Integrand size = 31, antiderivative size = 81 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+c \sinh (d+e x)} \, dx=\frac {C x}{c}-\frac {2 (A c-a C) \text {arctanh}\left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2+c^2}}\right )}{c \sqrt {a^2+c^2} e}+\frac {B \log (a+c \sinh (d+e x))}{c e} \]
C*x/c+B*ln(a+c*sinh(e*x+d))/c/e-2*(A*c-C*a)*arctanh((c-a*tanh(1/2*e*x+1/2* d))/(a^2+c^2)^(1/2))/c/e/(a^2+c^2)^(1/2)
Time = 2.33 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.05 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+c \sinh (d+e x)} \, dx=\frac {C (d+e x)+\frac {2 (A c-a C) \arctan \left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2-c^2}}\right )}{\sqrt {-a^2-c^2}}+B \log (a+c \sinh (d+e x))}{c e} \]
(C*(d + e*x) + (2*(A*c - a*C)*ArcTan[(c - a*Tanh[(d + e*x)/2])/Sqrt[-a^2 - c^2]])/Sqrt[-a^2 - c^2] + B*Log[a + c*Sinh[d + e*x]])/(c*e)
Time = 0.52 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.98, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {3042, 4876, 3042, 3147, 16, 3214, 3042, 3139, 1083, 217}
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.
| \(\displaystyle \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+c \sinh (d+e x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \cos (i d+i e x)-i C \sin (i d+i e x)}{a-i c \sin (i d+i e x)}dx\) |
| \(\Big \downarrow \) 4876 |
| \(\displaystyle \int \frac {A+C \sinh (d+e x)}{a+c \sinh (d+e x)}dx+B \int \frac {\cosh (d+e x)}{a+c \sinh (d+e x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A-i C \sin (i d+i e x)}{a-i c \sin (i d+i e x)}dx+B \int \frac {\cos (i d+i e x)}{a-i c \sin (i d+i e x)}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {B \int \frac {1}{a+c \sinh (d+e x)}d(c \sinh (d+e x))}{c e}+\int \frac {A-i C \sin (i d+i e x)}{a-i c \sin (i d+i e x)}dx\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {B \log (a+c \sinh (d+e x))}{c e}+\int \frac {A-i C \sin (i d+i e x)}{a-i c \sin (i d+i e x)}dx\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {(A c-a C) \int \frac {1}{a+c \sinh (d+e x)}dx}{c}+\frac {B \log (a+c \sinh (d+e x))}{c e}+\frac {C x}{c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A c-a C) \int \frac {1}{a-i c \sin (i d+i e x)}dx}{c}+\frac {B \log (a+c \sinh (d+e x))}{c e}+\frac {C x}{c}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {2 i (A c-a C) \int \frac {1}{-a \tanh ^2\left (\frac {1}{2} (d+e x)\right )+2 c \tanh \left (\frac {1}{2} (d+e x)\right )+a}d\left (i \tanh \left (\frac {1}{2} (d+e x)\right )\right )}{c e}+\frac {B \log (a+c \sinh (d+e x))}{c e}+\frac {C x}{c}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {4 i (A c-a C) \int \frac {1}{\tanh ^2\left (\frac {1}{2} (d+e x)\right )-4 \left (a^2+c^2\right )}d\left (2 i a \tanh \left (\frac {1}{2} (d+e x)\right )-2 i c\right )}{c e}+\frac {B \log (a+c \sinh (d+e x))}{c e}+\frac {C x}{c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 (A c-a C) \text {arctanh}\left (\frac {\tanh \left (\frac {1}{2} (d+e x)\right )}{2 \sqrt {a^2+c^2}}\right )}{c e \sqrt {a^2+c^2}}+\frac {B \log (a+c \sinh (d+e x))}{c e}+\frac {C x}{c}\) |
