Integrand size = 20, antiderivative size = 609 \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=2 b^2 d^3 x+\frac {4 b^2 d^2 e x}{3 c^2}+\frac {16 b^2 d e^2 x}{25 c^4}+\frac {32 b^2 e^3 x}{245 c^6}+\frac {2}{9} b^2 d^2 e x^3+\frac {8 b^2 d e^2 x^3}{75 c^2}+\frac {16 b^2 e^3 x^3}{735 c^4}+\frac {6}{125} b^2 d e^2 x^5+\frac {12 b^2 e^3 x^5}{1225 c^2}+\frac {2}{343} b^2 e^3 x^7-\frac {2 b d^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c}-\frac {4 b d^2 e \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3}-\frac {16 b d e^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c^5}-\frac {32 b e^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{245 c^7}-\frac {2 b d^2 e x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c}-\frac {8 b d e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c^3}-\frac {16 b e^3 x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{245 c^5}-\frac {6 b d e^2 x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c}-\frac {12 b e^3 x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{245 c^3}-\frac {2 b e^3 x^6 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{49 c}+d^3 x (a+b \text {arccosh}(c x))^2+d^2 e x^3 (a+b \text {arccosh}(c x))^2+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))^2+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))^2 \]
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Time = 1.42 (sec) , antiderivative size = 609, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5909, 5879, 5915, 8, 5883, 5939, 30} \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=-\frac {32 b e^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{245 c^7}-\frac {16 b d e^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{25 c^5}-\frac {16 b e^3 x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{245 c^5}-\frac {4 b d^2 e \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{3 c^3}-\frac {8 b d e^2 x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{25 c^3}-\frac {12 b e^3 x^4 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{245 c^3}+d^3 x (a+b \text {arccosh}(c x))^2-\frac {2 b d^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c}+d^2 e x^3 (a+b \text {arccosh}(c x))^2-\frac {2 b d^2 e x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{3 c}+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))^2-\frac {6 b d e^2 x^4 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{25 c}+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))^2-\frac {2 b e^3 x^6 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{49 c}+\frac {32 b^2 e^3 x}{245 c^6}+\frac {16 b^2 d e^2 x}{25 c^4}+\frac {16 b^2 e^3 x^3}{735 c^4}+\frac {4 b^2 d^2 e x}{3 c^2}+\frac {8 b^2 d e^2 x^3}{75 c^2}+\frac {12 b^2 e^3 x^5}{1225 c^2}+2 b^2 d^3 x+\frac {2}{9} b^2 d^2 e x^3+\frac {6}{125} b^2 d e^2 x^5+\frac {2}{343} b^2 e^3 x^7 \]
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Rule 8
Rule 30
Rule 5879
Rule 5883
Rule 5909
Rule 5915
Rule 5939
Rubi steps \begin{align*} \text {integral}& = \int \left (d^3 (a+b \text {arccosh}(c x))^2+3 d^2 e x^2 (a+b \text {arccosh}(c x))^2+3 d e^2 x^4 (a+b \text {arccosh}(c x))^2+e^3 x^6 (a+b \text {arccosh}(c x))^2\right ) \, dx \\ & = d^3 \int (a+b \text {arccosh}(c x))^2 \, dx+\left (3 d^2 e\right ) \int x^2 (a+b \text {arccosh}(c x))^2 \, dx+\left (3 d e^2\right ) \int x^4 (a+b \text {arccosh}(c x))^2 \, dx+e^3 \int x^6 (a+b \text {arccosh}(c x))^2 \, dx \\ & = d^3 x (a+b \text {arccosh}(c x))^2+d^2 e x^3 (a+b \text {arccosh}(c x))^2+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))^2+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))^2-\left (2 b c d^3\right ) \int \frac {x (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\left (2 b c d^2 e\right ) \int \frac {x^3 (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {1}{5} \left (6 b c d e^2\right ) \int \frac {x^5 (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {1}{7} \left (2 b c e^3\right ) \int \frac {x^7 (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {2 b d^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c}-\frac {2 b d^2 e x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c}-\frac {6 b d e^2 x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c}-\frac {2 b e^3 x^6 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{49 c}+d^3 x (a+b \text {arccosh}(c x))^2+d^2 e x^3 (a+b \text {arccosh}(c x))^2+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))^2+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))^2+\left (2 b^2 d^3\right ) \int 1 \, dx+\frac {1}{3} \left (2 b^2 d^2 e\right ) \int x^2 \, dx-\frac {\left (4 b d^2 e\right ) \int \frac {x (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c}+\frac {1}{25} \left (6 b^2 d e^2\right ) \int x^4 \, dx-\frac {\left (24 b d e^2\right ) \int \frac {x^3 (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{25 c}+\frac {1}{49} \left (2 b^2 e^3\right ) \int x^6 \, dx-\frac {\left (12 b e^3\right ) \int \frac {x^5 (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{49 c} \\ & = 2 b^2 d^3 x+\frac {2}{9} b^2 d^2 e x^3+\frac {6}{125} b^2 d e^2 x^5+\frac {2}{343} b^2 e^3 x^7-\frac {2 b d^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c}-\frac {4 b d^2 e \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3}-\frac {2 b d^2 e x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c}-\frac {8 b d e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c^3}-\frac {6 b d e^2 x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c}-\frac {12 b e^3 x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{245 c^3}-\frac {2 b e^3 x^6 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{49 c}+d^3 x (a+b \text {arccosh}(c x))^2+d^2 e x^3 (a+b \text {arccosh}(c x))^2+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))^2+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))^2+\frac {\left (4 b^2 d^2 e\right ) \int 1 \, dx}{3 c^2}-\frac {\left (16 b d e^2\right ) \int \frac {x (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{25 c^3}+\frac {\left (8 b^2 d e^2\right ) \int x^2 \, dx}{25 c^2}-\frac {\left (48 b e^3\right ) \int \frac {x^3 (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{245 c^3}+\frac {\left (12 b^2 e^3\right ) \int x^4 \, dx}{245 c^2} \\ & = 2 b^2 d^3 x+\frac {4 b^2 d^2 e x}{3 c^2}+\frac {2}{9} b^2 d^2 e x^3+\frac {8 b^2 d e^2 x^3}{75 c^2}+\frac {6}{125} b^2 d e^2 x^5+\frac {12 b^2 e^3 x^5}{1225 c^2}+\frac {2}{343} b^2 e^3 x^7-\frac {2 b d^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c}-\frac {4 b d^2 e \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3}-\frac {16 b d e^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c^5}-\frac {2 b d^2 e x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c}-\frac {8 b d e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c^3}-\frac {16 b e^3 x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{245 c^5}-\frac {6 b d e^2 x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c}-\frac {12 b e^3 x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{245 c^3}-\frac {2 b e^3 x^6 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{49 c}+d^3 x (a+b \text {arccosh}(c x))^2+d^2 e x^3 (a+b \text {arccosh}(c x))^2+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))^2+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))^2+\frac {\left (16 b^2 d e^2\right ) \int 1 \, dx}{25 c^4}-\frac {\left (32 b e^3\right ) \int \frac {x (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{245 c^5}+\frac {\left (16 b^2 e^3\right ) \int x^2 \, dx}{245 c^4} \\ & = 2 b^2 d^3 x+\frac {4 b^2 d^2 e x}{3 c^2}+\frac {16 b^2 d e^2 x}{25 c^4}+\frac {2}{9} b^2 d^2 e x^3+\frac {8 b^2 d e^2 x^3}{75 c^2}+\frac {16 b^2 e^3 x^3}{735 c^4}+\frac {6}{125} b^2 d e^2 x^5+\frac {12 b^2 e^3 x^5}{1225 c^2}+\frac {2}{343} b^2 e^3 x^7-\frac {2 b d^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c}-\frac {4 b d^2 e \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3}-\frac {16 b d e^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c^5}-\frac {32 b e^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{245 c^7}-\frac {2 b d^2 e x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c}-\frac {8 b d e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c^3}-\frac {16 b e^3 