Integrand size = 31, antiderivative size = 250 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx=-\frac {\left (2 a^3 A-3 a A c^2+4 a^2 c C-c^3 C\right ) \text {arctanh}\left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2+c^2}}\right )}{\left (a^2+c^2\right )^{7/2} e}-\frac {B}{3 c e (a+c \sinh (d+e x))^3}-\frac {(A c-a C) \cosh (d+e x)}{3 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^3}-\frac {\left (5 a A c-2 a^2 C+3 c^2 C\right ) \cosh (d+e x)}{6 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))^2}-\frac {\left (11 a^2 A c-4 A c^3-2 a^3 C+13 a c^2 C\right ) \cosh (d+e x)}{6 \left (a^2+c^2\right )^3 e (a+c \sinh (d+e x))} \]
-(2*A*a^3-3*A*a*c^2+4*C*a^2*c-C*c^3)*arctanh((c-a*tanh(1/2*e*x+1/2*d))/(a^ 2+c^2)^(1/2))/(a^2+c^2)^(7/2)/e-1/3*B/c/e/(a+c*sinh(e*x+d))^3-1/3*(A*c-C*a )*cosh(e*x+d)/(a^2+c^2)/e/(a+c*sinh(e*x+d))^3-1/6*(5*A*a*c-2*C*a^2+3*C*c^2 )*cosh(e*x+d)/(a^2+c^2)^2/e/(a+c*sinh(e*x+d))^2-1/6*(11*A*a^2*c-4*A*c^3-2* C*a^3+13*C*a*c^2)*cosh(e*x+d)/(a^2+c^2)^3/e/(a+c*sinh(e*x+d))
Time = 1.83 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.94 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx=\frac {\frac {6 \left (2 a^3 A-3 a A c^2+4 a^2 c C-c^3 C\right ) \arctan \left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2-c^2}}\right )}{\sqrt {-a^2-c^2}}-\frac {2 \left (a^2+c^2\right )^2 \left (B \left (a^2+c^2\right )+c (A c-a C) \cosh (d+e x)\right )}{c (a+c \sinh (d+e x))^3}+\frac {\left (a^2+c^2\right ) \left (-5 a A c+2 a^2 C-3 c^2 C\right ) \cosh (d+e x)}{(a+c \sinh (d+e x))^2}+\frac {\left (-11 a^2 A c+4 A c^3+2 a^3 C-13 a c^2 C\right ) \cosh (d+e x)}{a+c \sinh (d+e x)}}{6 \left (a^2+c^2\right )^3 e} \]
((6*(2*a^3*A - 3*a*A*c^2 + 4*a^2*c*C - c^3*C)*ArcTan[(c - a*Tanh[(d + e*x) /2])/Sqrt[-a^2 - c^2]])/Sqrt[-a^2 - c^2] - (2*(a^2 + c^2)^2*(B*(a^2 + c^2) + c*(A*c - a*C)*Cosh[d + e*x]))/(c*(a + c*Sinh[d + e*x])^3) + ((a^2 + c^2 )*(-5*a*A*c + 2*a^2*C - 3*c^2*C)*Cosh[d + e*x])/(a + c*Sinh[d + e*x])^2 + ((-11*a^2*A*c + 4*A*c^3 + 2*a^3*C - 13*a*c^2*C)*Cosh[d + e*x])/(a + c*Sinh [d + e*x]))/(6*(a^2 + c^2)^3*e)
Time = 1.17 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.10, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.548, Rules used = {3042, 4876, 3042, 3147, 17, 3233, 25, 3042, 3233, 25, 3042, 3233, 27, 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.
