Integrand size = 23, antiderivative size = 107 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\frac {a b \arctan (\sinh (c+d x))}{d}+\frac {a^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {a^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {b^2 \text {sech}^3(c+d x)}{3 d}+\frac {b^2 \text {sech}^5(c+d x)}{5 d}+\frac {a b \text {sech}(c+d x) \tanh (c+d x)}{d} \]
a*b*arctan(sinh(d*x+c))/d+1/2*a^2*arctanh(cosh(d*x+c))/d-1/2*a^2*coth(d*x+ c)*csch(d*x+c)/d-1/3*b^2*sech(d*x+c)^3/d+1/5*b^2*sech(d*x+c)^5/d+a*b*sech( d*x+c)*tanh(d*x+c)/d
Time = 1.74 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.49 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\frac {2 a b \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{d}-\frac {a^2 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a^2 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a^2 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a^2 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {b^2 \text {sech}^3(c+d x)}{3 d}+\frac {b^2 \text {sech}^5(c+d x)}{5 d}+\frac {a b \text {sech}(c+d x) \tanh (c+d x)}{d} \]
(2*a*b*ArcTan[Tanh[(c + d*x)/2]])/d - (a^2*Csch[(c + d*x)/2]^2)/(8*d) + (a ^2*Log[Cosh[(c + d*x)/2]])/(2*d) - (a^2*Log[Sinh[(c + d*x)/2]])/(2*d) - (a ^2*Sech[(c + d*x)/2]^2)/(8*d) - (b^2*Sech[c + d*x]^3)/(3*d) + (b^2*Sech[c + d*x]^5)/(5*d) + (a*b*Sech[c + d*x]*Tanh[c + d*x])/d
Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 26, 4149, 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.
| \(\displaystyle \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \left (a+i b \tan (i c+i d x)^3\right )^2}{\sin (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\left (i b \tan (i c+i d x)^3+a\right )^2}{\sin (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 4149 |
| \(\displaystyle -i \int \left (i a^2 \text {csch}^3(c+d x)+2 i a b \text {sech}^3(c+d x)+i b^2 \text {sech}^3(c+d x) \tanh ^3(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -i \left (\frac {i a^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {i a^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {i a b \arctan (\sinh (c+d x))}{d}+\frac {i a b \tanh (c+d x) \text {sech}(c+d x)}{d}+\frac {i b^2 \text {sech}^5(c+d x)}{5 d}-\frac {i b^2 \text {sech}^3(c+d x)}{3 d}\right )\) |
(-I)*((I*a*b*ArcTan[Sinh[c + d*x]])/d + ((I/2)*a^2*ArcTanh[Cosh[c + d*x]]) /d - ((I/2)*a^2*Coth[c + d*x]*Csch[c + d*x])/d - ((I/3)*b^2*Sech[c + d*x]^ 3)/d + ((I/5)*b^2*Sech[c + d*x]^5)/d + (I*a*b*Sech[c + d*x]*Tanh[c + d*x]) /d)
3.1.63.3.1 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_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> Int[ExpandTrig[(d*sin[e + f*x])^m*(a + b*(c*tan[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
Time = 8.68 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.85
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+2 a b \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \left (-\frac {\sinh \left (d x +c \right )^{2}}{3 \cosh \left (d x +c \right )^{5}}-\frac {2}{15 \cosh \left (d x +c \right )^{5}}\right )}{d}\) | \(91\) |
