Integrand size = 23, antiderivative size = 101 \[ \int \text {csch}^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=-\frac {a (3 a+16 b) \text {arctanh}(\cosh (c+d x))}{8 d}-\frac {b^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh ^3(c+d x)}{3 d}+\frac {3 a^2 \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {a^2 \coth (c+d x) \text {csch}^3(c+d x)}{4 d} \]
-1/8*a*(3*a+16*b)*arctanh(cosh(d*x+c))/d-b^2*cosh(d*x+c)/d+1/3*b^2*cosh(d* x+c)^3/d+3/8*a^2*coth(d*x+c)*csch(d*x+c)/d-1/4*a^2*coth(d*x+c)*csch(d*x+c) ^3/d
Leaf count is larger than twice the leaf count of optimal. \(207\) vs. \(2(101)=202\).
Time = 0.02 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.05 \[ \int \text {csch}^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=-\frac {3 b^2 \cosh (c+d x)}{4 d}+\frac {b^2 \cosh (3 (c+d x))}{12 d}+\frac {3 a^2 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {a^2 \text {csch}^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {2 a b \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}-\frac {3 a^2 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {2 a b \log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {3 a^2 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {3 a^2 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {a^2 \text {sech}^4\left (\frac {1}{2} (c+d x)\right )}{64 d} \]
(-3*b^2*Cosh[c + d*x])/(4*d) + (b^2*Cosh[3*(c + d*x)])/(12*d) + (3*a^2*Csc h[(c + d*x)/2]^2)/(32*d) - (a^2*Csch[(c + d*x)/2]^4)/(64*d) - (2*a*b*Log[C osh[c/2 + (d*x)/2]])/d - (3*a^2*Log[Cosh[(c + d*x)/2]])/(8*d) + (2*a*b*Log [Sinh[c/2 + (d*x)/2]])/d + (3*a^2*Log[Sinh[(c + d*x)/2]])/(8*d) + (3*a^2*S ech[(c + d*x)/2]^2)/(32*d) + (a^2*Sech[(c + d*x)/2]^4)/(64*d)
Time = 0.44 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3042, 26, 3694, 1471, 25, 2345, 25, 1467, 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.
| \(\displaystyle \int \text {csch}^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \left (a+b \sin (i c+i d x)^4\right )^2}{\sin (i c+i d x)^5}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\left (b \sin (i c+i d x)^4+a\right )^2}{\sin (i c+i d x)^5}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {\left (b \cosh ^4(c+d x)-2 b \cosh ^2(c+d x)+a+b\right )^2}{\left (1-\cosh ^2(c+d x)\right )^3}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle -\frac {\frac {a^2 \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}-\frac {1}{4} \int -\frac {-4 b^2 \cosh ^6(c+d x)+12 b^2 \cosh ^4(c+d x)-4 b (2 a+3 b) \cosh ^2(c+d x)+(a+2 b) (3 a+2 b)}{\left (1-\cosh ^2(c+d x)\right )^2}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{4} \int \frac {-4 b^2 \cosh ^6(c+d x)+12 b^2 \cosh ^4(c+d x)-4 b (2 a+3 b) \cosh ^2(c+d x)+(a+2 b) (3 a+2 b)}{\left (1-\cosh ^2(c+d x)\right )^2}d\cosh (c+d x)+\frac {a^2 \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {3 a^2 \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}-\frac {1}{2} \int -\frac {8 b^2 \cosh ^4(c+d x)-16 b^2 \cosh ^2(c+d x)+3 a^2+8 b^2+16 a b}{1-\cosh ^2(c+d x)}d\cosh (c+d x)\right )+\frac {a^2 \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \int \frac {8 b^2 \cosh ^4(c+d x)-16 b^2 \cosh ^2(c+d x)+3 a^2+8 b^2+16 a b}{1-\cosh ^2(c+d x)}d\cosh (c+d x)+\frac {3 a^2 \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}\right )+\frac {a^2 \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \int \left (-8 \cosh ^2(c+d x) b^2+8 b^2+\frac {3 a^2+16 b a}{1-\cosh ^2(c+d x)}\right )d\cosh (c+d x)+\frac {3 a^2 \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}\right )+\frac {a^2 \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {3 a^2 \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}+\frac {1}{2} \left (a (3 a+16 b) \text {arctanh}(\cosh (c+d x))-\frac {8}{3} b^2 \cosh ^3(c+d x)+8 b^2 \cosh (c+d x)\right )\right )+\frac {a^2 \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}}{d}\) |
-(((a^2*Cosh[c + d*x])/(4*(1 - Cosh[c + d*x]^2)^2) + ((3*a^2*Cosh[c + d*x] )/(2*(1 - Cosh[c + d*x]^2)) + (a*(3*a + 16*b)*ArcTanh[Cosh[c + d*x]] + 8*b ^2*Cosh[c + d*x] - (8*b^2*Cosh[c + d*x]^3)/3)/2)/4)/d)