x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{245 c^5}-\frac {6 b d e^2 x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c}-\frac {12 b e^3 x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{245 c^3}-\frac {2 b e^3 x^6 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{49 c}+d^3 x (a+b \text {arccosh}(c x))^2+d^2 e x^3 (a+b \text {arccosh}(c x))^2+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))^2+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))^2+\frac {\left (32 b^2 e^3\right ) \int 1 \, dx}{245 c^6} \\ & = 2 b^2 d^3 x+\frac {4 b^2 d^2 e x}{3 c^2}+\frac {16 b^2 d e^2 x}{25 c^4}+\frac {32 b^2 e^3 x}{245 c^6}+\frac {2}{9} b^2 d^2 e x^3+\frac {8 b^2 d e^2 x^3}{75 c^2}+\frac {16 b^2 e^3 x^3}{735 c^4}+\frac {6}{125} b^2 d e^2 x^5+\frac {12 b^2 e^3 x^5}{1225 c^2}+\frac {2}{343} b^2 e^3 x^7-\frac {2 b d^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c}-\frac {4 b d^2 e \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3}-\frac {16 b d e^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c^5}-\frac {32 b e^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{245 c^7}-\frac {2 b d^2 e x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c}-\frac {8 b d e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c^3}-\frac {16 b e^3 x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{245 c^5}-\frac {6 b d e^2 x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c}-\frac {12 b e^3 x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{245 c^3}-\frac {2 b e^3 x^6 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{49 c}+d^3 x (a+b \text {arccosh}(c x))^2+d^2 e x^3 (a+b \text {arccosh}(c x))^2+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))^2+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))^2 \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 453, normalized size of antiderivative = 0.74 \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=\frac {11025 a^2 c^7 x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )-210 a b \sqrt {-1+c x} \sqrt {1+c x} \left (240 e^3+24 c^2 e^2 \left (49 d+5 e x^2\right )+2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+c^6 \left (3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right )\right )+2 b^2 c x \left (25200 e^3+840 c^2 e^2 \left (147 d+5 e x^2\right )+210 c^4 e \left (1225 d^2+98 d e x^2+9 e^2 x^4\right )+c^6 \left (385875 d^3+42875 d^2 e x^2+9261 d e^2 x^4+1125 e^3 x^6\right )\right )-210 b \left (-105 a c^7 x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )+b \sqrt {-1+c x} \sqrt {1+c x} \left (240 e^3+24 c^2 e^2 \left (49 d+5 e x^2\right )+2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+c^6 \left (3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right )\right )\right ) \text {arccosh}(c x)+11025 b^2 c^7 x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right ) \text {arccosh}(c x)^2}{385875 c^7} \]
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Time = 0.81 (sec) , antiderivative size = 632, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} \left (d^{3} c^{7} x +d^{2} c^{7} e \,x^{3}+\frac {3}{5} d \,c^{7} e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b^{2} \left (c^{6} d^{3} \left (\operatorname {arccosh}\left (c x \right )^{2} x c -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )+\frac {c^{4} d^{2} e \left (9 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}-6 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c^{3} x^{3}+12 c x \right )}{9}+\frac {c^{2} d \,e^{2} \left (225 \operatorname {arccosh}\left (c x \right )^{2} c^{5} x^{5}-90 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-120 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+18 c^{5} x^{5}-240 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+40 c^{3} x^{3}+240 c x \right )}{375}+\frac {e^{3} \left (3675 \operatorname {arccosh}\left (c x \right )^{2} c^{7} x^{7}-1050 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{6} x^{6}-1260 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+150 c^{7} x^{7}-1680 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+252 c^{5} x^{5}-3360 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+560 