| default | \(\frac {a^{2} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+2 a b \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \left (-\frac {\sinh \left (d x +c \right )^{2}}{3 \cosh \left (d x +c \right )^{5}}-\frac {2}{15 \cosh \left (d x +c \right )^{5}}\right )}{d}\) | \(91\) |
| risch | \(-\frac {{\mathrm e}^{d x +c} \left (15 a^{2} {\mathrm e}^{12 d x +12 c}-30 a b \,{\mathrm e}^{12 d x +12 c}+90 a^{2} {\mathrm e}^{10 d x +10 c}+40 b^{2} {\mathrm e}^{10 d x +10 c}+225 a^{2} {\mathrm e}^{8 d x +8 c}+90 a b \,{\mathrm e}^{8 d x +8 c}-96 b^{2} {\mathrm e}^{8 d x +8 c}+300 a^{2} {\mathrm e}^{6 d x +6 c}+112 b^{2} {\mathrm e}^{6 d x +6 c}+225 a^{2} {\mathrm e}^{4 d x +4 c}-90 a b \,{\mathrm e}^{4 d x +4 c}-96 \,{\mathrm e}^{4 d x +4 c} b^{2}+90 a^{2} {\mathrm e}^{2 d x +2 c}+40 \,{\mathrm e}^{2 d x +2 c} b^{2}+15 a^{2}+30 a b \right )}{15 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} \left ({\mathrm e}^{2 d x +2 c}+1\right )^{5}}+\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d}-\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d}+\frac {i b a \ln \left ({\mathrm e}^{d x +c}+i\right )}{d}-\frac {i b a \ln \left ({\mathrm e}^{d x +c}-i\right )}{d}\) | \(312\) |
1/d*(a^2*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+2*a*b*(1/2*sec h(d*x+c)*tanh(d*x+c)+arctan(exp(d*x+c)))+b^2*(-1/3*sinh(d*x+c)^2/cosh(d*x+ c)^5-2/15/cosh(d*x+c)^5))
Leaf count of result is larger than twice the leaf count of optimal. 4642 vs. \(2 (99) = 198\).
Time = 0.32 (sec) , antiderivative size = 4642, normalized size of antiderivative = 43.38 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\text {Too large to display} \]
-1/30*(30*(a^2 - 2*a*b)*cosh(d*x + c)^13 + 390*(a^2 - 2*a*b)*cosh(d*x + c) *sinh(d*x + c)^12 + 30*(a^2 - 2*a*b)*sinh(d*x + c)^13 + 20*(9*a^2 + 4*b^2) *cosh(d*x + c)^11 + 20*(117*(a^2 - 2*a*b)*cosh(d*x + c)^2 + 9*a^2 + 4*b^2) *sinh(d*x + c)^11 + 220*(39*(a^2 - 2*a*b)*cosh(d*x + c)^3 + (9*a^2 + 4*b^2 )*cosh(d*x + c))*sinh(d*x + c)^10 + 6*(75*a^2 + 30*a*b - 32*b^2)*cosh(d*x + c)^9 + 2*(10725*(a^2 - 2*a*b)*cosh(d*x + c)^4 + 550*(9*a^2 + 4*b^2)*cosh (d*x + c)^2 + 225*a^2 + 90*a*b - 96*b^2)*sinh(d*x + c)^9 + 6*(6435*(a^2 - 2*a*b)*cosh(d*x + c)^5 + 550*(9*a^2 + 4*b^2)*cosh(d*x + c)^3 + 9*(75*a^2 + 30*a*b - 32*b^2)*cosh(d*x + c))*sinh(d*x + c)^8 + 8*(75*a^2 + 28*b^2)*cos h(d*x + c)^7 + 8*(6435*(a^2 - 2*a*b)*cosh(d*x + c)^6 + 825*(9*a^2 + 4*b^2) *cosh(d*x + c)^4 + 27*(75*a^2 + 30*a*b - 32*b^2)*cosh(d*x + c)^2 + 75*a^2 + 28*b^2)*sinh(d*x + c)^7 + 8*(6435*(a^2 - 2*a*b)*cosh(d*x + c)^7 + 1155*( 9*a^2 + 4*b^2)*cosh(d*x + c)^5 + 63*(75*a^2 + 30*a*b - 32*b^2)*cosh(d*x + c)^3 + 7*(75*a^2 + 28*b^2)*cosh(d*x + c))*sinh(d*x + c)^6 + 6*(75*a^2 - 30 *a*b - 32*b^2)*cosh(d*x + c)^5 + 6*(6435*(a^2 - 2*a*b)*cosh(d*x + c)^8 + 1 540*(9*a^2 + 4*b^2)*cosh(d*x + c)^6 + 126*(75*a^2 + 30*a*b - 32*b^2)*cosh( d*x + c)^4 + 28*(75*a^2 + 28*b^2)*cosh(d*x + c)^2 + 75*a^2 - 30*a*b - 32*b ^2)*sinh(d*x + c)^5 + 2*(10725*(a^2 - 2*a*b)*cosh(d*x + c)^9 + 3300*(9*a^2 + 4*b^2)*cosh(d*x + c)^7 + 378*(75*a^2 + 30*a*b - 32*b^2)*cosh(d*x + c)^5 + 140*(75*a^2 + 28*b^2)*cosh(d*x + c)^3 + 15*(75*a^2 - 30*a*b - 32*b^2...