3.3.4.3.1 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[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 , x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x , 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 0.60 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.78
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (\left (-\frac {\operatorname {csch}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {csch}\left (d x +c \right )}{8}\right ) \coth \left (d x +c \right )-\frac {3 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )}{4}\right )-4 a b \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+b^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )}{d}\) | \(79\) |
| default | \(\frac {a^{2} \left (\left (-\frac {\operatorname {csch}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {csch}\left (d x +c \right )}{8}\right ) \coth \left (d x +c \right )-\frac {3 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )}{4}\right )-4 a b \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+b^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )}{d}\) | \(79\) |
| parallelrisch | \(\frac {192 \left (a +\frac {16 b}{3}\right ) a \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-11 \left (\cosh \left (d x +c \right )-\frac {27 \cosh \left (2 d x +2 c \right )}{44}-\frac {3 \cosh \left (3 d x +3 c \right )}{11}+\frac {27 \cosh \left (4 d x +4 c \right )}{176}+\frac {81}{176}\right ) a^{2} \operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-384 \left (\cosh \left (d x +c \right )-\frac {\cosh \left (3 d x +3 c \right )}{9}+\frac {8}{9}\right ) b^{2}}{512 d}\) | \(117\) |
| risch | \(\frac {{\mathrm e}^{3 d x +3 c} b^{2}}{24 d}-\frac {3 \,{\mathrm e}^{d x +c} b^{2}}{8 d}-\frac {3 \,{\mathrm e}^{-d x -c} b^{2}}{8 d}+\frac {{\mathrm e}^{-3 d x -3 c} b^{2}}{24 d}+\frac {a^{2} {\mathrm e}^{d x +c} \left (3 \,{\mathrm e}^{6 d x +6 c}-11 \,{\mathrm e}^{4 d x +4 c}-11 \,{\mathrm e}^{2 d x +2 c}+3\right )}{4 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{4}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{8 d}+\frac {2 a \ln \left ({\mathrm e}^{d x +c}-1\right ) b}{d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{8 d}-\frac {2 a \ln \left ({\mathrm e}^{d x +c}+1\right ) b}{d}\) | \(195\) |
1/d*(a^2*((-1/4*csch(d*x+c)^3+3/8*csch(d*x+c))*coth(d*x+c)-3/4*arctanh(exp (d*x+c)))-4*a*b*arctanh(exp(d*x+c))+b^2*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*x+ c))
Leaf count of result is larger than twice the leaf count of optimal. 3356 vs. \(2 (93) = 186\).
Time = 0.29 (sec) , antiderivative size = 3356, normalized size of antiderivative = 33.23 \[ \int \text {csch}^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=\text {Too large to display} \]
1/24*(b^2*cosh(d*x + c)^14 + 14*b^2*cosh(d*x + c)*sinh(d*x + c)^13 + b^2*s inh(d*x + c)^14 - 13*b^2*cosh(d*x + c)^12 + 13*(7*b^2*cosh(d*x + c)^2 - b^ 2)*sinh(d*x + c)^12 + 52*(7*b^2*cosh(d*x + c)^3 - 3*b^2*cosh(d*x + c))*sin h(d*x + c)^11 + 3*(6*a^2 + 11*b^2)*cosh(d*x + c)^10 + (1001*b^2*cosh(d*x + c)^4 - 858*b^2*cosh(d*x + c)^2 + 18*a^2 + 33*b^2)*sinh(d*x + c)^10 + 2*(1 001*b^2*cosh(d*x + c)^5 - 1430*b^2*cosh(d*x + c)^3 + 15*(6*a^2 + 11*b^2)*c osh(d*x + c))*sinh(d*x + c)^9 - 3*(22*a^2 + 7*b^2)*cosh(d*x + c)^8 + 3*(10 01*b^2*cosh(d*x + c)^6 - 2145*b^2*cosh(d*x + c)^4 + 45*(6*a^2 + 11*b^2)*co sh(d*x + c)^2 - 22*a^2 - 7*b^2)*sinh(d*x + c)^8 + 24*(143*b^2*cosh(d*x + c )^7 - 429*b^2*cosh(d*x + c)^5 + 15*(6*a^2 + 11*b^2)*cosh(d*x + c)^3 - (22* a^2 + 7*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 3*(22*a^2 + 7*b^2)*cosh(d*x + c)^6 + 3*(1001*b^2*cosh(d*x + c)^8 - 4004*b^2*cosh(d*x + c)^6 + 210*(6*a ^2 + 11*b^2)*cosh(d*x + c)^4 - 28*(22*a^2 + 7*b^2)*cosh(d*x + c)^2 - 22*a^ 2 - 7*b^2)*sinh(d*x + c)^6 + 2*(1001*b^2*cosh(d*x + c)^9 - 5148*b^2*cosh(d *x + c)^7 + 378*(6*a^2 + 11*b^2)*cosh(d*x + c)^5 - 84*(22*a^2 + 7*b^2)*cos h(d*x + c)^3 - 9*(22*a^2 + 7*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 3*(6*a^ 2 + 11*b^2)*cosh(d*x + c)^4 + (1001*b^2*cosh(d*x + c)^10 - 6435*b^2*cosh(d *x + c)^8 + 630*(6*a^2 + 11*b^2)*cosh(d*x + c)^6 - 210*(22*a^2 + 7*b^2)*co sh(d*x + c)^4 - 45*(22*a^2 + 7*b^2)*cosh(d*x + c)^2 + 18*a^2 + 33*b^2)*sin h(d*x + c)^4 - 13*b^2*cosh(d*x + c)^2 + 4*(91*b^2*cosh(d*x + c)^11 - 71...
