Integrand size = 21, antiderivative size = 92 \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=\frac {(5 a+8 b) \text {arctanh}(\cosh (c+d x))}{16 d}-\frac {(5 a+8 b) \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a \coth (c+d x) \text {csch}^5(c+d x)}{6 d} \]
1/16*(5*a+8*b)*arctanh(cosh(d*x+c))/d-1/16*(5*a+8*b)*coth(d*x+c)*csch(d*x+ c)/d+5/24*a*coth(d*x+c)*csch(d*x+c)^3/d-1/6*a*coth(d*x+c)*csch(d*x+c)^5/d
Leaf count is larger than twice the leaf count of optimal. \(237\) vs. \(2(92)=184\).
Time = 0.14 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.58 \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=-\frac {5 a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {b \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a \text {csch}^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \text {csch}^6\left (\frac {1}{2} (c+d x)\right )}{384 d}+\frac {5 a \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {b \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {5 a \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {b \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {5 a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {b \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a \text {sech}^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \text {sech}^6\left (\frac {1}{2} (c+d x)\right )}{384 d} \]
(-5*a*Csch[(c + d*x)/2]^2)/(64*d) - (b*Csch[(c + d*x)/2]^2)/(8*d) + (a*Csc h[(c + d*x)/2]^4)/(64*d) - (a*Csch[(c + d*x)/2]^6)/(384*d) + (5*a*Log[Cosh [(c + d*x)/2]])/(16*d) + (b*Log[Cosh[(c + d*x)/2]])/(2*d) - (5*a*Log[Sinh[ (c + d*x)/2]])/(16*d) - (b*Log[Sinh[(c + d*x)/2]])/(2*d) - (5*a*Sech[(c + d*x)/2]^2)/(64*d) - (b*Sech[(c + d*x)/2]^2)/(8*d) - (a*Sech[(c + d*x)/2]^4 )/(64*d) - (a*Sech[(c + d*x)/2]^6)/(384*d)
Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 26, 3694, 1471, 25, 298, 215, 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.
| \(\displaystyle \int \text {csch}^7(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \left (a+b \sin (i c+i d x)^4\right )}{\sin (i c+i d x)^7}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {b \sin (i c+i d x)^4+a}{\sin (i c+i d x)^7}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle \frac {\int \frac {b \cosh ^4(c+d x)-2 b \cosh ^2(c+d x)+a+b}{\left (1-\cosh ^2(c+d x)\right )^4}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle \frac {\frac {a \cosh (c+d x)}{6 \left (1-\cosh ^2(c+d x)\right )^3}-\frac {1}{6} \int -\frac {-6 b \cosh ^2(c+d x)+5 a+6 b}{\left (1-\cosh ^2(c+d x)\right )^3}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{6} \int \frac {-6 b \cosh ^2(c+d x)+5 a+6 b}{\left (1-\cosh ^2(c+d x)\right )^3}d\cosh (c+d x)+\frac {a \cosh (c+d x)}{6 \left (1-\cosh ^2(c+d x)\right )^3}}{d}\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{4} (5 a+8 b) \int \frac {1}{\left (1-\cosh ^2(c+d x)\right )^2}d\cosh (c+d x)+\frac {5 a \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}\right )+\frac {a \cosh (c+d x)}{6 \left (1-\cosh ^2(c+d x)\right )^3}}{d}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{4} (5 a+8 b) \left (\frac {1}{2} \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)+\frac {\cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}\right )+\frac {5 a \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}\right )+\frac {a \cosh (c+d x)}{6 \left (1-\cosh ^2(c+d x)\right )^3}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{4} (5 a+8 b) \left (\frac {1}{2} \text {arctanh}(\cosh (c+d x))+\frac {\cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}\right )+\frac {5 a \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}\right )+\frac {a \cosh (c+d x)}{6 \left (1-\cosh ^2(c+d x)\right )^3}}{d}\) |
((a*Cosh[c + d*x])/(6*(1 - Cosh[c + d*x]^2)^3) + ((5*a*Cosh[c + d*x])/(4*( 1 - Cosh[c + d*x]^2)^2) + (3*(5*a + 8*b)*(ArcTanh[Cosh[c + d*x]]/2 + Cosh[ c + d*x]/(2*(1 - Cosh[c + d*x]^2))))/4)/6)/d
3.2.95.3.1 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)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
