Integrand size = 24, antiderivative size = 115 \[ \int \frac {\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^{3/4} d}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^{3/4} d} \]
-1/2*arctan(b^(1/4)*cosh(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/b^(3/4)/d/(a^(1/2 )-b^(1/2))^(1/2)+1/2*arctanh(b^(1/4)*cosh(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/ b^(3/4)/d/(a^(1/2)+b^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 7.33 (sec) , antiderivative size = 365, normalized size of antiderivative = 3.17 \[ \int \frac {\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=-\frac {\text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-c-d x-2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+3 c \text {$\#$1}^2+3 d x \text {$\#$1}^2+6 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-3 c \text {$\#$1}^4-3 d x \text {$\#$1}^4-6 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+c \text {$\#$1}^6+d x \text {$\#$1}^6+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{8 d} \]
-1/8*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , ( -c - d*x - 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2 ]*#1 - Sinh[(c + d*x)/2]*#1] + 3*c*#1^2 + 3*d*x*#1^2 + 6*Log[-Cosh[(c + d* x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*# 1^2 - 3*c*#1^4 - 3*d*x*#1^4 - 6*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + c*#1^6 + d*x*#1^6 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sin h[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^ 7) & ]/d
Time = 0.30 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 26, 3694, 1480, 218, 221}
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.
| \(\displaystyle \int \frac {\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sin (i c+i d x)^3}{a-b \sin (i c+i d x)^4}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\sin (i c+i d x)^3}{a-b \sin (i c+i d x)^4}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {1-\cosh ^2(c+d x)}{-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {-\frac {1}{2} \int \frac {1}{-b \cosh ^2(c+d x)-\left (\sqrt {a}-\sqrt {b}\right ) \sqrt {b}}d\cosh (c+d x)-\frac {1}{2} \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cosh ^2(c+d x)}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {\arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{3/4} \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {1}{2} \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cosh ^2(c+d x)}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{3/4} \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 b^{3/4} \sqrt {\sqrt {a}+\sqrt {b}}}}{d}\) |
-((ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]]/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(3/4)) - ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt [b]]]/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(3/4)))/d)
3.3.31.3.1 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)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.36 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.77
| method | result | size |
| risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (-1+\left (256 a \,b^{3} d^{4}-256 b^{4} d^{4}\right ) \textit {\_Z}^{4}+32 d^{2} \textit {\_Z}^{2} b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (128 a \,b^{2} d^{3}-128 b^{3} d^{3}\right ) \textit {\_R}^{3}+16 b d \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\) | \(89\) |
| derivativedivides | \(\frac {8 a \left (-\frac {\sqrt {a b}\, \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{16 a b \sqrt {-a b +\sqrt {a b}\, a}}-\frac {\sqrt {a b}\, \arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{16 a b \sqrt {-a b -\sqrt {a b}\, a}}\right )}{d}\) | \(148\) |
| default | \(\frac {8 a \left (-\frac {\sqrt {a b}\, \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{16 a b \sqrt {-a b +\sqrt {a b}\, a}}-\frac {\sqrt {a b}\, \arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{16 a b \sqrt {-a b -\sqrt {a b}\, a}}\right )}{d}\) | \(148\) |
sum(_R*ln(exp(2*d*x+2*c)+((128*a*b^2*d^3-128*b^3*d^3)*_R^3+16*b*d*_R)*exp( d*x+c)+1),_R=RootOf(-1+(256*a*b^3*d^4-256*b^4*d^4)*_Z^4+32*d^2*_Z^2*b^2))
Leaf count of result is larger than twice the leaf count of optimal. 975 vs. \(2 (79) = 158\).
Time = 0.29 (sec) , antiderivative size = 975, normalized size of antiderivative = 8.48 \[ \int \frac {\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\text {Too large to display} \]
1/4*sqrt(-((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + 1)/(( a*b - b^2)*d^2))*log(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sin h(d*x + c)^2 + 2*(b*d*cosh(d*x + c) + b*d*sinh(d*x + c) - ((a*b^2 - b^3)*d ^3*cosh(d*x + c) + (a*b^2 - b^3)*d^3*sinh(d*x + c))*sqrt(a/((a^2*b^3 - 2*a *b^4 + b^5)*d^4)))*sqrt(-((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5 )*d^4)) + 1)/((a*b - b^2)*d^2)) + 1) - 1/4*sqrt(-((a*b - b^2)*d^2*sqrt(a/( (a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + 1)/((a*b - b^2)*d^2))*log(cosh(d*x + c)^ 2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 2*(b*d*cosh(d*x + c) + b*d*sinh(d*x + c) - ((a*b^2 - b^3)*d^3*cosh(d*x + c) + (a*b^2 - b^3)*d^ 3*sinh(d*x + c))*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)))*sqrt(-((a*b - b^ 2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + 1)/((a*b - b^2)*d^2)) + 1 ) + 1/4*sqrt(((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - 1) /((a*b - b^2)*d^2))*log(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 2*(b*d*cosh(d*x + c) + b*d*sinh(d*x + c) + ((a*b^2 - b^3 )*d^3*cosh(d*x + c) + (a*b^2 - b^3)*d^3*sinh(d*x + c))*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)))*sqrt(((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b ^5)*d^4)) - 1)/((a*b - b^2)*d^2)) + 1) - 1/4*sqrt(((a*b - b^2)*d^2*sqrt(a/ ((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - 1)/((a*b - b^2)*d^2))*log(cosh(d*x + c) ^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 2*(b*d*cosh(d*x + c ) + b*d*sinh(d*x + c) + ((a*b^2 - b^3)*d^3*cosh(d*x + c) + (a*b^2 - b^3...