c^{3} x^{3}+3360 c x \right )}{25725}\right )}{c^{6}}+\frac {2 a b \left (\operatorname {arccosh}\left (c x \right ) d^{3} c^{7} x +\operatorname {arccosh}\left (c x \right ) d^{2} c^{7} e \,x^{3}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) d \,c^{7} e^{2} x^{5}}{5}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{7} x^{7}}{7}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} e^{3} x^{6}+441 c^{6} d \,e^{2} x^{4}+1225 c^{6} d^{2} e \,x^{2}+90 c^{4} x^{4} e^{3}+3675 d^{3} c^{6}+588 c^{4} d \,e^{2} x^{2}+2450 c^{4} d^{2} e +120 c^{2} x^{2} e^{3}+1176 c^{2} d \,e^{2}+240 e^{3}\right )}{3675}\right )}{c^{6}}}{c}\) | \(632\) |
default | \(\frac {\frac {a^{2} \left (d^{3} c^{7} x +d^{2} c^{7} e \,x^{3}+\frac {3}{5} d \,c^{7} e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b^{2} \left (c^{6} d^{3} \left (\operatorname {arccosh}\left (c x \right )^{2} x c -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )+\frac {c^{4} d^{2} e \left (9 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}-6 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c^{3} x^{3}+12 c x \right )}{9}+\frac {c^{2} d \,e^{2} \left (225 \operatorname {arccosh}\left (c x \right )^{2} c^{5} x^{5}-90 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-120 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+18 c^{5} x^{5}-240 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+40 c^{3} x^{3}+240 c x \right )}{375}+\frac {e^{3} \left (3675 \operatorname {arccosh}\left (c x \right )^{2} c^{7} x^{7}-1050 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{6} x^{6}-1260 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+150 c^{7} x^{7}-1680 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+252 c^{5} x^{5}-3360 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+560 c^{3} x^{3}+3360 c x \right )}{25725}\right )}{c^{6}}+\frac {2 a b \left (\operatorname {arccosh}\left (c x \right ) d^{3} c^{7} x +\operatorname {arccosh}\left (c x \right ) d^{2} c^{7} e \,x^{3}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) d \,c^{7} e^{2} x^{5}}{5}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{7} x^{7}}{7}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} e^{3} x^{6}+441 c^{6} d \,e^{2} x^{4}+1225 c^{6} d^{2} e \,x^{2}+90 c^{4} x^{4} e^{3}+3675 d^{3} c^{6}+588 c^{4} d \,e^{2} x^{2}+2450 c^{4} d^{2} e +120 c^{2} x^{2} e^{3}+1176 c^{2} d \,e^{2}+240 e^{3}\right )}{3675}\right )}{c^{6}}}{c}\) | \(632\) |
parts | \(a^{2} \left (\frac {1}{7} e^{3} x^{7}+\frac {3}{5} d \,e^{2} x^{5}+d^{2} e \,x^{3}+d^{3} x \right )+\frac {b^{2} \left (-15750 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) c^{6} x^{6} e^{3}-92610 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) c^{6} x^{4} d \,e^{2}-257250 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) c^{6} x^{2} d^{2} e -771750 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) c^{6} d^{3}+55125 \operatorname {arccosh}\left (c x \right )^{2} c^{7} x^{7} e^{3}+231525 \operatorname {arccosh}\left (c x \right )^{2} c^{7} x^{5} d \,e^{2}+385875 \operatorname {arccosh}\left (c x \right )^{2} c^{7} x^{3} d^{2} e +385875 \operatorname {arccosh}\left (c x \right )^{2} c^{7} x \,d^{3}-18900 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4} e^{3}-123480 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{2} d \,e^{2}-514500 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) c^{4} d^{2} e +2250 e^{3} c^{7} x^{7}+18522 d \,c^{7} e^{2} x^{5}+85750 d^{2} c^{7} e \,x^{3}+771750 d^{3} c^{7} x -25200 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) c^{2} x^{2} e^{3}-246960 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) c^{2} d \,e^{2}+3780 c^{5} x^{5} e^{3}+41160 c^{5} x^{3} d \,e^{2}+514500 c^{5} x \,d^{2} e -50400 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) e^{3}+8400 c^{3} x^{3} e^{3}+246960 c^{3} x d \,e^{2}+50400 c x \,e^{3}\right )}{385875 c^{7}}+\frac {2 a b \left (\frac {c \,\operatorname {arccosh}\left (c x \right ) e^{3} x^{7}}{7}+\frac {3 c \,\operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{5}}{5}+c \,\operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{3}+\operatorname {arccosh}\left (c x \right ) c x \,d^{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} e^{3} x^{6}+441 c^{6} d \,e^{2} x^{4}+1225 c^{6} d^{2} e \,x^{2}+90 c^{4} x^{4} e^{3}+3675 d^{3} c^{6}+588 c^{4} d \,e^{2} x^{2}+2450 c^{4} d^{2} e +120 c^{2} x^{2} e^{3}+1176 c^{2} d \,e^{2}+240 e^{3}\right )}{3675 c^{6}}\right )}{c}\) | \(677\) |