| \(\displaystyle \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \cos (i d+i e x)-i C \sin (i d+i e x)}{(a-i c \sin (i d+i e x))^4}dx\) |
| \(\Big \downarrow \) 4876 |
| \(\displaystyle \int \frac {A+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4}dx+B \int \frac {\cosh (d+e x)}{(a+c \sinh (d+e x))^4}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A-i C \sin (i d+i e x)}{(a-i c \sin (i d+i e x))^4}dx+B \int \frac {\cos (i d+i e x)}{(a-i c \sin (i d+i e x))^4}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {B \int \frac {1}{(a+c \sinh (d+e x))^4}d(c \sinh (d+e x))}{c e}+\int \frac {A-i C \sin (i d+i e x)}{(a-i c \sin (i d+i e x))^4}dx\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {B}{3 c e (a+c \sinh (d+e x))^3}+\int \frac {A-i C \sin (i d+i e x)}{(a-i c \sin (i d+i e x))^4}dx\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {\int -\frac {3 (a A+c C)-2 (A c-a C) \sinh (d+e x)}{(a+c \sinh (d+e x))^3}dx}{3 \left (a^2+c^2\right )}-\frac {(A c-a C) \cosh (d+e x)}{3 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^3}-\frac {B}{3 c e (a+c \sinh (d+e x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 (a A+c C)-2 (A c-a C) \sinh (d+e x)}{(a+c \sinh (d+e x))^3}dx}{3 \left (a^2+c^2\right )}-\frac {(A c-a C) \cosh (d+e x)}{3 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^3}-\frac {B}{3 c e (a+c \sinh (d+e x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 (a A+c C)+2 i (A c-a C) \sin (i d+i e x)}{(a-i c \sin (i d+i e x))^3}dx}{3 \left (a^2+c^2\right )}-\frac {(A c-a C) \cosh (d+e x)}{3 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^3}-\frac {B}{3 c e (a+c \sinh (d+e x))^3}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 \left (3 A a^2+5 c C a-2 A c^2\right )-\left (-2 C a^2+5 A c a+3 c^2 C\right ) \sinh (d+e x)}{(a+c \sinh (d+e x))^2}dx}{2 \left (a^2+c^2\right )}-\frac {\left (-2 a^2 C+5 a A c+3 c^2 C\right ) \cosh (d+e x)}{2 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^2}}{3 \left (a^2+c^2\right )}-\frac {(A c-a C) \cosh (d+e x)}{3 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^3}-\frac {B}{3 c e (a+c \sinh (d+e x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {2 \left (3 A a^2+5 c C a-2 A c^2\right )-\left (-2 C a^2+5 A c a+3 c^2 C\right ) \sinh (d+e x)}{(a+c \sinh (d+e x))^2}dx}{2 \left (a^2+c^2\right )}-\frac {\left (-2 a^2 C+5 a A c+3 c^2 C\right ) \cosh (d+e x)}{2 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^2}}{3 \left (a^2+c^2\right )}-\frac {(A c-a C) \cosh (d+e x)}{3 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^3}-\frac {B}{3 c e (a+c \sinh (d+e x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\left (-2 a^2 C+5 a A c+3 c^2 C\right ) \cosh (d+e x)}{2 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^2}+\frac {\int \frac {2 \left (3 A a^2+5 c C a-2 A c^2\right )+i \left (-2 C a^2+5 A c a+3 c^2 C\right ) \sin (i d+i e x)}{(a-i c \sin (i d+i e x))^2}dx}{2 \left (a^2+c^2\right )}}{3 \left (a^2+c^2\right )}-\frac {(A c-a C) \cosh (d+e x)}{3 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^3}-\frac {B}{3 c e (a+c \sinh (d+e x))^3}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {3 \left (2 A a^3+4 c C a^2-3 A c^2 a-c^3 C\right )}{a+c \sinh (d+e x)}dx}{a^2+c^2}-\frac {\left (-2 a^3 C+11 a^2 A c+13 a c^2 C-4 A c^3\right ) \cosh (d+e x)}{e \left (a^2+c^2\right ) (a+c \sinh (d+e x))}}{2 \left (a^2+c^2\right )}-\frac {\left (-2 a^2 C+5 a A c+3 c^2 C\right ) \cosh (d+e x)}{2 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^2}}{3 \left (a^2+c^2\right )}-\frac {(A c-a C) \cosh (d+e x)}{3 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^3}-\frac {B}{3 c e (a+c \sinh (d+e x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \left (2 a^3 A+4 a^2 c C-3 a A c^2-c^3 C\right ) \int \frac {1}{a+c \sinh (d+e x)}dx}{a^2+c^2}-\frac {\left (-2 a^3 C+11 a^2 A c+13 a c^2 C-4 A