(C*x)/c + (2*(A*c - a*C)*ArcTanh[Tanh[(d + e*x)/2]/(2*Sqrt[a^2 + c^2])])/( c*Sqrt[a^2 + c^2]*e) + (B*Log[a + c*Sinh[d + e*x]])/(c*e)
3.3.53.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[(u_)*((v_) + (d_.)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_.)), x_Symbol] : > With[{e = FreeFactors[Sin[c*(a + b*x)], x]}, Int[ActivateTrig[u*v], x] + Simp[d Int[ActivateTrig[u]*Cos[c*(a + b*x)]^n, x], x] /; FunctionOfQ[Sin[ c*(a + b*x)]/e, u, x]] /; FreeQ[{a, b, c, d}, x] && !FreeQ[v, x] && Intege rQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Cos] || EqQ[F, cos])
Time = 2.62 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.37
| method | result | size |
| parts | \(\frac {-\frac {2 \left (-A c +C a \right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-2 c}{2 \sqrt {a^{2}+c^{2}}}\right )}{c \sqrt {a^{2}+c^{2}}}+\frac {C \ln \left (\tanh \left (\frac {e x}{2}+\frac {d}{2}\right )+1\right )}{c}-\frac {C \ln \left (\tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-1\right )}{c}}{e}+\frac {B \ln \left (a +c \sinh \left (e x +d \right )\right )}{c e}\) | \(111\) |
| derivativedivides | \(\frac {\frac {\left (-B -C \right ) \ln \left (\tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-1\right )}{c}+\frac {\left (-B +C \right ) \ln \left (\tanh \left (\frac {e x}{2}+\frac {d}{2}\right )+1\right )}{c}+\frac {B \ln \left (a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-2 c \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-a \right )-\frac {2 \left (-A c +C a \right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-2 c}{2 \sqrt {a^{2}+c^{2}}}\right )}{\sqrt {a^{2}+c^{2}}}}{c}}{e}\) | \(136\) |
| default | \(\frac {\frac {\left (-B -C \right ) \ln \left (\tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-1\right )}{c}+\frac {\left (-B +C \right ) \ln \left (\tanh \left (\frac {e x}{2}+\frac {d}{2}\right )+1\right )}{c}+\frac {B \ln \left (a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-2 c \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-a \right )-\frac {2 \left (-A c +C a \right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-2 c}{2 \sqrt {a^{2}+c^{2}}}\right )}{\sqrt {a^{2}+c^{2}}}}{c}}{e}\) | \(136\) |
| risch | \(\frac {x B}{c}+\frac {C x}{c}-\frac {2 B \,a^{2} c \,e^{2} x}{a^{2} c^{2} e^{2}+c^{4} e^{2}}-\frac {2 B \,c^{3} e^{2} x}{a^{2} c^{2} e^{2}+c^{4} e^{2}}-\frac {2 B \,a^{2} c d e}{a^{2} c^{2} e^{2}+c^{4} e^{2}}-\frac {2 B \,c^{3} d e}{a^{2} c^{2} e^{2}+c^{4} e^{2}}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {A a c -a^{2} C -\sqrt {A^{2} a^{2} c^{2}+A^{2} c^{4}-2 A C \,a^{3} c -2 A C a \,c^{3}+C^{2} a^{4}+C^{2} a^{2} c^{2}}}{c \left (A c -C a \right )}\right ) B \,a^{2}}{\left (a^{2}+c^{2}\right ) e c}+\frac {c \ln \left ({\mathrm e}^{e x +d}+\frac {A a c -a^{2} C -\sqrt {A^{2} a^{2} c^{2}+A^{2} c^{4}-2 A C \,a^{3} c -2 A C a \,c^{3}+C^{2} a^{4}+C^{2} a^{2} c^{2}}}{c \left (A c -C a \right )}\right ) B}{\left (a^{2}+c^{2}\right ) e}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {A a c -a^{2} C -\sqrt {A^{2} a^{2} c^{2}+A^{2} c^{4}-2 A C \,a^{3} c -2 A C a \,c^{3}+C^{2} a^{4}+C^{2} a^{2} c^{2}}}{c \left (A c -C a \right )}\right ) \sqrt {A^{2} a^{2} c^{2}+A^{2} c^{4}-2 A C \,a^{3} c -2 A C a \,c^{3}+C^{2} a^{4}+C^{2} a^{2} c^{2}}}{\left (a^{2}+c^{2}\right ) e c}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {A a c -a^{2} C +\sqrt {A^{2} a^{2} c^{2}+A^{2} c^{4}-2 A C \,a^{3} c -2 A C a \,c^{3}+C^{2} a^{4}+C^{2} a^{2} c^{2}}}{c \left (A c -C a \right )}\right ) B \,a^{2}}{\left (a^{2}+c^{2}\right ) e c}+\frac {c \ln \left ({\mathrm e}^{e x +d}+\frac {A a c -a^{2} C +\sqrt {A^{2} a^{2} c^{2}+A^{2} c^{4}-2 A C \,a^{3} c -2 A C a \,c^{3}+C^{2} a^{4}+C^{2} a^{2} c^{2}}}{c \left (A c -C a \right )}\right ) B}{\left (a^{2}+c^{2}\right ) e}-\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {A a c -a^{2} C +\sqrt {A^{2} a^{2} c^{2}+A^{2} c^{4}-2 A C \,a^{3} c -2 A C a \,c^{3}+C^{2} a^{4}+C^{2} a^{2} c^{2}}}{c \left (A c -C a \right )}\right ) \sqrt {A^{2} a^{2} c^{2}+A^{2} c^{4}-2 A C \,a^{3} c -2 A C a \,c^{3}+C^{2} a^{4}+C^{2} a^{2} c^{2}}}{\left (a^{2}+c^{2}\right ) e c}\) | \(863\) |
1/e*(-2*(-A*c+C*a)/c/(a^2+c^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2*e*x+1/2*d)- 2*c)/(a^2+c^2)^(1/2))+C/c*ln(tanh(1/2*e*x+1/2*d)+1)-C/c*ln(tanh(1/2*e*x+1/ 2*d)-1))+B*ln(a+c*sinh(e*x+d))/c/e
Leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (78) = 156\).
Time = 0.28 (sec) , antiderivative size = 249, normalized size of antiderivative = 3.07 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+c \sinh (d+e x)} \, dx=-\frac {{\left ({\left (B - C\right )} a^{2} + {\left (B - C\right )} c^{2}\right )} e x + {\left (C a - A c\right )} \sqrt {a^{2} + c^{2}} \log \left (\frac {c^{2} \cosh \left (e x + d\right )^{2} + c^{2} \sinh \left (e x + d\right )^{2} + 2 \, a c \cosh \left (e x + d\right ) + 2 \, a^{2} + c^{2} + 2 \, {\left (c^{2} \cosh \left (e x + d\right ) + a c\right )} \sinh \left (e x + d\right ) - 2 \, \sqrt {a^{2} + c^{2}} {\left (c \cosh \left (e x + d\right ) + c \sinh \left (e x + d\right ) + a\right )}}{c \cosh \left (e x + d\right )^{2} + c \sinh \left (e x + d\right )^{2} + 2 \, a \cosh \left (e x + d\right ) + 2 \, {\left (c \cosh \left (e x + d\right ) + a\right )} \sinh \left (e x + d\right ) - c}\right ) - {\left (B a^{2} + B c^{2}\right )} \log \left (\frac {2 \, {\left (c \sinh \left (e x + d\right ) + a\right )}}{\cosh \left (e x + d\right ) - \sinh \left (e x + d\right )}\right )}{{\left (a^{2} c + c^{3}\right )} e} \]
-(((B - C)*a^2 + (B - C)*c^2)*e*x + (C*a - A*c)*sqrt(a^2 + c^2)*log((c^2*c osh(e*x + d)^2 + c^2*sinh(e*x + d)^2 + 2*a*c*cosh(e*x + d) + 2*a^2 + c^2 + 2*(c^2*cosh(e*x + d) + a*c)*sinh(e*x + d) - 2*sqrt(a^2 + c^2)*(c*cosh(e*x + d) + c*sinh(e*x + d) + a))/(c*cosh(e*x + d)^2 + c*sinh(e*x + d)^2 + 2*a *cosh(e*x + d) + 2*(c*cosh(e*x + d) + a)*sinh(e*x + d) - c)) - (B*a^2 + B* c^2)*log(2*(c*sinh(e*x + d) + a)/(cosh(e*x + d) - sinh(e*x + d))))/((a^2*c + c^3)*e)
Result contains complex when optimal does not.