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[((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[((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[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 = 0.38 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.85
| method | result | size |
| derivativedivides | \(\frac {a \left (\left (-\frac {\operatorname {csch}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {csch}\left (d x +c \right )^{3}}{24}-\frac {5 \,\operatorname {csch}\left (d x +c \right )}{16}\right ) \coth \left (d x +c \right )+\frac {5 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )}{8}\right )+b \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(78\) |
| default | \(\frac {a \left (\left (-\frac {\operatorname {csch}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {csch}\left (d x +c \right )^{3}}{24}-\frac {5 \,\operatorname {csch}\left (d x +c \right )}{16}\right ) \coth \left (d x +c \right )+\frac {5 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )}{8}\right )+b \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(78\) |
| parallelrisch | \(\frac {24 \left (-5 a -8 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-9\right ) \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -9 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +45 a +48 b \right ) \left (\coth \left (\frac {d x}{2}+\frac {c}{2}\right )-\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\coth \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}\) | \(141\) |
| risch | \(-\frac {{\mathrm e}^{d x +c} \left (15 a \,{\mathrm e}^{10 d x +10 c}+24 b \,{\mathrm e}^{10 d x +10 c}-85 \,{\mathrm e}^{8 d x +8 c} a -72 b \,{\mathrm e}^{8 d x +8 c}+198 \,{\mathrm e}^{6 d x +6 c} a +48 b \,{\mathrm e}^{6 d x +6 c}+198 \,{\mathrm e}^{4 d x +4 c} a +48 b \,{\mathrm e}^{4 d x +4 c}-85 a \,{\mathrm e}^{2 d x +2 c}-72 b \,{\mathrm e}^{2 d x +2 c}+15 a +24 b \right )}{24 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{6}}-\frac {5 a \ln \left ({\mathrm e}^{d x +c}-1\right )}{16 d}-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) b}{2 d}+\frac {5 a \ln \left ({\mathrm e}^{d x +c}+1\right )}{16 d}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) b}{2 d}\) | \(213\) |
1/d*(a*((-1/6*csch(d*x+c)^5+5/24*csch(d*x+c)^3-5/16*csch(d*x+c))*coth(d*x+ c)+5/8*arctanh(exp(d*x+c)))+b*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d* x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 3115 vs. \(2 (84) = 168\).
Time = 0.31 (sec) , antiderivative size = 3115, normalized size of antiderivative = 33.86 \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=\text {Too large to display} \]
-1/48*(6*(5*a + 8*b)*cosh(d*x + c)^11 + 66*(5*a + 8*b)*cosh(d*x + c)*sinh( d*x + c)^10 + 6*(5*a + 8*b)*sinh(d*x + c)^11 - 2*(85*a + 72*b)*cosh(d*x + c)^9 + 2*(165*(5*a + 8*b)*cosh(d*x + c)^2 - 85*a - 72*b)*sinh(d*x + c)^9 + 18*(55*(5*a + 8*b)*cosh(d*x + c)^3 - (85*a + 72*b)*cosh(d*x + c))*sinh(d* x + c)^8 + 12*(33*a + 8*b)*cosh(d*x + c)^7 + 12*(165*(5*a + 8*b)*cosh(d*x + c)^4 - 6*(85*a + 72*b)*cosh(d*x + c)^2 + 33*a + 8*b)*sinh(d*x + c)^7 + 8 4*(33*(5*a + 8*b)*cosh(d*x + c)^5 - 2*(85*a + 72*b)*cosh(d*x + c)^3 + (33* a + 8*b)*cosh(d*x + c))*sinh(d*x + c)^6 + 12*(33*a + 8*b)*cosh(d*x + c)^5 + 12*(231*(5*a + 8*b)*cosh(d*x + c)^6 - 21*(85*a + 72*b)*cosh(d*x + c)^4 + 21*(33*a + 8*b)*cosh(d*x + c)^2 + 33*a + 8*b)*sinh(d*x + c)^5 + 12*(165*( 5*a + 8*b)*cosh(d*x + c)^7 - 21*(85*a + 72*b)*cosh(d*x + c)^5 + 35*(33*a + 8*b)*cosh(d*x + c)^3 + 5*(33*a + 8*b)*cosh(d*x + c))*sinh(d*x + c)^4 - 2* (85*a + 72*b)*cosh(d*x + c)^3 + 2*(495*(5*a + 8*b)*cosh(d*x + c)^8 - 84*(8 5*a + 72*b)*cosh(d*x + c)^6 + 210*(33*a + 8*b)*cosh(d*x + c)^4 + 60*(33*a + 8*b)*cosh(d*x + c)^2 - 85*a - 72*b)*sinh(d*x + c)^3 + 6*(55*(5*a + 8*b)* cosh(d*x + c)^9 - 12*(85*a + 72*b)*cosh(d*x + c)^7 + 42*(33*a + 8*b)*cosh( d*x + c)^5 + 20*(33*a + 8*b)*cosh(d*x + c)^3 - (85*a + 72*b)*cosh(d*x + c) )*sinh(d*x + c)^2 + 6*(5*a + 8*b)*cosh(d*x + c) - 3*((5*a + 8*b)*cosh(d*x + c)^12 + 12*(5*a + 8*b)*cosh(d*x + c)*sinh(d*x + c)^11 + (5*a + 8*b)*sinh (d*x + c)^12 - 6*(5*a + 8*b)*cosh(d*x + c)^10 + 6*(11*(5*a + 8*b)*cosh(...