Timed out. \[ \int \frac {\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\int { -\frac {\sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right )^{4} - a} \,d x } \]
\[ \int \frac {\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\int { -\frac {\sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right )^{4} - a} \,d x } \]
Time = 6.95 (sec) , antiderivative size = 975, normalized size of antiderivative = 8.48 \[ \int \frac {\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^4\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^7\,{\left (a-b\right )}^2}-\frac {8388608\,a^4\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )\,\sqrt {-\frac {b^2-\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{b^7\,\left (a-b\right )}\right )\,\sqrt {-\frac {b^2-\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{4}+\frac {2097152\,a^4\,d\,{\mathrm {e}}^{c+d\,x}}{b^8\,\left (a-b\right )}\right )\,\sqrt {-\frac {b^2-\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a^4\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^9\,{\left (a-b\right )}^2}\right )\,\sqrt {\frac {b^2-\sqrt {a\,b^3}}{16\,\left (b^4\,d^2-a\,b^3\,d^2\right )}}-\ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^4\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^7\,{\left (a-b\right )}^2}+\frac {8388608\,a^4\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )\,\sqrt {-\frac {b^2-\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{b^7\,\left (a-b\right )}\right )\,\sqrt {-\frac {b^2-\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{4}-\frac {2097152\,a^4\,d\,{\mathrm {e}}^{c+d\,x}}{b^8\,\left (a-b\right )}\right )\,\sqrt {-\frac {b^2-\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a^4\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^9\,{\left (a-b\right )}^2}\right )\,\sqrt {\frac {b^2-\sqrt {a\,b^3}}{16\,\left (b^4\,d^2-a\,b^3\,d^2\right )}}+\ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^4\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^7\,{\left (a-b\right )}^2}-\frac {8388608\,a^4\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )\,\sqrt {-\frac {b^2+\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{b^7\,\left (a-b\right )}\right )\,\sqrt {-\frac {b^2+\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{4}+\frac {2097152\,a^4\,d\,{\mathrm {e}}^{c+d\,x}}{b^8\,\left (a-b\right )}\right )\,\sqrt {-\frac {b^2+\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a^4\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^9\,{\left (a-b\right )}^2}\right )\,\sqrt {\frac {b^2+\sqrt {a\,b^3}}{16\,\left (b^4\,d^2-a\,b^3\,d^2\right )}}-\ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^4\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^7\,{\left (a-b\right )}^2}+\frac {8388608\,a^4\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )\,\sqrt {-\frac {b^2+\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{b^7\,\left (a-b\right )}\right )\,\sqrt {-\frac {b^2+\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{4}-\frac {2097152\,a^4\,d\,{\mathrm {e}}^{c+d\,x}}{b^8\,\left (a-b\right )}\right )\,\sqrt {-\frac {b^2+\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a^4\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^9\,{\left (a-b\right )}^2}\right )\,\sqrt {\frac {b^2+\sqrt {a\,b^3}}{16\,\left (b^4\,d^2-a\,b^3\,d^2\right )}} \]
log((((((4194304*a^4*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^7*(a - b)^2) - (8388608*a^4*d^3*exp(c + d*x)*(a + b)*(-(b^2 - (a*b^3)^(1/2))/(b^3*d^2* (a - b)))^(1/2))/(b^7*(a - b)))*(-(b^2 - (a*b^3)^(1/2))/(b^3*d^2*(a - b))) ^(1/2))/4 + (2097152*a^4*d*exp(c + d*x))/(b^8*(a - b)))*(-(b^2 - (a*b^3)^( 1/2))/(b^3*d^2*(a - b)))^(1/2))/4 - (262144*a^4*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^9*(a - b)^2))*((b^2 - (a*b^3)^(1/2))/(16*(b^4*d^2 - a*b^3*d^2)))^ (1/2) - log((((((4194304*a^4*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^7*(a - b)^2) + (8388608*a^4*d^3*exp(c + d*x)*(a + b)*(-(b^2 - (a*b^3)^(1/2))/( b^3*d^2*(a - b)))^(1/2))/(b^7*(a - b)))*(-(b^2 - (a*b^3)^(1/2))/(b^3*d^2*( a - b)))^(1/2))/4 - (2097152*a^4*d*exp(c + d*x))/(b^8*(a - b)))*(-(b^2 - ( a*b^3)^(1/2))/(b^3*d^2*(a - b)))^(1/2))/4 - (262144*a^4*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^9*(a - b)^2))*((b^2 - (a*b^3)^(1/2))/(16*(b^4*d^2 - a*b^3 *d^2)))^(1/2) + log((((((4194304*a^4*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b)) /(b^7*(a - b)^2) - (8388608*a^4*d^3*exp(c + d*x)*(a + b)*(-(b^2 + (a*b^3)^ (1/2))/(b^3*d^2*(a - b)))^(1/2))/(b^7*(a - b)))*(-(b^2 + (a*b^3)^(1/2))/(b ^3*d^2*(a - b)))^(1/2))/4 + (2097152*a^4*d*exp(c + d*x))/(b^8*(a - b)))*(- (b^2 + (a*b^3)^(1/2))/(b^3*d^2*(a - b)))^(1/2))/4 - (262144*a^4*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^9*(a - b)^2))*((b^2 + (a*b^3)^(1/2))/(16*(b^4*d^2 - a*b^3*d^2)))^(1/2) - log((((((4194304*a^4*d^2*(exp(2*c + 2*d*x) + 1)*(3 *a + b))/(b^7*(a - b)^2) + (8388608*a^4*d^3*exp(c + d*x)*(a + b)*(-(b^2...