Time = 18.12 (sec) , antiderivative size = 1318, normalized size of antiderivative = 16.27 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+c \sinh (d+e x)} \, dx=\text {Too large to display} \]
Piecewise((zoo*x*(A + B*cosh(d) + C*sinh(d))/sinh(d), Eq(a, 0) & Eq(c, 0) & Eq(e, 0)), ((A*log(tanh(d/2 + e*x/2))/e + B*x - 2*B*log(tanh(d/2 + e*x/2 ) + 1)/e + B*log(tanh(d/2 + e*x/2))/e + C*x)/c, Eq(a, 0)), (2*I*A/(c*e*tan h(d/2 + e*x/2) - I*c*e) + B*e*x*tanh(d/2 + e*x/2)/(c*e*tanh(d/2 + e*x/2) - I*c*e) - I*B*e*x/(c*e*tanh(d/2 + e*x/2) - I*c*e) - 2*B*log(tanh(d/2 + e*x /2) + 1)*tanh(d/2 + e*x/2)/(c*e*tanh(d/2 + e*x/2) - I*c*e) + 2*I*B*log(tan h(d/2 + e*x/2) + 1)/(c*e*tanh(d/2 + e*x/2) - I*c*e) + 2*B*log(tanh(d/2 + e *x/2) - I)*tanh(d/2 + e*x/2)/(c*e*tanh(d/2 + e*x/2) - I*c*e) - 2*I*B*log(t anh(d/2 + e*x/2) - I)/(c*e*tanh(d/2 + e*x/2) - I*c*e) + C*e*x*tanh(d/2 + e *x/2)/(c*e*tanh(d/2 + e*x/2) - I*c*e) - I*C*e*x/(c*e*tanh(d/2 + e*x/2) - I *c*e) - 2*C/(c*e*tanh(d/2 + e*x/2) - I*c*e), Eq(a, -I*c)), (-2*I*A/(c*e*ta nh(d/2 + e*x/2) + I*c*e) + B*e*x*tanh(d/2 + e*x/2)/(c*e*tanh(d/2 + e*x/2) + I*c*e) + I*B*e*x/(c*e*tanh(d/2 + e*x/2) + I*c*e) - 2*B*log(tanh(d/2 + e* x/2) + 1)*tanh(d/2 + e*x/2)/(c*e*tanh(d/2 + e*x/2) + I*c*e) - 2*I*B*log(ta nh(d/2 + e*x/2) + 1)/(c*e*tanh(d/2 + e*x/2) + I*c*e) + 2*B*log(tanh(d/2 + e*x/2) + I)*tanh(d/2 + e*x/2)/(c*e*tanh(d/2 + e*x/2) + I*c*e) + 2*I*B*log( tanh(d/2 + e*x/2) + I)/(c*e*tanh(d/2 + e*x/2) + I*c*e) + C*e*x*tanh(d/2 + e*x/2)/(c*e*tanh(d/2 + e*x/2) + I*c*e) + I*C*e*x/(c*e*tanh(d/2 + e*x/2) + I*c*e) - 2*C/(c*e*tanh(d/2 + e*x/2) + I*c*e), Eq(a, I*c)), ((A*x + B*sinh( d + e*x)/e + C*cosh(d + e*x)/e)/a, Eq(c, 0)), (x*(A + B*cosh(d) + C*sin...
Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (78) = 156\).