c^3\right ) \cosh (d+e x)}{e \left (a^2+c^2\right ) (a+c \sinh (d+e x))}}{2 \left (a^2+c^2\right )}-\frac {\left (-2 a^2 C+5 a A c+3 c^2 C\right ) \cosh (d+e x)}{2 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^2}}{3 \left (a^2+c^2\right )}-\frac {(A c-a C) \cosh (d+e x)}{3 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^3}-\frac {B}{3 c e (a+c \sinh (d+e x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\left (-2 a^2 C+5 a A c+3 c^2 C\right ) \cosh (d+e x)}{2 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^2}+\frac {-\frac {\left (-2 a^3 C+11 a^2 A c+13 a c^2 C-4 A c^3\right ) \cosh (d+e x)}{e \left (a^2+c^2\right ) (a+c \sinh (d+e x))}+\frac {3 \left (2 a^3 A+4 a^2 c C-3 a A c^2-c^3 C\right ) \int \frac {1}{a-i c \sin (i d+i e x)}dx}{a^2+c^2}}{2 \left (a^2+c^2\right )}}{3 \left (a^2+c^2\right )}-\frac {(A c-a C) \cosh (d+e x)}{3 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^3}-\frac {B}{3 c e (a+c \sinh (d+e x))^3}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {-\frac {\left (-2 a^2 C+5 a A c+3 c^2 C\right ) \cosh (d+e x)}{2 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^2}+\frac {-\frac {\left (-2 a^3 C+11 a^2 A c+13 a c^2 C-4 A c^3\right ) \cosh (d+e x)}{e \left (a^2+c^2\right ) (a+c \sinh (d+e x))}-\frac {6 i \left (2 a^3 A+4 a^2 c C-3 a A c^2-c^3 C\right ) \int \frac {1}{-a \tanh ^2\left (\frac {1}{2} (d+e x)\right )+2 c \tanh \left (\frac {1}{2} (d+e x)\right )+a}d\left (i \tanh \left (\frac {1}{2} (d+e x)\right )\right )}{e \left (a^2+c^2\right )}}{2 \left (a^2+c^2\right )}}{3 \left (a^2+c^2\right )}-\frac {(A c-a C) \cosh (d+e x)}{3 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^3}-\frac {B}{3 c e (a+c \sinh (d+e x))^3}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-\frac {\left (-2 a^2 C+5 a A c+3 c^2 C\right ) \cosh (d+e x)}{2 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^2}+\frac {-\frac {\left (-2 a^3 C+11 a^2 A c+13 a c^2 C-4 A c^3\right ) \cosh (d+e x)}{e \left (a^2+c^2\right ) (a+c \sinh (d+e x))}+\frac {12 i \left (2 a^3 A+4 a^2 c C-3 a A c^2-c^3 C\right ) \int \frac {1}{\tanh ^2\left (\frac {1}{2} (d+e x)\right )-4 \left (a^2+c^2\right )}d\left (2 i a \tanh \left (\frac {1}{2} (d+e x)\right )-2 i c\right )}{e \left (a^2+c^2\right )}}{2 \left (a^2+c^2\right )}}{3 \left (a^2+c^2\right )}-\frac {(A c-a C) \cosh (d+e x)}{3 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^3}-\frac {B}{3 c e (a+c \sinh (d+e x))^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {(A c-a C) \cosh (d+e x)}{3 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^3}+\frac {\frac {\frac {6 \left (2 a^3 A+4 a^2 c C-3 a A c^2-c^3 C\right ) \text {arctanh}\left (\frac {\tanh \left (\frac {1}{2} (d+e x)\right )}{2 \sqrt {a^2+c^2}}\right )}{e \left (a^2+c^2\right )^{3/2}}-\frac {\left (-2 a^3 C+11 a^2 A c+13 a c^2 C-4 A c^3\right ) \cosh (d+e x)}{e \left (a^2+c^2\right ) (a+c \sinh (d+e x))}}{2 \left (a^2+c^2\right )}-\frac {\left (-2 a^2 C+5 a A c+3 c^2 C\right ) \cosh (d+e x)}{2 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^2}}{3 \left (a^2+c^2\right )}-\frac {B}{3 c e (a+c \sinh (d+e x))^3}\) |
-1/3*B/(c*e*(a + c*Sinh[d + e*x])^3) - ((A*c - a*C)*Cosh[d + e*x])/(3*(a^2 + c^2)*e*(a + c*Sinh[d + e*x])^3) + (-1/2*((5*a*A*c - 2*a^2*C + 3*c^2*C)* Cosh[d + e*x])/((a^2 + c^2)*e*(a + c*Sinh[d + e*x])^2) + ((6*(2*a^3*A - 3* a*A*c^2 + 4*a^2*c*C - c^3*C)*ArcTanh[Tanh[(d + e*x)/2]/(2*Sqrt[a^2 + c^2]) ])/((a^2 + c^2)^(3/2)*e) - ((11*a^2*A*c - 4*A*c^3 - 2*a^3*C + 13*a*c^2*C)* Cosh[d + e*x])/((a^2 + c^2)*e*(a + c*Sinh[d + e*x])))/(2*(a^2 + c^2)))/(3* (a^2 + c^2))
3.3.56.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( m + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
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])
Leaf count of result is larger than twice the leaf count of optimal. \(698\) vs. \(2(237)=474\).