\[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right )^{2} \operatorname {csch}^{3}{\left (c + d x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 378 vs. \(2 (99) = 198\).
Time = 0.29 (sec) , antiderivative size = 378, normalized size of antiderivative = 3.53 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=-2 \, a b {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac {1}{2} \, a^{2} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - \frac {8}{15} \, b^{2} {\left (\frac {5 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} - \frac {2 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} \]
-2*a*b*(arctan(e^(-d*x - c))/d - (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))) + 1/2*a^2*(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))) - 8/15*b^2*(5*e^(-3*d*x - 3*c)/( d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(- 8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)) - 2*e^(-5*d*x - 5*c)/(d*(5*e^(-2*d *x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)) + 5*e^(-7*d*x - 7*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d* x - 10*c) + 1)))
Time = 0.39 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.79 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\frac {60 \, a b \arctan \left (e^{\left (d x + c\right )}\right ) + 15 \, a^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) - 15 \, a^{2} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {30 \, {\left (a^{2} e^{\left (3 \, d x + 3 \, c\right )} + a^{2} e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}} + \frac {4 \, {\left (15 \, a b e^{\left (9 \, d x + 9 \, c\right )} + 30 \, a b e^{\left (7 \, d x + 7 \, c\right )} - 20 \, b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 8 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 30 \, a b e^{\left (3 \, d x + 3 \, c\right )} - 20 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 15 \, a b e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{30 \, d} \]
1/30*(60*a*b*arctan(e^(d*x + c)) + 15*a^2*log(e^(d*x + c) + 1) - 15*a^2*lo g(abs(e^(d*x + c) - 1)) - 30*(a^2*e^(3*d*x + 3*c) + a^2*e^(d*x + c))/(e^(2 *d*x + 2*c) - 1)^2 + 4*(15*a*b*e^(9*d*x + 9*c) + 30*a*b*e^(7*d*x + 7*c) - 20*b^2*e^(7*d*x + 7*c) + 8*b^2*e^(5*d*x + 5*c) - 30*a*b*e^(3*d*x + 3*c) - 20*b^2*e^(3*d*x + 3*c) - 15*a*b*e^(d*x + c))/(e^(2*d*x + 2*c) + 1)^5)/d
Time = 3.77 (sec) , antiderivative size = 561, normalized size of antiderivative = 5.24 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\frac {a^2\,{\mathrm {e}}^{c+d\,x}}{d-d\,{\mathrm {e}}^{2\,c+2\,d\,x}}+\frac {136\,b^2\,{\mathrm {e}}^{c+d\,x}}{15\,\left (d+3\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,d\,{\mathrm {e}}^{4\,c+4\,d\,x}+d\,{\mathrm {e}}^{6\,c+6\,d\,x}\right )}+\frac {32\,b^2\,{\mathrm {e}}^{c+d\,x}}{5\,\left (d+5\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,d\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,d\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,d\,{\mathrm {e}}^{8\,c+8\,d\,x}+d\,{\mathrm {e}}^{10\,c+10\,d\,x}\right )}-\frac {a^2\,\ln \left (4\,a^6\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-16\,a^4\,b^2-4\,a^6+16\,a^4\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{2\,d}+\frac {a^2\,\ln \left (4\,a^6+16\,a^4\,b^2+4\,a^6\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+16\,a^4\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{2\,d}-\frac {2\,a^2\,{\mathrm {e}}^{c+d\,x}}{d-2\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+d\,{\mathrm {e}}^{4\,c+4\,d\,x}}-\frac {8\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,\left (d+2\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+d\,{\mathrm {e}}^{4\,c+4\,d\,x}\right )}-\frac {64\,b^2\,{\mathrm {e}}^{c+d\,x}}{5\,\left (d+4\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,d\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,d\,{\mathrm {e}}^{6\,c+6\,d\,x}+d\,{\mathrm {e}}^{8\,c+8\,d\,x}\right )}+\frac {2\,a\,b\,{\mathrm {e}}^{c+d\,x}}{d+d\,{\mathrm {e}}^{2\,c+2\,d\,x}}-\frac {4\,a\,b\,{\mathrm {e}}^{c+d\,x}}{d+2\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+d\,{\mathrm {e}}^{4\,c+4\,d\,x}}-\frac {a\,b\,\left (\ln \left (32\,a^3\,b^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+8\,a^5\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-a^5\,b\,8{}\mathrm {i}-a^3\,b^3\,32{}\mathrm {i}\right )\,1{}\mathrm {i}-\ln \left (32\,a^3\,b^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+8\,a^5\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+a^5\,b\,8{}\mathrm {i}+a^3\,b^3\,32{}\mathrm {i}\right )\,1{}\mathrm {i}\right )}{d} \]
(a^2*exp(c + d*x))/(d - d*exp(2*c + 2*d*x)) + (136*b^2*exp(c + d*x))/(15*( d + 3*d*exp(2*c + 2*d*x) + 3*d*exp(4*c + 4*d*x) + d*exp(6*c + 6*d*x))) + ( 32*b^2*exp(c + d*x))/(5*(d + 5*d*exp(2*c + 2*d*x) + 10*d*exp(4*c + 4*d*x) + 10*d*exp(6*c + 6*d*x) + 5*d*exp(8*c + 8*d*x) + d*exp(10*c + 10*d*x))) - (a^2*log(4*a^6*exp(d*x)*exp(c) - 16*a^4*b^2 - 4*a^6 + 16*a^4*b^2*exp(d*x)* exp(c)))/(2*d) + (a^2*log(4*a^6 + 16*a^4*b^2 + 4*a^6*exp(d*x)*exp(c) + 16* a^4*b^2*exp(d*x)*exp(c)))/(2*d) - (2*a^2*exp(c + d*x))/(d - 2*d*exp(2*c + 2*d*x) + d*exp(4*c + 4*d*x)) - (8*b^2*exp(c + d*x))/(3*(d + 2*d*exp(2*c + 2*d*x) + d*exp(4*c + 4*d*x))) - (64*b^2*exp(c + d*x))/(5*(d + 4*d*exp(2*c + 2*d*x) + 6*d*exp(4*c + 4*d*x) + 4*d*exp(6*c + 6*d*x) + d*exp(8*c + 8*d*x ))) + (2*a*b*exp(c + d*x))/(d + d*exp(2*c + 2*d*x)) - (4*a*b*exp(c + d*x)) /(d + 2*d*exp(2*c + 2*d*x) + d*exp(4*c + 4*d*x)) - (a*b*(log(32*a^3*b^3*ex p(d*x)*exp(c) - a^3*b^3*32i - a^5*b*8i + 8*a^5*b*exp(d*x)*exp(c))*1i - log (a^5*b*8i + a^3*b^3*32i + 32*a^3*b^3*exp(d*x)*exp(c) + 8*a^5*b*exp(d*x)*ex p(c))*1i))/d