Integrand size = 18, antiderivative size = 112 \[ \int \left (c-a^2 c x^2\right ) \text {arccosh}(a x)^2 \, dx=\frac {14 c x}{9}-\frac {2}{27} a^2 c x^3-\frac {4 c \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{3 a}+\frac {2 c (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)}{9 a}+\frac {2}{3} c x \text {arccosh}(a x)^2+\frac {1}{3} c x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^2 \]
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Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5897, 5879, 5915, 8, 41} \[ \int \left (c-a^2 c x^2\right ) \text {arccosh}(a x)^2 \, dx=\frac {1}{3} c x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^2-\frac {2}{27} a^2 c x^3+\frac {2}{3} c x \text {arccosh}(a x)^2+\frac {2 c (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)}{9 a}-\frac {4 c \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{3 a}+\frac {14 c x}{9} \]
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Rule 8
Rule 41
Rule 5879
Rule 5897
Rule 5915
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} c x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^2+\frac {1}{3} (2 c) \int \text {arccosh}(a x)^2 \, dx+\frac {1}{3} (2 a c) \int x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \, dx \\ & = \frac {2 c (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)}{9 a}+\frac {2}{3} c x \text {arccosh}(a x)^2+\frac {1}{3} c x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^2-\frac {1}{9} (2 c) \int (-1+a x) (1+a x) \, dx-\frac {1}{3} (4 a c) \int \frac {x \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {4 c \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{3 a}+\frac {2 c (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)}{9 a}+\frac {2}{3} c x \text {arccosh}(a x)^2+\frac {1}{3} c x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^2-\frac {1}{9} (2 c) \int \left (-1+a^2 x^2\right ) \, dx+\frac {1}{3} (4 c) \int 1 \, dx \\ & = \frac {14 c x}{9}-\frac {2}{27} a^2 c x^3-\frac {4 c \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{3 a}+\frac {2 c (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)}{9 a}+\frac {2}{3} c x \text {arccosh}(a x)^2+\frac {1}{3} c x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^2 \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.65 \[ \int \left (c-a^2 c x^2\right ) \text {arccosh}(a x)^2 \, dx=\frac {c \left (42 a x-2 a^3 x^3+6 \sqrt {-1+a x} \sqrt {1+a x} \left (-7+a^2 x^2\right ) \text {arccosh}(a x)-9 a x \left (-3+a^2 x^2\right ) \text {arccosh}(a x)^2\right )}{27 a} \]
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Time = 0.14 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(-\frac {c \left (9 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{2}-6 a^{2} x^{2} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}-27 a x \operatorname {arccosh}\left (a x \right )^{2}+42 \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )+2 a^{3} x^{3}-42 a x \right )}{27 a}\) | \(90\) |
default | \(-\frac {c \left (9 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{2}-6 a^{2} x^{2} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}-27 a x \operatorname {arccosh}\left (a x \right )^{2}+42 \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )+2 a^{3} x^{3}-42 a x \right )}{27 a}\) | \(90\) |
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Time = 0.26 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.85 \[ \int \left (c-a^2 c x^2\right ) \text {arccosh}(a x)^2 \, dx=-\frac {2 \, a^{3} c x^{3} - 42 \, a c x + 9 \, {\left (a^{3} c x^{3} - 3 \, a c x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 6 \, {\left (a^{2} c x^{2} - 7 \, c\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{27 \, a} \]
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\[ \int \left (c-a^2 c x^2\right ) \text {arccosh}(a x)^2 \, dx=- c \left (\int a^{2} x^{2} \operatorname {acosh}^{2}{\left (a x \right )}\, dx + \int \left (- \operatorname {acosh}^{2}{\left (a x \right )}\right )\, dx\right ) \]
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Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.68 \[ \int \left (c-a^2 c x^2\right ) \text {arccosh}(a x)^2 \, dx=-\frac {2}{27} \, a^{2} c x^{3} + \frac {2}{9} \, {\left (\sqrt {a^{2} x^{2} - 1} c x^{2} - \frac {7 \, \sqrt {a^{2} x^{2} - 1} c}{a^{2}}\right )} a \operatorname {arcosh}\left (a x\right ) - \frac {1}{3} \, {\left (a^{2} c x^{3} - 3 \, c x\right )} \operatorname {arcosh}\left (a x\right )^{2} + \frac {14}{9} \, c x \]
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Exception generated. \[ \int \left (c-a^2 c x^2\right ) \text {arccosh}(a x)^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (c-a^2 c x^2\right ) \text {arccosh}(a x)^2 \, dx=\int {\mathrm {acosh}\left (a\,x\right )}^2\,\left (c-a^2\,c\,x^2\right ) \,d x \]
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