Integrand size = 23, antiderivative size = 133 \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\frac {5 a^2 \text {arctanh}(\cosh (c+d x))}{16 d}-\frac {b^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {2 a b \coth (c+d x)}{d}-\frac {2 a b \coth ^3(c+d x)}{3 d}-\frac {5 a^2 \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^2 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^2 \coth (c+d x) \text {csch}^5(c+d x)}{6 d} \]
5/16*a^2*arctanh(cosh(d*x+c))/d-b^2*arctanh(cosh(d*x+c))/d+2*a*b*coth(d*x+ c)/d-2/3*a*b*coth(d*x+c)^3/d-5/16*a^2*coth(d*x+c)*csch(d*x+c)/d+5/24*a^2*c oth(d*x+c)*csch(d*x+c)^3/d-1/6*a^2*coth(d*x+c)*csch(d*x+c)^5/d
Time = 0.22 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.92 \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\frac {4 a b \coth (c+d x)}{3 d}-\frac {5 a^2 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a^2 \text {csch}^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a^2 \text {csch}^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {2 a b \coth (c+d x) \text {csch}^2(c+d x)}{3 d}-\frac {b^2 \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {5 a^2 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {b^2 \log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}-\frac {5 a^2 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {5 a^2 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a^2 \text {sech}^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a^2 \text {sech}^6\left (\frac {1}{2} (c+d x)\right )}{384 d} \]
(4*a*b*Coth[c + d*x])/(3*d) - (5*a^2*Csch[(c + d*x)/2]^2)/(64*d) + (a^2*Cs ch[(c + d*x)/2]^4)/(64*d) - (a^2*Csch[(c + d*x)/2]^6)/(384*d) - (2*a*b*Cot h[c + d*x]*Csch[c + d*x]^2)/(3*d) - (b^2*Log[Cosh[c/2 + (d*x)/2]])/d + (5* a^2*Log[Cosh[(c + d*x)/2]])/(16*d) + (b^2*Log[Sinh[c/2 + (d*x)/2]])/d - (5 *a^2*Log[Sinh[(c + d*x)/2]])/(16*d) - (5*a^2*Sech[(c + d*x)/2]^2)/(64*d) - (a^2*Sech[(c + d*x)/2]^4)/(64*d) - (a^2*Sech[(c + d*x)/2]^6)/(384*d)
Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.14, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 26, 3699, 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.
| \(\displaystyle \int \text {csch}^7(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \left (a+i b \sin (i c+i d x)^3\right )^2}{\sin (i c+i d x)^7}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\left (i b \sin (i c+i d x)^3+a\right )^2}{\sin (i c+i d x)^7}dx\) |
| \(\Big \downarrow \) 3699 |
| \(\displaystyle -i \int \left (i a^2 \text {csch}^7(c+d x)+2 i a b \text {csch}^4(c+d x)+i b^2 \text {csch}(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -i \left (\frac {5 i a^2 \text {arctanh}(\cosh (c+d x))}{16 d}-\frac {i a^2 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}+\frac {5 i a^2 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {5 i a^2 \coth (c+d x) \text {csch}(c+d x)}{16 d}-\frac {2 i a b \coth ^3(c+d x)}{3 d}+\frac {2 i a b \coth (c+d x)}{d}-\frac {i b^2 \text {arctanh}(\cosh (c+d x))}{d}\right )\) |
(-I)*((((5*I)/16)*a^2*ArcTanh[Cosh[c + d*x]])/d - (I*b^2*ArcTanh[Cosh[c + d*x]])/d + ((2*I)*a*b*Coth[c + d*x])/d - (((2*I)/3)*a*b*Coth[c + d*x]^3)/d - (((5*I)/16)*a^2*Coth[c + d*x]*Csch[c + d*x])/d + (((5*I)/24)*a^2*Coth[c + d*x]*Csch[c + d*x]^3)/d - ((I/6)*a^2*Coth[c + d*x]*Csch[c + d*x]^5)/d)
3.2.60.3.1 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[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_ ))^(p_.), x_Symbol] :> Int[ExpandTrig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n) ^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4] || Gt Q[p, 0] || (EqQ[p, -1] && IntegerQ[n]))
Time = 0.38 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.68
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (\left (-\frac {\operatorname {csch}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {csch}\left (d x +c \right )^{3}}{24}-\frac {5 \,\operatorname {csch}\left (d x +c \right )}{16}\right ) \coth \left (d x +c \right )+\frac {5 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )}{8}\right )+2 a b \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )-2 b^{2} \operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )}{d}\) | \(90\) |