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Time = 0.27 (sec) , antiderivative size = 586, normalized size of antiderivative = 0.96 \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=\frac {1125 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{7} e^{3} x^{7} + 189 \, {\left (49 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{7} d e^{2} + 20 \, b^{2} c^{5} e^{3}\right )} x^{5} + 35 \, {\left (1225 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{7} d^{2} e + 1176 \, b^{2} c^{5} d e^{2} + 240 \, b^{2} c^{3} e^{3}\right )} x^{3} + 11025 \, {\left (5 \, b^{2} c^{7} e^{3} x^{7} + 21 \, b^{2} c^{7} d e^{2} x^{5} + 35 \, b^{2} c^{7} d^{2} e x^{3} + 35 \, b^{2} c^{7} d^{3} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 105 \, {\left (3675 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{7} d^{3} + 4900 \, b^{2} c^{5} d^{2} e + 2352 \, b^{2} c^{3} d e^{2} + 480 \, b^{2} c e^{3}\right )} x + 210 \, {\left (525 \, a b c^{7} e^{3} x^{7} + 2205 \, a b c^{7} d e^{2} x^{5} + 3675 \, a b c^{7} d^{2} e x^{3} + 3675 \, a b c^{7} d^{3} x - {\left (75 \, b^{2} c^{6} e^{3} x^{6} + 3675 \, b^{2} c^{6} d^{3} + 2450 \, b^{2} c^{4} d^{2} e + 1176 \, b^{2} c^{2} d e^{2} + 240 \, b^{2} e^{3} + 9 \, {\left (49 \, b^{2} c^{6} d e^{2} + 10 \, b^{2} c^{4} e^{3}\right )} x^{4} + {\left (1225 \, b^{2} c^{6} d^{2} e + 588 \, b^{2} c^{4} d e^{2} + 120 \, b^{2} c^{2} e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 210 \, {\left (75 \, a b c^{6} e^{3} x^{6} + 3675 \, a b c^{6} d^{3} + 2450 \, a b c^{4} d^{2} e + 1176 \, a b c^{2} d e^{2} + 240 \, a b e^{3} + 9 \, {\left (49 \, a b c^{6} d e^{2} + 10 \, a b c^{4} e^{3}\right )} x^{4} + {\left (1225 \, a b c^{6} d^{2} e + 588 \, a b c^{4} d e^{2} + 120 \, a b c^{2} e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{385875 \, c^{7}} \]
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\[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=\int \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )^{3}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 684, normalized size of antiderivative = 1.12 \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=\frac {1}{7} \, b^{2} e^{3} x^{7} \operatorname {arcosh}\left (c x\right )^{2} + \frac {1}{7} \, a^{2} e^{3} x^{7} + \frac {3}{5} \, b^{2} d e^{2} x^{5} \operatorname {arcosh}\left (c x\right )^{2} + \frac {3}{5} \, a^{2} d e^{2} x^{5} + b^{2} d^{2} e x^{3} \operatorname {arcosh}\left (c x\right )^{2} + a^{2} d^{2} e x^{3} + b^{2} d^{3} x \operatorname {arcosh}\left (c x\right )^{2} + \frac {2}{3} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} a b d^{2} e - \frac {2}{9} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} d^{2} e + \frac {2}{25} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} a b d e^{2} - \frac {2}{375} \, {\left (15 \, {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c \operatorname {arcosh}\left (c x\right ) - \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} d e^{2} + \frac {2}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} a b e^{3} - \frac {2}{25725} \, {\left (105 \, {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c \operatorname {arcosh}\left (c x\right ) - \frac {75 \, c^{6} x^{7} + 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} + 1680 \, x}{c^{6}}\right )} b^{2} e^{3} + 2 \, b^{2} d^{3} {\left (x - \frac {\sqrt {c^{2} x^{2} - 1} \operatorname {arcosh}\left (c x\right )}{c}\right )} + a^{2} d^{3} x + \frac {2 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} a b d^{3}}{c} \]
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Exception generated. \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^3 \,d x \]
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