Time = 75.69 (sec) , antiderivative size = 699, normalized size of antiderivative = 2.80
| method | result | size |
| parts | \(\frac {-\frac {2 \left (-\frac {c \left (9 A \,a^{4} c +6 A \,a^{2} c^{3}+2 A \,c^{5}-4 C \,a^{5}+C \,a^{3} c^{2}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{2 a \left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right )}-\frac {\left (6 A \,a^{6} c -27 A \,a^{4} c^{3}-12 A \,a^{2} c^{5}-4 A \,c^{7}-2 C \,a^{7}+14 C \,a^{5} c^{2}-11 C \,a^{3} c^{4}-2 C a \,c^{6}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{2 \left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right ) a^{2}}+\frac {c \left (54 A \,a^{6} c -21 A \,a^{4} c^{3}-4 A \,a^{2} c^{5}-4 A \,c^{7}-18 C \,a^{7}+42 C \,a^{5} c^{2}-17 C \,a^{3} c^{4}-2 C a \,c^{6}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{3 a^{3} \left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right )}+\frac {\left (6 A \,a^{6} c -20 A \,a^{4} c^{3}-3 A \,a^{2} c^{5}-2 A \,c^{7}-2 C \,a^{7}+10 C \,a^{5} c^{2}-14 C \,a^{3} c^{4}-C a \,c^{6}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{a^{2} \left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right )}-\frac {c \left (27 A \,a^{4} c +4 A \,a^{2} c^{3}+2 A \,c^{5}-8 C \,a^{5}+19 C \,a^{3} c^{2}+2 C a \,c^{4}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{2 a \left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right )}-\frac {18 A \,a^{4} c +5 A \,a^{2} c^{3}+2 A \,c^{5}-6 C \,a^{5}+10 C \,a^{3} c^{2}+C a \,c^{4}}{6 \left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right )}\right )}{\left (a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-2 c \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-a \right )^{3}}+\frac {\left (2 a^{3} A -3 A a \,c^{2}+4 c C \,a^{2}-C \,c^{3}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-2 c}{2 \sqrt {a^{2}+c^{2}}}\right )}{\left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right ) \sqrt {a^{2}+c^{2}}}}{e}-\frac {B}{3 c e \left (a +c \sinh \left (e x +d \right )\right )^{3}}\) | \(699\) |
| derivativedivides | \(\frac {-\frac {2 \left (-\frac {\left (9 A \,a^{4} c^{2}+6 A \,a^{2} c^{4}+2 A \,c^{6}-2 B \,a^{6}-6 c^{2} B \,a^{4}-6 c^{4} B \,a^{2}-2 c^{6} B -4 C \,a^{5} c +C \,a^{3} c^{3}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{2 a \left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right )}-\frac {\left (6 A \,a^{6} c -27 A \,a^{4} c^{3}-12 A \,a^{2} c^{5}-4 A \,c^{7}+4 B \,a^{6} c +12 B \,a^{4} c^{3}+12 B \,a^{2} c^{5}+4 B \,c^{7}-2 C \,a^{7}+14 C \,a^{5} c^{2}-11 C \,a^{3} c^{4}-2 C a \,c^{6}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{2 \left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right ) a^{2}}+\frac {\left (54 c^{2} a^{6} A -21 c^{4} a^{4} A -4 c^{6} a^{2} A -4 c^{8} A -6 B \,a^{8}-14 B \,a^{6} c^{2}-6 B \,a^{4} c^{4}+6 B \,a^{2} c^{6}+4 B \,c^{8}-18 c \,a^{7} C +42 c^{3} a^{5} C -17 c^{5} a^{3} C -2 c^{7} a C \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{3 a^{3} \left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right )}+\frac {\left (6 A \,a^{6} c -20 A \,a^{4} c^{3}-3 A \,a^{2} c^{5}-2 A \,c^{7}+2 B \,a^{6} c +6 B \,a^{4} c^{3}+6 B \,a^{2} c^{5}+2 B \,c^{7}-2 C \,a^{7}+10 C \,a^{5} c^{2}-14 C \,a^{3} c^{4}-C a \,c^{6}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{a^{2} \left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right )}-\frac {\left (27 A \,a^{4} c^{2}+4 A \,a^{2} c^{4}+2 A \,c^{6}-2 B \,a^{6}-6 c^{2} B \,a^{4}-6 c^{4} B \,a^{2}-2 c^{6} B -8 C \,a^{5} c +19 C \,a^{3} c^{3}+2 C a \,c^{5}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{2 a \left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right )}-\frac {18 A \,a^{4} c +5 A \,a^{2} c^{3}+2 A \,c^{5}-6 C \,a^{5}+10 C \,a^{3} c^{2}+C a \,c^{4}}{6 \left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right )}\right )}{\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 )^{3}}+\frac {\left (2 a^{3} A -3 A a \,c^{2}+4 c C \,a^{2}-C \,c^{3}\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 )}{\left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right ) \sqrt {a^{2}+c^{2}}}}{e}\) | \(844\) |
