Integrand size = 23, antiderivative size = 382 \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x))^3 \, dx=\frac {16}{25} a b^2 e^4 x-\frac {4144 b^3 e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{5625 d}-\frac {272 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5625 d}-\frac {6 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{625 d}+\frac {16 b^3 e^4 (c+d x) \text {arccosh}(c+d x)}{25 d}+\frac {8 b^2 e^4 (c+d x)^3 (a+b \text {arccosh}(c+d x))}{75 d}+\frac {6 b^2 e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))}{125 d}-\frac {8 b e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}-\frac {3 b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))^3}{5 d} \]
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Time = 0.48 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5996, 12, 5883, 5939, 5915, 5879, 75, 102} \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x))^3 \, dx=\frac {6 b^2 e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))}{125 d}+\frac {8 b^2 e^4 (c+d x)^3 (a+b \text {arccosh}(c+d x))}{75 d}+\frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))^3}{5 d}-\frac {3 b e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4 (a+b \text {arccosh}(c+d x))^2}{25 d}-\frac {4 b e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))^2}{25 d}-\frac {8 b e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2}{25 d}+\frac {16}{25} a b^2 e^4 x+\frac {16 b^3 e^4 (c+d x) \text {arccosh}(c+d x)}{25 d}-\frac {6 b^3 e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{625 d}-\frac {272 b^3 e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5625 d}-\frac {4144 b^3 e^4 \sqrt {c+d x-1} \sqrt {c+d x+1}}{5625 d} \]
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Rule 12
Rule 75
Rule 102
Rule 5879
Rule 5883
Rule 5915
Rule 5939
Rule 5996
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^4 x^4 (a+b \text {arccosh}(x))^3 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^4 \text {Subst}\left (\int x^4 (a+b \text {arccosh}(x))^3 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))^3}{5 d}-\frac {\left (3 b e^4\right ) \text {Subst}\left (\int \frac {x^5 (a+b \text {arccosh}(x))^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{5 d} \\ & = -\frac {3 b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))^3}{5 d}-\frac {\left (12 b e^4\right ) \text {Subst}\left (\int \frac {x^3 (a+b \text {arccosh}(x))^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{25 d}+\frac {\left (6 b^2 e^4\right ) \text {Subst}\left (\int x^4 (a+b \text {arccosh}(x)) \, dx,x,c+d x\right )}{25 d} \\ & = \frac {6 b^2 e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))}{125 d}-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}-\frac {3 b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))^3}{5 d}-\frac {\left (8 b e^4\right ) \text {Subst}\left (\int \frac {x (a+b \text {arccosh}(x))^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{25 d}+\frac {\left (8 b^2 e^4\right ) \text {Subst}\left (\int x^2 (a+b \text {arccosh}(x)) \, dx,x,c+d x\right )}{25 d}-\frac {\left (6 b^3 e^4\right ) \text {Subst}\left (\int \frac {x^5}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{125 d} \\ & = -\frac {6 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{625 d}+\frac {8 b^2 e^4 (c+d x)^3 (a+b \text {arccosh}(c+d x))}{75 d}+\frac {6 b^2 e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))}{125 d}-\frac {8 b e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}-\frac {3 b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))^3}{5 d}+\frac {\left (16 b^2 e^4\right ) \text {Subst}(\int (a+b \text {arccosh}(x)) \, dx,x,c+d x)}{25 d}-\frac {\left (6 b^3 e^4\right ) \text {Subst}\left (\int \frac {4 x^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{625 d}-\frac {\left (8 b^3 e^4\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{75 d} \\ & = \frac {16}{25} a b^2 e^4 x-\frac {8 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{225 d}-\frac {6 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{625 d}+\frac {8 b^2 e^4 (c+d x)^3 (a+b \text {arccosh}(c+d x))}{75 d}+\frac {6 b^2 e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))}{125 d}-\frac {8 b e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}-\frac {3 b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))^3}{5 d}-\frac {\left (8 b^3 e^4\right ) \text {Subst}\left (\int \frac {2 x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{225 d}-\frac {\left (24 b^3 e^4\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{625 d}+\frac {\left (16 b^3 e^4\right ) \text {Subst}(\int \text {arccosh}(x) \, dx,x,c+d x)}{25 d} \\ & = \frac {16}{25} a b^2 e^4 x-\frac {272 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5625 d}-\frac {6 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{625 d}+\frac {16 b^3 e^4 (c+d x) \text {arccosh}(c+d x)}{25 d}+\frac {8 b^2 e^4 (c+d x)^3 (a+b \text {arccosh}(c+d x))}{75 d}+\frac {6 b^2 e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))}{125 d}-\frac {8 b e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}-\frac {3 b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))^3}{5 d}-\frac {\left (8 b^3 e^4\right ) \text {Subst}\left (\int \frac {2 x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{625 d}-\frac {\left (16 b^3 e^4\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{225 d}-\frac {\left (16 b^3 e^4\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{25 d} \\ & = \frac {16}{25} a b^2 e^4 x-\frac {32 b^3 e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{45 d}-\frac {272 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5625 d}-\frac {6 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{625 d}+\frac {16 b^3 e^4 (c+d x) \text {arccosh}(c+d x)}{25 d}+\frac {8 b^2 e^4 (c+d x)^3 (a+b \text {arccosh}(c+d x))}{75 d}+\frac {6 b^2 e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))}{125 d}-\frac {8 b e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}-\frac {3 b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))^3}{5 d}-\frac {\left (16 b^3 e^4\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{625 d} \\ & = \frac {16}{25} a b^2 e^4 x-\frac {4144 b^3 e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{5625 d}-\frac {272 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5625 d}-\frac {6 b^3 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{625 d}+\frac {16 b^3 e^4 (c+d x) \text {arccosh}(c+d x)}{25 d}+\frac {8 b^2 e^4 (c+d x)^3 (a+b \text {arccosh}(c+d x))}{75 d}+\frac {6 b^2 e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))}{125 d}-\frac {8 b e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}-\frac {3 b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^2}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))^3}{5 d} \\ \end{align*}