Timed out. \[ \int \text {csch}^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (93) = 186\).
Time = 0.20 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.32 \[ \int \text {csch}^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=\frac {1}{24} \, b^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} - \frac {1}{8} \, a^{2} {\left (\frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (3 \, e^{\left (-d x - c\right )} - 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} + 3 \, e^{\left (-7 \, d x - 7 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} - 2 \, a b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d}\right )} \]
1/24*b^2*(e^(3*d*x + 3*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d *x - 3*c)/d) - 1/8*a^2*(3*log(e^(-d*x - c) + 1)/d - 3*log(e^(-d*x - c) - 1 )/d + 2*(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))) - 2*a*b*(log(e^(-d*x - c) + 1)/d - log(e^ (-d*x - c) - 1)/d)
Time = 0.36 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.77 \[ \int \text {csch}^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=\frac {2 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 24 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 3 \, {\left (3 \, a^{2} + 16 \, a b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) + 3 \, {\left (3 \, a^{2} + 16 \, a b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) + \frac {12 \, {\left (3 \, a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 20 \, a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )}^{2}}}{48 \, d} \]
1/48*(2*b^2*(e^(d*x + c) + e^(-d*x - c))^3 - 24*b^2*(e^(d*x + c) + e^(-d*x - c)) - 3*(3*a^2 + 16*a*b)*log(e^(d*x + c) + e^(-d*x - c) + 2) + 3*(3*a^2 + 16*a*b)*log(e^(d*x + c) + e^(-d*x - c) - 2) + 12*(3*a^2*(e^(d*x + c) + e^(-d*x - c))^3 - 20*a^2*(e^(d*x + c) + e^(-d*x - c)))/((e^(d*x + c) + e^( -d*x - c))^2 - 4)^2)/d
Time = 0.23 (sec) , antiderivative size = 328, normalized size of antiderivative = 3.25 \[ \int \text {csch}^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=\frac {b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}-\frac {3\,b^2\,{\mathrm {e}}^{-c-d\,x}}{8\,d}-\frac {3\,b^2\,{\mathrm {e}}^{c+d\,x}}{8\,d}+\frac {b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (3\,a^2\,\sqrt {-d^2}+16\,a\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {9\,a^4+96\,a^3\,b+256\,a^2\,b^2}}\right )\,\sqrt {9\,a^4+96\,a^3\,b+256\,a^2\,b^2}}{4\,\sqrt {-d^2}}-\frac {6\,a^2\,{\mathrm {e}}^{c+d\,x}}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {4\,a^2\,{\mathrm {e}}^{c+d\,x}}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {3\,a^2\,{\mathrm {e}}^{c+d\,x}}{4\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {a^2\,{\mathrm {e}}^{c+d\,x}}{2\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
(b^2*exp(- 3*c - 3*d*x))/(24*d) - (3*b^2*exp(- c - d*x))/(8*d) - (3*b^2*ex p(c + d*x))/(8*d) + (b^2*exp(3*c + 3*d*x))/(24*d) - (atan((exp(d*x)*exp(c) *(3*a^2*(-d^2)^(1/2) + 16*a*b*(-d^2)^(1/2)))/(d*(96*a^3*b + 9*a^4 + 256*a^ 2*b^2)^(1/2)))*(96*a^3*b + 9*a^4 + 256*a^2*b^2)^(1/2))/(4*(-d^2)^(1/2)) - (6*a^2*exp(c + d*x))/(d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) - (4*a^2*exp(c + d*x))/(d*(6*exp(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) + (3*a^2*exp(c + d* x))/(4*d*(exp(2*c + 2*d*x) - 1)) - (a^2*exp(c + d*x))/(2*d*(exp(4*c + 4*d* x) - 2*exp(2*c + 2*d*x) + 1))