| default | \(\frac {-\frac {2 \left (-\frac {\left (9 A \,a^{4} c^{2}+6 A \,a^{2} c^{4}+2 A \,c^{6}-2 B \,a^{6}-6 c^{2} B \,a^{4}-6 c^{4} B \,a^{2}-2 c^{6} B -4 C \,a^{5} c +C \,a^{3} c^{3}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{2 a \left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right )}-\frac {\left (6 A \,a^{6} c -27 A \,a^{4} c^{3}-12 A \,a^{2} c^{5}-4 A \,c^{7}+4 B \,a^{6} c +12 B \,a^{4} c^{3}+12 B \,a^{2} c^{5}+4 B \,c^{7}-2 C \,a^{7}+14 C \,a^{5} c^{2}-11 C \,a^{3} c^{4}-2 C a \,c^{6}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{2 \left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right ) a^{2}}+\frac {\left (54 c^{2} a^{6} A -21 c^{4} a^{4} A -4 c^{6} a^{2} A -4 c^{8} A -6 B \,a^{8}-14 B \,a^{6} c^{2}-6 B \,a^{4} c^{4}+6 B \,a^{2} c^{6}+4 B \,c^{8}-18 c \,a^{7} C +42 c^{3} a^{5} C -17 c^{5} a^{3} C -2 c^{7} a C \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{3 a^{3} \left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right )}+\frac {\left (6 A \,a^{6} c -20 A \,a^{4} c^{3}-3 A \,a^{2} c^{5}-2 A \,c^{7}+2 B \,a^{6} c +6 B \,a^{4} c^{3}+6 B \,a^{2} c^{5}+2 B \,c^{7}-2 C \,a^{7}+10 C \,a^{5} c^{2}-14 C \,a^{3} c^{4}-C a \,c^{6}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{a^{2} \left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right )}-\frac {\left (27 A \,a^{4} c^{2}+4 A \,a^{2} c^{4}+2 A \,c^{6}-2 B \,a^{6}-6 c^{2} B \,a^{4}-6 c^{4} B \,a^{2}-2 c^{6} B -8 C \,a^{5} c +19 C \,a^{3} c^{3}+2 C a \,c^{5}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{2 a \left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right )}-\frac {18 A \,a^{4} c +5 A \,a^{2} c^{3}+2 A \,c^{5}-6 C \,a^{5}+10 C \,a^{3} c^{2}+C a \,c^{4}}{6 \left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right )}\right )}{\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 )^{3}}+\frac {\left (2 a^{3} A -3 A a \,c^{2}+4 c C \,a^{2}-C \,c^{3}\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 )}{\left (a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right ) \sqrt {a^{2}+c^{2}}}}{e}\) | \(844\) |
| risch | \(\text {Expression too large to display}\) | \(1219\) |
1/e*(-2*(-1/2*c*(9*A*a^4*c+6*A*a^2*c^3+2*A*c^5-4*C*a^5+C*a^3*c^2)/a/(a^6+3 *a^4*c^2+3*a^2*c^4+c^6)*tanh(1/2*e*x+1/2*d)^5-1/2*(6*A*a^6*c-27*A*a^4*c^3- 12*A*a^2*c^5-4*A*c^7-2*C*a^7+14*C*a^5*c^2-11*C*a^3*c^4-2*C*a*c^6)/(a^6+3*a ^4*c^2+3*a^2*c^4+c^6)/a^2*tanh(1/2*e*x+1/2*d)^4+1/3/a^3*c*(54*A*a^6*c-21*A *a^4*c^3-4*A*a^2*c^5-4*A*c^7-18*C*a^7+42*C*a^5*c^2-17*C*a^3*c^4-2*C*a*c^6) /(a^6+3*a^4*c^2+3*a^2*c^4+c^6)*tanh(1/2*e*x+1/2*d)^3+1/a^2*(6*A*a^6*c-20*A *a^4*c^3-3*A*a^2*c^5-2*A*c^7-2*C*a^7+10*C*a^5*c^2-14*C*a^3*c^4-C*a*c^6)/(a ^6+3*a^4*c^2+3*a^2*c^4+c^6)*tanh(1/2*e*x+1/2*d)^2-1/2/a*c*(27*A*a^4*c+4*A* a^2*c^3+2*A*c^5-8*C*a^5+19*C*a^3*c^2+2*C*a*c^4)/(a^6+3*a^4*c^2+3*a^2*c^4+c ^6)*tanh(1/2*e*x+1/2*d)-1/6*(18*A*a^4*c+5*A*a^2*c^3+2*A*c^5-6*C*a^5+10*C*a ^3*c^2+C*a*c^4)/(a^6+3*a^4*c^2+3*a^2*c^4+c^6))/(a*tanh(1/2*e*x+1/2*d)^2-2* c*tanh(1/2*e*x+1/2*d)-a)^3+(2*A*a^3-3*A*a*c^2+4*C*a^2*c-C*c^3)/(a^6+3*a^4* c^2+3*a^2*c^4+c^6)/(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)))-1/3*B/c/e/(a+c*sinh(e*x+d))^3
Leaf count of result is larger than twice the leaf count of optimal. 4350 vs. \(2 (239) = 478\).