Time = 0.65 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.06 \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x))^3 \, dx=\frac {e^4 \left (240 a b^2 (c+d x)+40 a b^2 (c+d x)^3+3 a \left (25 a^2+6 b^2\right ) (c+d x)^5+\frac {1}{15} b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (-8 \left (225 a^2+518 b^2\right )-4 \left (225 a^2+68 b^2\right ) (c+d x)^2-27 \left (25 a^2+2 b^2\right ) (c+d x)^4\right )-b \left (-240 b^2 (c+d x)-40 b^2 (c+d x)^3-225 a^2 (c+d x)^5-18 b^2 (c+d x)^5+240 a b \sqrt {-1+c+d x} \sqrt {1+c+d x}+120 a b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}+90 a b \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)-15 b^2 \left (-15 a (c+d x)^5+8 b \sqrt {-1+c+d x} \sqrt {1+c+d x}+4 b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}+3 b \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)^2+75 b^3 (c+d x)^5 \text {arccosh}(c+d x)^3\right )}{375 d} \]
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Time = 0.66 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.18
method | result | size |
derivativedivides | \(\frac {\frac {e^{4} a^{3} \left (d x +c \right )^{5}}{5}+e^{4} b^{3} \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )^{3}}{5}-\frac {8 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{25}-\frac {3 \left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{25}-\frac {4 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{25}+\frac {16 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )}{25}-\frac {4144 \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{5625}+\frac {6 \left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )}{125}-\frac {6 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{4}}{625}-\frac {272 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{5625}+\frac {8 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{75}\right )+3 e^{4} a \,b^{2} \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )^{2}}{5}-\frac {16 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{75}-\frac {2 \left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{25}-\frac {8 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{75}+\frac {16 d x}{75}+\frac {16 c}{75}+\frac {2 \left (d x +c \right )^{5}}{125}+\frac {8 \left (d x +c \right )^{3}}{225}\right )+3 e^{4} a^{2} b \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )}{5}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right )}{75}\right )}{d}\) | \(450\) |
default | \(\frac {\frac {e^{4} a^{3} \left (d x +c \right )^{5}}{5}+e^{4} b^{3} \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )^{3}}{5}-\frac {8 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{25}-\frac {3 \left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{25}-\frac {4 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{25}+\frac {16 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )}{25}-\frac {4144 \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{5625}+\frac {6 \left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )}{125}-\frac {6 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{4}}{625}-\frac {272 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{5625}+\frac {8 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{75}\right )+3 e^{4} a \,b^{2} \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )^{2}}{5}-\frac {16 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{75}-\frac {2 \left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{25}-\frac {8 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{75}+\frac {16 d x}{75}+\frac {16 c}{75}+\frac {2 \left (d x +c \right )^{5}}{125}+\frac {8 \left (d x +c \right )^{3}}{225}\right )+3 e^{4} a^{2} b \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )}{5}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right )}{75}\right )}{d}\) | \(450\) |
parts | \(\frac {e^{4} a^{3} \left (d x +c \right )^{5}}{5 d}+\frac {e^{4} b^{3} \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )^{3}}{5}-\frac {8 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{25}-\frac {3 \left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{25}-\frac {4 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{25}+\frac {16 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )}{25}-\frac {4144 \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{5625}+\frac {6 \left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )}{125}-\frac {6 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{4}}{625}-\frac {272 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{5625}+\frac {8 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{75}\right )}{d}+\frac {3 e^{4} a \,b^{2} \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )^{2}}{5}-\frac {16 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{75}-\frac {2 \left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{25}-\frac {8 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{75}+\frac {16 d x}{75}+\frac {16 c}{75}+\frac {2 \left (d x +c \right )^{5}}{125}+\frac {8 \left (d x +c \right )^{3}}{225}\right )}{d}+\frac {3 e^{4} a^{2} b \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )}{5}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right )}{75}\right )}{d}\) | \(458\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1074 vs. \(2 (336) = 672\).
Time = 0.30 (sec) , antiderivative size = 1074, normalized size of antiderivative = 2.81 \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x))^3 \, dx=\text {Too large to display} \]
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\[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x))^3 \, dx=e^{4} \left (\int a^{3} c^{4}\, dx + \int a^{3} d^{4} x^{4}\, dx + \int b^{3} c^{4} \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 3 a b^{2} c^{4} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 3 a^{2} b c^{4} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 4 a^{3} c d^{3} x^{3}\, dx + \int 6 a^{3} c^{2} d^{2} x^{2}\, dx + \int 4 a^{3} c^{3} d x\, dx + \int b^{3} d^{4} x^{4} \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 3 a b^{2} d^{4} x^{4} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 3 a^{2} b d^{4} x^{4} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 4 b^{3} c d^{3} x^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 6 b^{3} c^{2} d^{2} x^{2} \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 4 b^{3} c^{3} d x \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 12 a b^{2} c d^{3} x^{3} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 18 a b^{2} c^{2} d^{2} x^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 12 a b^{2} c^{3} d x \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 12 a^{2} b c d^{3} x^{3} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 18 a^{2} b c^{2} d^{2} x^{2} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 12 a^{2} b c^{3} d x \operatorname {acosh}{\left (c + d x \right )}\, dx\right ) \]
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\[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{4} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
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\[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{4} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
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Timed out. \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x))^3 \, dx=\int {\left (c\,e+d\,e\,x\right )}^4\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3 \,d x \]
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