Integrand size = 19, antiderivative size = 138 \[ \int x^2 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {2 b \left (25 c^2 d+12 e\right ) \sqrt {-1+c x} \sqrt {1+c x}}{225 c^5}-\frac {b \left (25 c^2 d+12 e\right ) x^2 \sqrt {-1+c x} \sqrt {1+c x}}{225 c^3}-\frac {b e x^4 \sqrt {-1+c x} \sqrt {1+c x}}{25 c}+\frac {1}{3} d x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e x^5 (a+b \text {arccosh}(c x)) \]
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Time = 0.09 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5956, 471, 102, 12, 75} \[ \int x^2 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{3} d x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e x^5 (a+b \text {arccosh}(c x))-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (25 c^2 d+12 e\right )}{225 c^5}-\frac {b x^2 \sqrt {c x-1} \sqrt {c x+1} \left (25 c^2 d+12 e\right )}{225 c^3}-\frac {b e x^4 \sqrt {c x-1} \sqrt {c x+1}}{25 c} \]
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Rule 12
Rule 75
Rule 102
Rule 471
Rule 5956
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} d x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e x^5 (a+b \text {arccosh}(c x))-\frac {1}{15} (b c) \int \frac {x^3 \left (5 d+3 e x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {b e x^4 \sqrt {-1+c x} \sqrt {1+c x}}{25 c}+\frac {1}{3} d x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e x^5 (a+b \text {arccosh}(c x))-\frac {1}{75} \left (b c \left (25 d+\frac {12 e}{c^2}\right )\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {b \left (25 c^2 d+12 e\right ) x^2 \sqrt {-1+c x} \sqrt {1+c x}}{225 c^3}-\frac {b e x^4 \sqrt {-1+c x} \sqrt {1+c x}}{25 c}+\frac {1}{3} d x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e x^5 (a+b \text {arccosh}(c x))-\frac {\left (b \left (25 c^2 d+12 e\right )\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{225 c^3} \\ & = -\frac {b \left (25 c^2 d+12 e\right ) x^2 \sqrt {-1+c x} \sqrt {1+c x}}{225 c^3}-\frac {b e x^4 \sqrt {-1+c x} \sqrt {1+c x}}{25 c}+\frac {1}{3} d x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e x^5 (a+b \text {arccosh}(c x))-\frac {\left (2 b \left (25 c^2 d+12 e\right )\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{225 c^3} \\ & = -\frac {2 b \left (25 c^2 d+12 e\right ) \sqrt {-1+c x} \sqrt {1+c x}}{225 c^5}-\frac {b \left (25 c^2 d+12 e\right ) x^2 \sqrt {-1+c x} \sqrt {1+c x}}{225 c^3}-\frac {b e x^4 \sqrt {-1+c x} \sqrt {1+c x}}{25 c}+\frac {1}{3} d x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e x^5 (a+b \text {arccosh}(c x)) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.73 \[ \int x^2 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{225} \left (15 a x^3 \left (5 d+3 e x^2\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (24 e+2 c^2 \left (25 d+6 e x^2\right )+c^4 \left (25 d x^2+9 e x^4\right )\right )}{c^5}+15 b x^3 \left (5 d+3 e x^2\right ) \text {arccosh}(c x)\right ) \]
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Time = 0.31 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.76
method | result | size |
parts | \(a \left (\frac {1}{5} e \,x^{5}+\frac {1}{3} d \,x^{3}\right )+\frac {b \left (\frac {c^{3} \operatorname {arccosh}\left (c x \right ) e \,x^{5}}{5}+\frac {\operatorname {arccosh}\left (c x \right ) c^{3} x^{3} d}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} e \,x^{4}+25 c^{4} d \,x^{2}+12 c^{2} e \,x^{2}+50 c^{2} d +24 e \right )}{225 c^{2}}\right )}{c^{3}}\) | \(105\) |
derivativedivides | \(\frac {\frac {a \left (\frac {1}{3} d \,c^{5} x^{3}+\frac {1}{5} e \,c^{5} x^{5}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) d \,c^{5} x^{3}}{3}+\frac {\operatorname {arccosh}\left (c x \right ) e \,c^{5} x^{5}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} e \,x^{4}+25 c^{4} d \,x^{2}+12 c^{2} e \,x^{2}+50 c^{2} d +24 e \right )}{225}\right )}{c^{2}}}{c^{3}}\) | \(115\) |
default | \(\frac {\frac {a \left (\frac {1}{3} d \,c^{5} x^{3}+\frac {1}{5} e \,c^{5} x^{5}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) d \,c^{5} x^{3}}{3}+\frac {\operatorname {arccosh}\left (c x \right ) e \,c^{5} x^{5}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} e \,x^{4}+25 c^{4} d \,x^{2}+12 c^{2} e \,x^{2}+50 c^{2} d +24 e \right )}{225}\right )}{c^{2}}}{c^{3}}\) | \(115\) |
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Time = 0.25 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.86 \[ \int x^2 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {45 \, a c^{5} e x^{5} + 75 \, a c^{5} d x^{3} + 15 \, {\left (3 \, b c^{5} e x^{5} + 5 \, b c^{5} d x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (9 \, b c^{4} e x^{4} + 50 \, b c^{2} d + {\left (25 \, b c^{4} d + 12 \, b c^{2} e\right )} x^{2} + 24 \, b e\right )} \sqrt {c^{2} x^{2} - 1}}{225 \, c^{5}} \]
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\[ \int x^2 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\int x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \]
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Time = 0.18 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.01 \[ \int x^2 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{5} \, a e x^{5} + \frac {1}{3} \, a d x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b e \]
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Exception generated. \[ \int x^2 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int x^2 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x \]
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