Time = 0.40 (sec) , antiderivative size = 4350, normalized size of antiderivative = 17.40 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx=\text {Too large to display} \]
1/6*(4*C*a^5*c^3 - 22*A*a^4*c^4 - 22*C*a^3*c^5 - 14*A*a^2*c^6 - 26*C*a*c^7 + 8*A*c^8 + 6*(2*A*a^5*c^3 + 4*C*a^4*c^4 - A*a^3*c^5 + 3*C*a^2*c^6 - 3*A* a*c^7 - C*c^8)*cosh(e*x + d)^5 + 6*(2*A*a^5*c^3 + 4*C*a^4*c^4 - A*a^3*c^5 + 3*C*a^2*c^6 - 3*A*a*c^7 - C*c^8)*sinh(e*x + d)^5 + 30*(2*A*a^6*c^2 + 4*C *a^5*c^3 - A*a^4*c^4 + 3*C*a^3*c^5 - 3*A*a^2*c^6 - C*a*c^7)*cosh(e*x + d)^ 4 + 30*(2*A*a^6*c^2 + 4*C*a^5*c^3 - A*a^4*c^4 + 3*C*a^3*c^5 - 3*A*a^2*c^6 - C*a*c^7 + (2*A*a^5*c^3 + 4*C*a^4*c^4 - A*a^3*c^5 + 3*C*a^2*c^6 - 3*A*a*c ^7 - C*c^8)*cosh(e*x + d))*sinh(e*x + d)^4 - 4*(4*(B + C)*a^8 - 22*A*a^7*c + 4*(4*B - 7*C)*a^6*c^2 + 19*A*a^5*c^3 + (24*B + 7*C)*a^4*c^4 + 29*A*a^3* c^5 + (16*B + 39*C)*a^2*c^6 - 12*A*a*c^7 + 4*B*c^8)*cosh(e*x + d)^3 - 4*(4 *(B + C)*a^8 - 22*A*a^7*c + 4*(4*B - 7*C)*a^6*c^2 + 19*A*a^5*c^3 + (24*B + 7*C)*a^4*c^4 + 29*A*a^3*c^5 + (16*B + 39*C)*a^2*c^6 - 12*A*a*c^7 + 4*B*c^ 8 - 15*(2*A*a^5*c^3 + 4*C*a^4*c^4 - A*a^3*c^5 + 3*C*a^2*c^6 - 3*A*a*c^7 - C*c^8)*cosh(e*x + d)^2 - 30*(2*A*a^6*c^2 + 4*C*a^5*c^3 - A*a^4*c^4 + 3*C*a ^3*c^5 - 3*A*a^2*c^6 - C*a*c^7)*cosh(e*x + d))*sinh(e*x + d)^3 + 12*(4*C*a ^7*c - 17*A*a^6*c^2 - 13*C*a^5*c^3 - 11*A*a^4*c^4 - 13*C*a^3*c^5 + 4*A*a^2 *c^6 + 4*C*a*c^7 - 2*A*c^8)*cosh(e*x + d)^2 + 12*(4*C*a^7*c - 17*A*a^6*c^2 - 13*C*a^5*c^3 - 11*A*a^4*c^4 - 13*C*a^3*c^5 + 4*A*a^2*c^6 + 4*C*a*c^7 - 2*A*c^8 + 5*(2*A*a^5*c^3 + 4*C*a^4*c^4 - A*a^3*c^5 + 3*C*a^2*c^6 - 3*A*a*c ^7 - C*c^8)*cosh(e*x + d)^3 + 15*(2*A*a^6*c^2 + 4*C*a^5*c^3 - A*a^4*c^4...
Timed out. \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1263 vs. \(2 (239) = 478\).
