12\,a\,b^3+9\,b^4}}{4\,\sqrt {d^2}}-\frac {6\,{\mathrm {e}}^{c+d\,x}\,\left (a^2-2\,a\,b+b^2\right )}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {4\,{\mathrm {e}}^{c+d\,x}\,\left (a^2-2\,a\,b+b^2\right )}{d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^2-10\,a\,b+9\,b^2\right )}{2\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,a^2+2\,a\,b-5\,b^2\right )}{4\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
(atan((exp(d*x)*exp(c)*(3*a^2*(d^2)^(1/2) + 3*b^2*(d^2)^(1/2) + 2*a*b*(d^2 )^(1/2)))/(d*(12*a*b^3 + 12*a^3*b + 9*a^4 + 9*b^4 + 22*a^2*b^2)^(1/2)))*(1 2*a*b^3 + 12*a^3*b + 9*a^4 + 9*b^4 + 22*a^2*b^2)^(1/2))/(4*(d^2)^(1/2)) - (6*exp(c + d*x)*(a^2 - 2*a*b + b^2))/(d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) + (4*exp(c + d*x)*(a^2 - 2*a*b + b^2))/(d* (4*exp(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)) + (exp(c + d*x)*(a^2 - 10*a*b + 9*b^2))/(2*d*(2*exp(2*c + 2*d *x) + exp(4*c + 4*d*x) + 1)) + (exp(c + d*x)*(2*a*b + 3*a^2 - 5*b^2))/(4*d *(exp(2*c + 2*d*x) + 1))