Timed out. \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (84) = 168\).
Time = 0.22 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.91 \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=\frac {1}{48} \, a {\left (\frac {15 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {15 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (15 \, e^{\left (-d x - c\right )} - 85 \, e^{\left (-3 \, d x - 3 \, c\right )} + 198 \, e^{\left (-5 \, d x - 5 \, c\right )} + 198 \, e^{\left (-7 \, d x - 7 \, c\right )} - 85 \, e^{\left (-9 \, d x - 9 \, c\right )} + 15 \, e^{\left (-11 \, d x - 11 \, c\right )}\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} + \frac {1}{2} \, b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \]
1/48*a*(15*log(e^(-d*x - c) + 1)/d - 15*log(e^(-d*x - c) - 1)/d + 2*(15*e^ (-d*x - c) - 85*e^(-3*d*x - 3*c) + 198*e^(-5*d*x - 5*c) + 198*e^(-7*d*x - 7*c) - 85*e^(-9*d*x - 9*c) + 15*e^(-11*d*x - 11*c))/(d*(6*e^(-2*d*x - 2*c) - 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) - 15*e^(-8*d*x - 8*c) + 6*e^( -10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1))) + 1/2*b*(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. 207 vs. \(2 (84) = 168\).
Time = 0.30 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.25 \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=\frac {3 \, {\left (5 \, a + 8 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - 3 \, {\left (5 \, a + 8 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {4 \, {\left (15 \, a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 24 \, b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 160 \, a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 192 \, b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 528 \, a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 384 \, b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )}^{3}}}{96 \, d} \]
1/96*(3*(5*a + 8*b)*log(e^(d*x + c) + e^(-d*x - c) + 2) - 3*(5*a + 8*b)*lo g(e^(d*x + c) + e^(-d*x - c) - 2) - 4*(15*a*(e^(d*x + c) + e^(-d*x - c))^5 + 24*b*(e^(d*x + c) + e^(-d*x - c))^5 - 160*a*(e^(d*x + c) + e^(-d*x - c) )^3 - 192*b*(e^(d*x + c) + e^(-d*x - c))^3 + 528*a*(e^(d*x + c) + e^(-d*x - c)) + 384*b*(e^(d*x + c) + e^(-d*x - c)))/((e^(d*x + c) + e^(-d*x - c))^ 2 - 4)^3)/d
Time = 1.51 (sec) , antiderivative size = 472, normalized size of antiderivative = 5.13 \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (5\,a\,\sqrt {-d^2}+8\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {25\,a^2+80\,a\,b+64\,b^2}}\right )\,\sqrt {25\,a^2+80\,a\,b+64\,b^2}}{8\,\sqrt {-d^2}}-\frac {\frac {2\,b\,{\mathrm {e}}^{9\,c+9\,d\,x}}{3\,d}-\frac {8\,b\,{\mathrm {e}}^{7\,c+7\,d\,x}}{3\,d}-\frac {8\,b\,{\mathrm {e}}^{3\,c+3\,d\,x}}{3\,d}+\frac {4\,{\mathrm {e}}^{5\,c+5\,d\,x}\,\left (8\,a+3\,b\right )}{3\,d}+\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{3\,d}}{15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{2\,c+2\,d\,x}-20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}-6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (5\,a-16\,b\right )}{12\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {a\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {22\,a\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (5\,a+8\,b\right )}{8\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {16\,a\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1\right )} \]
(atan((exp(d*x)*exp(c)*(5*a*(-d^2)^(1/2) + 8*b*(-d^2)^(1/2)))/(d*(80*a*b + 25*a^2 + 64*b^2)^(1/2)))*(80*a*b + 25*a^2 + 64*b^2)^(1/2))/(8*(-d^2)^(1/2 )) - ((2*b*exp(9*c + 9*d*x))/(3*d) - (8*b*exp(7*c + 7*d*x))/(3*d) - (8*b*e xp(3*c + 3*d*x))/(3*d) + (4*exp(5*c + 5*d*x)*(8*a + 3*b))/(3*d) + (2*b*exp (c + d*x))/(3*d))/(15*exp(4*c + 4*d*x) - 6*exp(2*c + 2*d*x) - 20*exp(6*c + 6*d*x) + 15*exp(8*c + 8*d*x) - 6*exp(10*c + 10*d*x) + exp(12*c + 12*d*x) + 1) + (exp(c + d*x)*(5*a - 16*b))/(12*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2 *d*x) + 1)) - (a*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)) - (22*a*exp(c + d*x))/(3*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)) - (exp (c + d*x)*(5*a + 8*b))/(8*d*(exp(2*c + 2*d*x) - 1)) - (16*a*exp(c + d*x))/ (3*d*(5*exp(2*c + 2*d*x) - 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) - 5*e xp(8*c + 8*d*x) + exp(10*c + 10*d*x) - 1))