Time = 0.34 (sec) , antiderivative size = 1263, normalized size of antiderivative = 5.05 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx=\text {Too large to display} \]
1/6*A*(3*(2*a^2 - 3*c^2)*a*log((c*e^(-e*x - d) - a - sqrt(a^2 + c^2))/(c*e ^(-e*x - d) - a + sqrt(a^2 + c^2)))/((a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6)*s qrt(a^2 + c^2)*e) - 2*(11*a^2*c^3 - 4*c^5 + 15*(4*a^3*c^2 - a*c^4)*e^(-e*x - d) + 6*(17*a^4*c - 6*a^2*c^3 + 2*c^5)*e^(-2*e*x - 2*d) + 2*(22*a^5 - 41 *a^3*c^2 + 12*a*c^4)*e^(-3*e*x - 3*d) - 15*(2*a^4*c - 3*a^2*c^3)*e^(-4*e*x - 4*d) + 3*(2*a^3*c^2 - 3*a*c^4)*e^(-5*e*x - 5*d))/((a^6*c^3 + 3*a^4*c^5 + 3*a^2*c^7 + c^9 + 6*(a^7*c^2 + 3*a^5*c^4 + 3*a^3*c^6 + a*c^8)*e^(-e*x - d) + 3*(4*a^8*c + 11*a^6*c^3 + 9*a^4*c^5 + a^2*c^7 - c^9)*e^(-2*e*x - 2*d) + 4*(2*a^9 + 3*a^7*c^2 - 3*a^5*c^4 - 7*a^3*c^6 - 3*a*c^8)*e^(-3*e*x - 3*d ) - 3*(4*a^8*c + 11*a^6*c^3 + 9*a^4*c^5 + a^2*c^7 - c^9)*e^(-4*e*x - 4*d) + 6*(a^7*c^2 + 3*a^5*c^4 + 3*a^3*c^6 + a*c^8)*e^(-5*e*x - 5*d) - (a^6*c^3 + 3*a^4*c^5 + 3*a^2*c^7 + c^9)*e^(-6*e*x - 6*d))*e)) + 1/6*C*(3*(4*a^2*c - c^3)*log((c*e^(-e*x - d) - a - sqrt(a^2 + c^2))/(c*e^(-e*x - d) - a + sqr t(a^2 + c^2)))/((a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6)*sqrt(a^2 + c^2)*e) + 2 *(2*a^3*c^3 - 13*a*c^5 + 3*(4*a^4*c^2 - 22*a^2*c^4 - c^6)*e^(-e*x - d) + 6 *(4*a^5*c - 17*a^3*c^3 + 4*a*c^5)*e^(-2*e*x - 2*d) + 2*(4*a^6 - 32*a^4*c^2 + 39*a^2*c^4)*e^(-3*e*x - 3*d) + 15*(4*a^3*c^3 - a*c^5)*e^(-4*e*x - 4*d) - 3*(4*a^2*c^4 - c^6)*e^(-5*e*x - 5*d))/((a^6*c^4 + 3*a^4*c^6 + 3*a^2*c^8 + c^10 + 6*(a^7*c^3 + 3*a^5*c^5 + 3*a^3*c^7 + a*c^9)*e^(-e*x - d) + 3*(4*a ^8*c^2 + 11*a^6*c^4 + 9*a^4*c^6 + a^2*c^8 - c^10)*e^(-2*e*x - 2*d) + 4*...
Leaf count of result is larger than twice the leaf count of optimal. 685 vs. \(2 (239) = 478\).
Time = 0.35 (sec) , antiderivative size = 685, normalized size of antiderivative = 2.74 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx=\frac {\frac {3 \, {\left (2 \, A a^{3} + 4 \, C a^{2} c - 3 \, A a c^{2} - C c^{3}\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 )}{{\left (a^{6} + 3 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6}\right )} \sqrt {a^{2} + c^{2}}} + \frac {2 \, {\left (6 \, A a^{3} c^{3} e^{\left (5 \, e x + 5 \, d\right )} + 12 \, C a^{2} c^{4} e^{\left (5 \, e x + 5 \, d\right )} - 9 \, A a c^{5} e^{\left (5 \, e x + 5 \, d\right )} - 3 \, C c^{6} e^{\left (5 \, e x + 5 \, d\right )} + 30 \, A a^{4} c^{2} e^{\left (4 \, e x + 4 \, d\right )} + 60 \, C a^{3} c^{3} e^{\left (4 \, e x + 4 \, d\right )} - 45 \, A a^{2} c^{4} e^{\left (4 \, e x + 4 \, d\right )} - 15 \, C a c^{5} e^{\left (4 \, e x + 4 \, d\right )} - 8 \, B a^{6} e^{\left (3 \, e x + 3 \, d\right )} - 8 \, C