| default | \(\frac {a^{2} \left (\left (-\frac {\operatorname {csch}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {csch}\left (d x +c \right )^{3}}{24}-\frac {5 \,\operatorname {csch}\left (d x +c \right )}{16}\right ) \coth \left (d x +c \right )+\frac {5 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )}{8}\right )+2 a b \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )-2 b^{2} \operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )}{d}\) | \(90\) |
| parallelrisch | \(\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a^{2}-a^{2} \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-9 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{2}+9 a^{2} \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-32 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b -32 a b \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+45 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{2}-45 a^{2} \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-120 a^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+384 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}+288 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a b +288 a b \coth \left (\frac {d x}{2}+\frac {c}{2}\right )}{384 d}\) | \(188\) |
| risch | \(-\frac {a \left (15 a \,{\mathrm e}^{11 d x +11 c}-85 a \,{\mathrm e}^{9 d x +9 c}+192 b \,{\mathrm e}^{8 d x +8 c}+198 a \,{\mathrm e}^{7 d x +7 c}-640 b \,{\mathrm e}^{6 d x +6 c}+198 a \,{\mathrm e}^{5 d x +5 c}+768 b \,{\mathrm e}^{4 d x +4 c}-85 a \,{\mathrm e}^{3 d x +3 c}-384 b \,{\mathrm e}^{2 d x +2 c}+15 \,{\mathrm e}^{d x +c} a +64 b \right )}{24 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{6}}-\frac {5 a^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{16 d}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) b^{2}}{d}+\frac {5 a^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{16 d}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) b^{2}}{d}\) | \(209\) |
1/d*(a^2*((-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)))+2*a*b*(2/3-1/3*csch(d*x+c)^2)*coth(d*x+c)-2* b^2*arctanh(exp(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 3607 vs. \(2 (123) = 246\).
Time = 0.28 (sec) , antiderivative size = 3607, normalized size of antiderivative = 27.12 \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\text {Too large to display} \]
-1/48*(30*a^2*cosh(d*x + c)^11 + 330*a^2*cosh(d*x + c)*sinh(d*x + c)^10 + 30*a^2*sinh(d*x + c)^11 - 170*a^2*cosh(d*x + c)^9 + 384*a*b*cosh(d*x + c)^ 8 + 10*(165*a^2*cosh(d*x + c)^2 - 17*a^2)*sinh(d*x + c)^9 + 396*a^2*cosh(d *x + c)^7 + 6*(825*a^2*cosh(d*x + c)^3 - 255*a^2*cosh(d*x + c) + 64*a*b)*s inh(d*x + c)^8 - 1280*a*b*cosh(d*x + c)^6 + 12*(825*a^2*cosh(d*x + c)^4 - 510*a^2*cosh(d*x + c)^2 + 256*a*b*cosh(d*x + c) + 33*a^2)*sinh(d*x + c)^7 + 396*a^2*cosh(d*x + c)^5 + 4*(3465*a^2*cosh(d*x + c)^5 - 3570*a^2*cosh(d* x + c)^3 + 2688*a*b*cosh(d*x + c)^2 + 693*a^2*cosh(d*x + c) - 320*a*b)*sin h(d*x + c)^6 + 1536*a*b*cosh(d*x + c)^4 + 12*(1155*a^2*cosh(d*x + c)^6 - 1 785*a^2*cosh(d*x + c)^4 + 1792*a*b*cosh(d*x + c)^3 + 693*a^2*cosh(d*x + c) ^2 - 640*a*b*cosh(d*x + c) + 33*a^2)*sinh(d*x + c)^5 - 170*a^2*cosh(d*x + c)^3 + 12*(825*a^2*cosh(d*x + c)^7 - 1785*a^2*cosh(d*x + c)^5 + 2240*a*b*c osh(d*x + c)^4 + 1155*a^2*cosh(d*x + c)^3 - 1600*a*b*cosh(d*x + c)^2 + 165 *a^2*cosh(d*x + c) + 128*a*b)*sinh(d*x + c)^4 - 768*a*b*cosh(d*x + c)^2 + 2*(2475*a^2*cosh(d*x + c)^8 - 7140*a^2*cosh(d*x + c)^6 + 10752*a*b*cosh(d* x + c)^5 + 6930*a^2*cosh(d*x + c)^4 - 12800*a*b*cosh(d*x + c)^3 + 1980*a^2 *cosh(d*x + c)^2 + 3072*a*b*cosh(d*x + c) - 85*a^2)*sinh(d*x + c)^3 + 30*a ^2*cosh(d*x + c) + 6*(275*a^2*cosh(d*x + c)^9 - 1020*a^2*cosh(d*x + c)^7 + 1792*a*b*cosh(d*x + c)^6 + 1386*a^2*cosh(d*x + c)^5 - 3200*a*b*cosh(d*x + c)^4 + 660*a^2*cosh(d*x + c)^3 + 1536*a*b*cosh(d*x + c)^2 - 85*a^2*cos...
