Integrand size = 19, antiderivative size = 161 \[ \int x^3 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {b \left (9 c^2 d+5 e\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{96 c^5}-\frac {b \left (9 c^2 d+5 e\right ) x^3 \sqrt {-1+c x} \sqrt {1+c x}}{144 c^3}-\frac {b e x^5 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}-\frac {b \left (9 c^2 d+5 e\right ) \text {arccosh}(c x)}{96 c^6}+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e x^6 (a+b \text {arccosh}(c x)) \]
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Time = 0.10 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5956, 471, 102, 12, 92, 54} \[ \int x^3 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e x^6 (a+b \text {arccosh}(c x))-\frac {b \text {arccosh}(c x) \left (9 c^2 d+5 e\right )}{96 c^6}-\frac {b x \sqrt {c x-1} \sqrt {c x+1} \left (9 c^2 d+5 e\right )}{96 c^5}-\frac {b x^3 \sqrt {c x-1} \sqrt {c x+1} \left (9 c^2 d+5 e\right )}{144 c^3}-\frac {b e x^5 \sqrt {c x-1} \sqrt {c x+1}}{36 c} \]
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Rule 12
Rule 54
Rule 92
Rule 102
Rule 471
Rule 5956
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} d x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e x^6 (a+b \text {arccosh}(c x))-\frac {1}{24} (b c) \int \frac {x^4 \left (6 d+4 e x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {b e x^5 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e x^6 (a+b \text {arccosh}(c x))-\frac {1}{36} \left (b c \left (9 d+\frac {5 e}{c^2}\right )\right ) \int \frac {x^4}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {b \left (9 c^2 d+5 e\right ) x^3 \sqrt {-1+c x} \sqrt {1+c x}}{144 c^3}-\frac {b e x^5 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e x^6 (a+b \text {arccosh}(c x))-\frac {\left (b \left (9 c^2 d+5 e\right )\right ) \int \frac {3 x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{144 c^3} \\ & = -\frac {b \left (9 c^2 d+5 e\right ) x^3 \sqrt {-1+c x} \sqrt {1+c x}}{144 c^3}-\frac {b e x^5 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e x^6 (a+b \text {arccosh}(c x))-\frac {\left (b \left (9 c^2 d+5 e\right )\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{48 c^3} \\ & = -\frac {b \left (9 c^2 d+5 e\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{96 c^5}-\frac {b \left (9 c^2 d+5 e\right ) x^3 \sqrt {-1+c x} \sqrt {1+c x}}{144 c^3}-\frac {b e x^5 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e x^6 (a+b \text {arccosh}(c x))-\frac {\left (b \left (9 c^2 d+5 e\right )\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{96 c^5} \\ & = -\frac {b \left (9 c^2 d+5 e\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{96 c^5}-\frac {b \left (9 c^2 d+5 e\right ) x^3 \sqrt {-1+c x} \sqrt {1+c x}}{144 c^3}-\frac {b e x^5 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}-\frac {b \left (9 c^2 d+5 e\right ) \text {arccosh}(c x)}{96 c^6}+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))+\frac {1}{6} e x^6 (a+b \text {arccosh}(c x)) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.87 \[ \int x^3 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {24 a c^6 x^4 \left (3 d+2 e x^2\right )-b c x \sqrt {-1+c x} \sqrt {1+c x} \left (15 e+c^2 \left (27 d+10 e x^2\right )+2 c^4 \left (9 d x^2+4 e x^4\right )\right )+24 b c^6 x^4 \left (3 d+2 e x^2\right ) \text {arccosh}(c x)-6 b \left (9 c^2 d+5 e\right ) \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )}{288 c^6} \]
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Time = 0.38 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.34
method | result | size |
parts | \(a \left (\frac {1}{6} e \,x^{6}+\frac {1}{4} d \,x^{4}\right )+\frac {b \left (\frac {c^{4} \operatorname {arccosh}\left (c x \right ) e \,x^{6}}{6}+\frac {\operatorname {arccosh}\left (c x \right ) c^{4} x^{4} d}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (18 c^{5} d \sqrt {c^{2} x^{2}-1}\, x^{3}+8 e \sqrt {c^{2} x^{2}-1}\, c^{5} x^{5}+27 d \,c^{3} x \sqrt {c^{2} x^{2}-1}+10 e \,c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+27 d \,c^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+15 e c x \sqrt {c^{2} x^{2}-1}+15 e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{288 c^{2} \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\) | \(215\) |
derivativedivides | \(\frac {\frac {a \left (\frac {1}{4} c^{6} d \,x^{4}+\frac {1}{6} c^{6} e \,x^{6}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) d \,c^{6} x^{4}}{4}+\frac {\operatorname {arccosh}\left (c x \right ) e \,c^{6} x^{6}}{6}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (18 c^{5} d \sqrt {c^{2} x^{2}-1}\, x^{3}+8 e \sqrt {c^{2} x^{2}-1}\, c^{5} x^{5}+27 d \,c^{3} x \sqrt {c^{2} x^{2}-1}+10 e \,c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+27 d \,c^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+15 e c x \sqrt {c^{2} x^{2}-1}+15 e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{288 \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}}{c^{4}}\) | \(225\) |
default | \(\frac {\frac {a \left (\frac {1}{4} c^{6} d \,x^{4}+\frac {1}{6} c^{6} e \,x^{6}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) d \,c^{6} x^{4}}{4}+\frac {\operatorname {arccosh}\left (c x \right ) e \,c^{6} x^{6}}{6}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (18 c^{5} d \sqrt {c^{2} x^{2}-1}\, x^{3}+8 e \sqrt {c^{2} x^{2}-1}\, c^{5} x^{5}+27 d \,c^{3} x \sqrt {c^{2} x^{2}-1}+10 e \,c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+27 d \,c^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+15 e c x \sqrt {c^{2} x^{2}-1}+15 e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{288 \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}}{c^{4}}\) | \(225\) |
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Time = 0.30 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.84 \[ \int x^3 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {48 \, a c^{6} e x^{6} + 72 \, a c^{6} d x^{4} + 3 \, {\left (16 \, b c^{6} e x^{6} + 24 \, b c^{6} d x^{4} - 9 \, b c^{2} d - 5 \, b e\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (8 \, b c^{5} e x^{5} + 2 \, {\left (9 \, b c^{5} d + 5 \, b c^{3} e\right )} x^{3} + 3 \, {\left (9 \, b c^{3} d + 5 \, b c e\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{288 \, c^{6}} \]
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\[ \int x^3 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\int x^{3} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \]
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Time = 0.20 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.22 \[ \int x^3 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{6} \, a e x^{6} + \frac {1}{4} \, a d x^{4} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b d + \frac {1}{288} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b e \]
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Exception generated. \[ \int x^3 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x^3 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x \]
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