a^{6} e^{\left (3 \, e x + 3 \, d\right )} + 44 \, A a^{5} c e^{\left (3 \, e x + 3 \, d\right )} - 24 \, B a^{4} c^{2} e^{\left (3 \, e x + 3 \, d\right )} + 64 \, C a^{4} c^{2} e^{\left (3 \, e x + 3 \, d\right )} - 82 \, A a^{3} c^{3} e^{\left (3 \, e x + 3 \, d\right )} - 24 \, B a^{2} c^{4} e^{\left (3 \, e x + 3 \, d\right )} - 78 \, C a^{2} c^{4} e^{\left (3 \, e x + 3 \, d\right )} + 24 \, A a c^{5} e^{\left (3 \, e x + 3 \, d\right )} - 8 \, B c^{6} e^{\left (3 \, e x + 3 \, d\right )} + 24 \, C a^{5} c e^{\left (2 \, e x + 2 \, d\right )} - 102 \, A a^{4} c^{2} e^{\left (2 \, e x + 2 \, d\right )} - 102 \, C a^{3} c^{3} e^{\left (2 \, e x + 2 \, d\right )} + 36 \, A a^{2} c^{4} e^{\left (2 \, e x + 2 \, d\right )} + 24 \, C a c^{5} e^{\left (2 \, e x + 2 \, d\right )} - 12 \, A c^{6} e^{\left (2 \, e x + 2 \, d\right )} - 12 \, C a^{4} c^{2} e^{\left (e x + d\right )} + 60 \, A a^{3} c^{3} e^{\left (e x + d\right )} + 66 \, C a^{2} c^{4} e^{\left (e x + d\right )} - 15 \, A a c^{5} e^{\left (e x + d\right )} + 3 \, C c^{6} e^{\left (e x + d\right )} + 2 \, C a^{3} c^{3} - 11 \, A a^{2} c^{4} - 13 \, C a c^{5} + 4 \, A c^{6}\right )}}{{\left (a^{6} c + 3 \, a^{4} c^{3} + 3 \, a^{2} c^{5} + c^{7}\right )} {\left (c e^{\left (2 \, e x + 2 \, d\right )} + 2 \, a e^{\left (e x + d\right )} - c\right )}^{3}}}{6 \, e} \]
1/6*(3*(2*A*a^3 + 4*C*a^2*c - 3*A*a*c^2 - C*c^3)*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)))/ ((a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6)*sqrt(a^2 + c^2)) + 2*(6*A*a^3*c^3*e^( 5*e*x + 5*d) + 12*C*a^2*c^4*e^(5*e*x + 5*d) - 9*A*a*c^5*e^(5*e*x + 5*d) - 3*C*c^6*e^(5*e*x + 5*d) + 30*A*a^4*c^2*e^(4*e*x + 4*d) + 60*C*a^3*c^3*e^(4 *e*x + 4*d) - 45*A*a^2*c^4*e^(4*e*x + 4*d) - 15*C*a*c^5*e^(4*e*x + 4*d) - 8*B*a^6*e^(3*e*x + 3*d) - 8*C*a^6*e^(3*e*x + 3*d) + 44*A*a^5*c*e^(3*e*x + 3*d) - 24*B*a^4*c^2*e^(3*e*x + 3*d) + 64*C*a^4*c^2*e^(3*e*x + 3*d) - 82*A* a^3*c^3*e^(3*e*x + 3*d) - 24*B*a^2*c^4*e^(3*e*x + 3*d) - 78*C*a^2*c^4*e^(3 *e*x + 3*d) + 24*A*a*c^5*e^(3*e*x + 3*d) - 8*B*c^6*e^(3*e*x + 3*d) + 24*C* a^5*c*e^(2*e*x + 2*d) - 102*A*a^4*c^2*e^(2*e*x + 2*d) - 102*C*a^3*c^3*e^(2 *e*x + 2*d) + 36*A*a^2*c^4*e^(2*e*x + 2*d) + 24*C*a*c^5*e^(2*e*x + 2*d) - 12*A*c^6*e^(2*e*x + 2*d) - 12*C*a^4*c^2*e^(e*x + d) + 60*A*a^3*c^3*e^(e*x + d) + 66*C*a^2*c^4*e^(e*x + d) - 15*A*a*c^5*e^(e*x + d) + 3*C*c^6*e^(e*x + d) + 2*C*a^3*c^3 - 11*A*a^2*c^4 - 13*C*a*c^5 + 4*A*c^6)/((a^6*c + 3*a^4* c^3 + 3*a^2*c^5 + c^7)*(c*e^(2*e*x + 2*d) + 2*a*e^(e*x + d) - c)^3))/e
Timed out. \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx=\int \frac {A+B\,\mathrm {cosh}\left (d+e\,x\right )+C\,\mathrm {sinh}\left (d+e\,x\right )}{{\left (a+c\,\mathrm {sinh}\left (d+e\,x\right )\right )}^4} \,d x \]