Timed out. \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (123) = 246\).
Time = 0.22 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.38 \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\frac {1}{48} \, a^{2} {\left (\frac {15 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {15 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (15 \, e^{\left (-d x - c\right )} - 85 \, e^{\left (-3 \, d x - 3 \, c\right )} + 198 \, e^{\left (-5 \, d x - 5 \, c\right )} + 198 \, e^{\left (-7 \, d x - 7 \, c\right )} - 85 \, e^{\left (-9 \, d x - 9 \, c\right )} + 15 \, e^{\left (-11 \, d x - 11 \, c\right )}\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} - b^{2} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d}\right )} + \frac {8}{3} \, a b {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} \]
1/48*a^2*(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))) - b^2*(log(e^(-d*x - c) + 1) /d - log(e^(-d*x - c) - 1)/d) + 8/3*a*b*(3*e^(-2*d*x - 2*c)/(d*(3*e^(-2*d* x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1)) - 1/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1)))
Time = 0.34 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.53 \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\frac {3 \, {\left (5 \, a^{2} - 16 \, b^{2}\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) - 3 \, {\left (5 \, a^{2} - 16 \, b^{2}\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {2 \, {\left (15 \, a^{2} e^{\left (11 \, d x + 11 \, c\right )} - 85 \, a^{2} e^{\left (9 \, d x + 9 \, c\right )} + 192 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 198 \, a^{2} e^{\left (7 \, d x + 7 \, c\right )} - 640 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 198 \, a^{2} e^{\left (5 \, d x + 5 \, c\right )} + 768 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 85 \, a^{2} e^{\left (3 \, d x + 3 \, c\right )} - 384 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 15 \, a^{2} e^{\left (d x + c\right )} + 64 \, a b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{6}}}{48 \, d} \]
1/48*(3*(5*a^2 - 16*b^2)*log(e^(d*x + c) + 1) - 3*(5*a^2 - 16*b^2)*log(abs (e^(d*x + c) - 1)) - 2*(15*a^2*e^(11*d*x + 11*c) - 85*a^2*e^(9*d*x + 9*c) + 192*a*b*e^(8*d*x + 8*c) + 198*a^2*e^(7*d*x + 7*c) - 640*a*b*e^(6*d*x + 6 *c) + 198*a^2*e^(5*d*x + 5*c) + 768*a*b*e^(4*d*x + 4*c) - 85*a^2*e^(3*d*x + 3*c) - 384*a*b*e^(2*d*x + 2*c) + 15*a^2*e^(d*x + c) + 64*a*b)/(e^(2*d*x + 2*c) - 1)^6)/d
Time = 0.17 (sec) , antiderivative size = 434, normalized size of antiderivative = 3.26 \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\frac {\frac {5\,a^2\,{\mathrm {e}}^{c+d\,x}}{12\,d}-\frac {8\,a\,b}{d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {\frac {a^2\,{\mathrm {e}}^{c+d\,x}}{3\,d}+\frac {16\,a\,b}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}+\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (5\,a^2\,\sqrt {-d^2}-16\,b^2\,\sqrt {-d^2}\right )}{d\,\sqrt {25\,a^4-160\,a^2\,b^2+256\,b^4}}\right )\,\sqrt {25\,a^4-160\,a^2\,b^2+256\,b^4}}{8\,\sqrt {-d^2}}-\frac {18\,a^2\,{\mathrm {e}}^{c+d\,x}}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {80\,a^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1\right )}-\frac {32\,a^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{2\,c+2\,d\,x}-20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}-6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}-\frac {5\,a^2\,{\mathrm {e}}^{c+d\,x}}{8\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \]
((5*a^2*exp(c + d*x))/(12*d) - (8*a*b)/d)/(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1) - ((a^2*exp(c + d*x))/(3*d) + (16*a*b)/(3*d))/(3*exp(2*c + 2*d *x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1) + (atan((exp(d*x)*exp(c)* (5*a^2*(-d^2)^(1/2) - 16*b^2*(-d^2)^(1/2)))/(d*(25*a^4 + 256*b^4 - 160*a^2 *b^2)^(1/2)))*(25*a^4 + 256*b^4 - 160*a^2*b^2)^(1/2))/(8*(-d^2)^(1/2)) - ( 18*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)) - (80*a^2*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*exp(8*c + 8*d* x) + exp(10*c + 10*d*x) - 1)) - (32*a^2*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)) - (5*a^2*exp(c + d*x))/(8*d *(exp(2*c + 2*d*x) - 1))