Integrand size = 31, antiderivative size = 250 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx=-\frac {\left (2 a^3 A-3 a A c^2+4 a^2 c C-c^3 C\right ) \text {arctanh}\left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2+c^2}}\right )}{\left (a^2+c^2\right )^{7/2} e}-\frac {B}{3 c e (a+c \sinh (d+e x))^3}-\frac {(A c-a C) \cosh (d+e x)}{3 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^3}-\frac {\left (5 a A c-2 a^2 C+3 c^2 C\right ) \cosh (d+e x)}{6 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))^2}-\frac {\left (11 a^2 A c-4 A c^3-2 a^3 C+13 a c^2 C\right ) \cosh (d+e x)}{6 \left (a^2+c^2\right )^3 e (a+c \sinh (d+e x))} \]
[Out]
Time = 0.31 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {4461, 2833, 12, 2739, 632, 210, 2747, 32} \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx=-\frac {\left (-2 a^2 C+5 a A c+3 c^2 C\right ) \cosh (d+e x)}{6 e \left (a^2+c^2\right )^2 (a+c \sinh (d+e x))^2}-\frac {(A c-a C) \cosh (d+e x)}{3 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^3}-\frac {\left (2 a^3 A+4 a^2 c C-3 a A c^2-c^3 C\right ) \text {arctanh}\left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2+c^2}}\right )}{e \left (a^2+c^2\right )^{7/2}}-\frac {\left (-2 a^3 C+11 a^2 A c+13 a c^2 C-4 A c^3\right ) \cosh (d+e x)}{6 e \left (a^2+c^2\right )^3 (a+c \sinh (d+e x))}-\frac {B}{3 c e (a+c \sinh (d+e x))^3} \]
[In]
[Out]
Rule 12
Rule 32
Rule 210
Rule 632
Rule 2739
Rule 2747
Rule 2833
Rule 4461
Rubi steps \begin{align*} \text {integral}& = B \int \frac {\cosh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx+\int \frac {A+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx \\ & = -\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 {\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 {B \text {Subst}\left (\int \frac {1}{(a+x)^4} \, dx,x,c \sinh (d+e x)\right )}{c 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 {\int \frac {2 \left (3 a^2 A-2 A c^2+5 a c C\right )-\left (5 a A c-2 a^2 C+3 c^2 C\right ) \sinh (d+e x)}{(a+c \sinh (d+e x))^2} \, dx}{6 \left (a^2+c^2\right )^2} \\ & = -\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))}-\frac {\int -\frac {3 \left (2 a^3 A-3 a A c^2+4 a^2 c C-c^3 C\right )}{a+c \sinh (d+e x)} \, dx}{6 \left (a^2+c^2\right )^3} \\ & = -\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))}+\frac {\left (2 a^3 A-3 a A c^2+4 a^2 c C-c^3 C\right ) \int \frac {1}{a+c \sinh (d+e x)} \, dx}{2 \left (a^2+c^2\right )^3} \\ & = -\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))}-\frac {\left (i \left (2 a^3 A-3 a A c^2+4 a^2 c C-c^3 C\right )\right ) \text {Subst}\left (\int \frac {1}{a-2 i c x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (i d+i e x)\right )\right )}{\left (a^2+c^2\right )^3 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))}+\frac {\left (2 i \left (2 a^3 A-3 a A c^2+4 a^2 c C-c^3 C\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2+c^2\right )-x^2} \, dx,x,-2 i c+2 a \tan \left (\frac {1}{2} (i d+i e x)\right )\right )}{\left (a^2+c^2\right )^3 e} \\ & = -\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))} \\ \end{align*}
Time = 1.83 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.94 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx=\frac {\frac {6 \left (2 a^3 A-3 a A c^2+4 a^2 c C-c^3 C\right ) \arctan \left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2-c^2}}\right )}{\sqrt {-a^2-c^2}}-\frac {2 \left (a^2+c^2\right )^2 \left (B \left (a^2+c^2\right )+c (A c-a C) \cosh (d+e x)\right )}{c (a+c \sinh (d+e x))^3}+\frac {\left (a^2+c^2\right ) \left (-5 a A c+2 a^2 C-3 c^2 C\right ) \cosh (d+e x)}{(a+c \sinh (d+e x))^2}+\frac {\left (-11 a^2 A c+4 A c^3+2 a^3 C-13 a c^2 C\right ) \cosh (d+e x)}{a+c \sinh (d+e x)}}{6 \left (a^2+c^2\right )^3 e} \]
[In]
[Out]
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 | \(\text {Expression too large to display}\) | \(1219\) |
[In]
[Out]
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 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx=\text {Timed out} \]
[In]
[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 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx=\text {Too large to display} \]
[In]
[Out]
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} \]
[In]
[Out]
Timed out. \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^4} \, dx=\int \frac {A+B\,\mathrm {cosh}\left (d+e\,x\right )+C\,\mathrm {sinh}\left (d+e\,x\right )}{{\left (a+c\,\mathrm {sinh}\left (d+e\,x\right )\right )}^4} \,d x \]
[In]
[Out]