Integrand size = 23, antiderivative size = 135 \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {b d x \sqrt {-1+c x} \sqrt {1+c x}}{24 c^3}-\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}-\frac {b d \text {arccosh}(c x)}{24 c^4}+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))-\frac {1}{6} c^2 d x^6 (a+b \text {arccosh}(c x)) \]
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Time = 0.10 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {14, 5921, 12, 471, 102, 92, 54} \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{6} c^2 d x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))-\frac {b d \text {arccosh}(c x)}{24 c^4}-\frac {b d x \sqrt {c x-1} \sqrt {c x+1}}{24 c^3}+\frac {1}{36} b c d x^5 \sqrt {c x-1} \sqrt {c x+1}-\frac {b d x^3 \sqrt {c x-1} \sqrt {c x+1}}{36 c} \]
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Rule 12
Rule 14
Rule 54
Rule 92
Rule 102
Rule 471
Rule 5921
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} d x^4 (a+b \text {arccosh}(c x))-\frac {1}{6} c^2 d x^6 (a+b \text {arccosh}(c x))-(b c) \int \frac {d x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {1}{4} d x^4 (a+b \text {arccosh}(c x))-\frac {1}{6} c^2 d x^6 (a+b \text {arccosh}(c x))-\frac {1}{12} (b c d) \int \frac {x^4 \left (3-2 c^2 x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))-\frac {1}{6} c^2 d x^6 (a+b \text {arccosh}(c x))-\frac {1}{9} (b c d) \int \frac {x^4}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))-\frac {1}{6} c^2 d x^6 (a+b \text {arccosh}(c x))-\frac {(b d) \int \frac {3 x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{36 c} \\ & = -\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))-\frac {1}{6} c^2 d x^6 (a+b \text {arccosh}(c x))-\frac {(b d) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{12 c} \\ & = -\frac {b d x \sqrt {-1+c x} \sqrt {1+c x}}{24 c^3}-\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))-\frac {1}{6} c^2 d x^6 (a+b \text {arccosh}(c x))-\frac {(b d) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{24 c^3} \\ & = -\frac {b d x \sqrt {-1+c x} \sqrt {1+c x}}{24 c^3}-\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}-\frac {b d \text {arccosh}(c x)}{24 c^4}+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))-\frac {1}{6} c^2 d x^6 (a+b \text {arccosh}(c x)) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.23 \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{4} a d x^4-\frac {1}{6} a c^2 d x^6-\frac {b d x \sqrt {-1+c x} \sqrt {1+c x}}{24 c^3}-\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{4} b d x^4 \text {arccosh}(c x)-\frac {1}{6} b c^2 d x^6 \text {arccosh}(c x)-\frac {b d \text {arctanh}\left (\frac {\sqrt {-1+c x}}{\sqrt {1+c x}}\right )}{12 c^4} \]
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Time = 0.16 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.12
method | result | size |
parts | \(-d a \left (\frac {1}{6} c^{2} x^{6}-\frac {1}{4} x^{4}\right )-\frac {d b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \operatorname {arccosh}\left (c x \right )}{4}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (-2 c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+2 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+3 c x \sqrt {c^{2} x^{2}-1}+3 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{72 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\) | \(151\) |
derivativedivides | \(\frac {-d a \left (\frac {1}{6} c^{6} x^{6}-\frac {1}{4} c^{4} x^{4}\right )-d b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \operatorname {arccosh}\left (c x \right )}{4}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (-2 c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+2 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+3 c x \sqrt {c^{2} x^{2}-1}+3 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{72 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\) | \(155\) |
default | \(\frac {-d a \left (\frac {1}{6} c^{6} x^{6}-\frac {1}{4} c^{4} x^{4}\right )-d b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \operatorname {arccosh}\left (c x \right )}{4}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (-2 c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+2 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+3 c x \sqrt {c^{2} x^{2}-1}+3 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{72 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\) | \(155\) |
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Time = 0.27 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.80 \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {12 \, a c^{6} d x^{6} - 18 \, a c^{4} d x^{4} + 3 \, {\left (4 \, b c^{6} d x^{6} - 6 \, b c^{4} d x^{4} + b d\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (2 \, b c^{5} d x^{5} - 2 \, b c^{3} d x^{3} - 3 \, b c d x\right )} \sqrt {c^{2} x^{2} - 1}}{72 \, c^{4}} \]
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\[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=- d \left (\int \left (- a x^{3}\right )\, dx + \int a c^{2} x^{5}\, dx + \int \left (- b x^{3} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{2} x^{5} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
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Time = 0.21 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.50 \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{6} \, a c^{2} d x^{6} + \frac {1}{4} \, a d x^{4} - \frac {1}{288} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b c^{2} d + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b d \]
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Exception generated. \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right ) \,d x \]
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