Time = 0.32 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.17 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+c \sinh (d+e x)} \, dx=-C {\left (\frac {a \log \left (\frac {c e^{\left (-e x - d\right )} - a - \sqrt {a^{2} + c^{2}}}{c e^{\left (-e x - d\right )} - a + \sqrt {a^{2} + c^{2}}}\right )}{\sqrt {a^{2} + c^{2}} c e} - \frac {e x + d}{c e}\right )} + \frac {A \log \left (\frac {c e^{\left (-e x - d\right )} - a - \sqrt {a^{2} + c^{2}}}{c e^{\left (-e x - d\right )} - a + \sqrt {a^{2} + c^{2}}}\right )}{\sqrt {a^{2} + c^{2}} e} + \frac {B \log \left (c \sinh \left (e x + d\right ) + a\right )}{c e} \]
-C*(a*log((c*e^(-e*x - d) - a - sqrt(a^2 + c^2))/(c*e^(-e*x - d) - a + sqr t(a^2 + c^2)))/(sqrt(a^2 + c^2)*c*e) - (e*x + d)/(c*e)) + A*log((c*e^(-e*x - d) - a - sqrt(a^2 + c^2))/(c*e^(-e*x - d) - a + sqrt(a^2 + c^2)))/(sqrt (a^2 + c^2)*e) + B*log(c*sinh(e*x + d) + a)/(c*e)
Time = 0.29 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.57 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+c \sinh (d+e x)} \, dx=-\frac {\frac {{\left (e x + d\right )} {\left (B - C\right )}}{c} - \frac {B \log \left ({\left | c e^{\left (2 \, e x + 2 \, d\right )} + 2 \, a e^{\left (e x + d\right )} - c \right |}\right )}{c} + \frac {{\left (C a - A c\right )} \log \left (\frac {{\left | 2 \, c e^{\left (e x + d\right )} + 2 \, a - 2 \, \sqrt {a^{2} + c^{2}} \right |}}{{\left | 2 \, c e^{\left (e x + d\right )} + 2 \, a + 2 \, \sqrt {a^{2} + c^{2}} \right |}}\right )}{\sqrt {a^{2} + c^{2}} c}}{e} \]
-((e*x + d)*(B - C)/c - B*log(abs(c*e^(2*e*x + 2*d) + 2*a*e^(e*x + d) - c) )/c + (C*a - A*c)*log(abs(2*c*e^(e*x + d) + 2*a - 2*sqrt(a^2 + c^2))/abs(2 *c*e^(e*x + d) + 2*a + 2*sqrt(a^2 + c^2)))/(sqrt(a^2 + c^2)*c))/e
Time = 2.63 (sec) , antiderivative size = 656, normalized size of antiderivative = 8.10 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+c \sinh (d+e x)} \, dx=\frac {C\,x}{c}-\frac {B\,x}{c}-\frac {2\,\mathrm {atan}\left (\frac {a\,\sqrt {-a^2\,c^2\,e^2-c^4\,e^2}\,\sqrt {A^2\,c^2-2\,A\,C\,a\,c+C^2\,a^2}}{-C\,e\,a^3\,c+A\,e\,a^2\,c^2-C\,e\,a\,c^3+A\,e\,c^4}-\frac {a^2\,c^2\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d\,\sqrt {-a^2\,c^2\,e^2-c^4\,e^2}\,\sqrt {A^2\,c^2-2\,A\,C\,a\,c+C^2\,a^2}}{-C\,e\,a^3\,c^4+A\,e\,a^2\,c^5-C\,e\,a\,c^6+A\,e\,c^7}+\frac {A\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d\,\sqrt {-a^2\,c^2\,e^2-c^4\,e^2}}{c\,e\,\sqrt {A^2\,c^2-2\,A\,C\,a\,c+C^2\,a^2}}-\frac {C\,a\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d\,\sqrt {-a^2\,c^2\,e^2-c^4\,e^2}}{c^2\,e\,\sqrt {A^2\,c^2-2\,A\,C\,a\,c+C^2\,a^2}}\right )\,\sqrt {A^2\,c^2-2\,A\,C\,a\,c+C^2\,a^2}}{\sqrt {-a^2\,c^2\,e^2-c^4\,e^2}}+\frac {B\,c^3\,e\,\ln \left (8\,A\,C\,a\,c^2-4\,C^2\,a^2\,c-4\,A^2\,c^3+8\,C^2\,a^3\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d+4\,A^2\,c^3\,{\mathrm {e}}^{2\,d}\,{\mathrm {e}}^{2\,e\,x}+8\,A^2\,a\,c^2\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d+4\,C^2\,a^2\,c\,{\mathrm {e}}^{2\,d}\,{\mathrm {e}}^{2\,e\,x}-16\,A\,C\,a^2\,c\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d-8\,A\,C\,a\,c^2\,{\mathrm {e}}^{2\,d}\,{\mathrm {e}}^{2\,e\,x}\right )}{a^2\,c^2\,e^2+c^4\,e^2}+\frac {B\,a^2\,c\,e\,\ln \left (8\,A\,C\,a\,c^2-4\,C^2\,a^2\,c-4\,A^2\,c^3+8\,C^2\,a^3\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d+4\,A^2\,c^3\,{\mathrm {e}}^{2\,d}\,{\mathrm {e}}^{2\,e\,x}+8\,A^2\,a\,c^2\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d+4\,C^2\,a^2\,c\,{\mathrm {e}}^{2\,d}\,{\mathrm {e}}^{2\,e\,x}-16\,A\,C\,a^2\,c\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d-8\,A\,C\,a\,c^2\,{\mathrm {e}}^{2\,d}\,{\mathrm {e}}^{2\,e\,x}\right )}{a^2\,c^2\,e^2+c^4\,e^2} \]
(C*x)/c - (B*x)/c - (2*atan((a*(- c^4*e^2 - a^2*c^2*e^2)^(1/2)*(A^2*c^2 + C^2*a^2 - 2*A*C*a*c)^(1/2))/(A*c^4*e - C*a*c^3*e - C*a^3*c*e + A*a^2*c^2*e ) - (a^2*c^2*exp(e*x)*exp(d)*(- c^4*e^2 - a^2*c^2*e^2)^(1/2)*(A^2*c^2 + C^ 2*a^2 - 2*A*C*a*c)^(1/2))/(A*c^7*e - C*a*c^6*e + A*a^2*c^5*e - C*a^3*c^4*e ) + (A*exp(e*x)*exp(d)*(- c^4*e^2 - a^2*c^2*e^2)^(1/2))/(c*e*(A^2*c^2 + C^ 2*a^2 - 2*A*C*a*c)^(1/2)) - (C*a*exp(e*x)*exp(d)*(- c^4*e^2 - a^2*c^2*e^2) ^(1/2))/(c^2*e*(A^2*c^2 + C^2*a^2 - 2*A*C*a*c)^(1/2)))*(A^2*c^2 + C^2*a^2 - 2*A*C*a*c)^(1/2))/(- c^4*e^2 - a^2*c^2*e^2)^(1/2) + (B*c^3*e*log(8*A*C*a *c^2 - 4*C^2*a^2*c - 4*A^2*c^3 + 8*C^2*a^3*exp(e*x)*exp(d) + 4*A^2*c^3*exp (2*d)*exp(2*e*x) + 8*A^2*a*c^2*exp(e*x)*exp(d) + 4*C^2*a^2*c*exp(2*d)*exp( 2*e*x) - 16*A*C*a^2*c*exp(e*x)*exp(d) - 8*A*C*a*c^2*exp(2*d)*exp(2*e*x)))/ (c^4*e^2 + a^2*c^2*e^2) + (B*a^2*c*e*log(8*A*C*a*c^2 - 4*C^2*a^2*c - 4*A^2 *c^3 + 8*C^2*a^3*exp(e*x)*exp(d) + 4*A^2*c^3*exp(2*d)*exp(2*e*x) + 8*A^2*a *c^2*exp(e*x)*exp(d) + 4*C^2*a^2*c*exp(2*d)*exp(2*e*x) - 16*A*C*a^2*c*exp( e*x)*exp(d) - 8*A*C*a*c^2*exp(2*d)*exp(2*e*x)))/(c^4*e^2 + a^2*c^2*e^2)