Integrand size = 12, antiderivative size = 84 \[ \int \left (c+d x^2\right ) \text {arccosh}(a x) \, dx=-\frac {\left (9 a^2 c+2 d\right ) \sqrt {-1+a x} \sqrt {1+a x}}{9 a^3}-\frac {d x^2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a}+c x \text {arccosh}(a x)+\frac {1}{3} d x^3 \text {arccosh}(a x) \]
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Time = 0.05 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5908, 471, 75} \[ \int \left (c+d x^2\right ) \text {arccosh}(a x) \, dx=-\frac {\sqrt {a x-1} \sqrt {a x+1} \left (9 a^2 c+2 d\right )}{9 a^3}+c x \text {arccosh}(a x)+\frac {1}{3} d x^3 \text {arccosh}(a x)-\frac {d x^2 \sqrt {a x-1} \sqrt {a x+1}}{9 a} \]
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Rule 75
Rule 471
Rule 5908
Rubi steps \begin{align*} \text {integral}& = c x \text {arccosh}(a x)+\frac {1}{3} d x^3 \text {arccosh}(a x)-a \int \frac {x \left (c+\frac {d x^2}{3}\right )}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {d x^2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a}+c x \text {arccosh}(a x)+\frac {1}{3} d x^3 \text {arccosh}(a x)+\frac {1}{9} \left (a \left (-9 c-\frac {2 d}{a^2}\right )\right ) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {\left (9 a^2 c+2 d\right ) \sqrt {-1+a x} \sqrt {1+a x}}{9 a^3}-\frac {d x^2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a}+c x \text {arccosh}(a x)+\frac {1}{3} d x^3 \text {arccosh}(a x) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71 \[ \int \left (c+d x^2\right ) \text {arccosh}(a x) \, dx=-\frac {\sqrt {-1+a x} \sqrt {1+a x} \left (2 d+a^2 \left (9 c+d x^2\right )\right )}{9 a^3}+\left (c x+\frac {d x^3}{3}\right ) \text {arccosh}(a x) \]
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Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.67
method | result | size |
parts | \(\frac {d \,x^{3} \operatorname {arccosh}\left (a x \right )}{3}+c x \,\operatorname {arccosh}\left (a x \right )-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (a^{2} d \,x^{2}+9 c \,a^{2}+2 d \right )}{9 a^{3}}\) | \(56\) |
derivativedivides | \(\frac {\operatorname {arccosh}\left (a x \right ) c a x +\frac {a \,\operatorname {arccosh}\left (a x \right ) d \,x^{3}}{3}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (a^{2} d \,x^{2}+9 c \,a^{2}+2 d \right )}{9 a^{2}}}{a}\) | \(62\) |
default | \(\frac {\operatorname {arccosh}\left (a x \right ) c a x +\frac {a \,\operatorname {arccosh}\left (a x \right ) d \,x^{3}}{3}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (a^{2} d \,x^{2}+9 c \,a^{2}+2 d \right )}{9 a^{2}}}{a}\) | \(62\) |
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Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.85 \[ \int \left (c+d x^2\right ) \text {arccosh}(a x) \, dx=\frac {3 \, {\left (a^{3} d x^{3} + 3 \, a^{3} c x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (a^{2} d x^{2} + 9 \, a^{2} c + 2 \, d\right )} \sqrt {a^{2} x^{2} - 1}}{9 \, a^{3}} \]
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\[ \int \left (c+d x^2\right ) \text {arccosh}(a x) \, dx=\int \left (c + d x^{2}\right ) \operatorname {acosh}{\left (a x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.88 \[ \int \left (c+d x^2\right ) \text {arccosh}(a x) \, dx=-\frac {1}{9} \, {\left (\frac {\sqrt {a^{2} x^{2} - 1} d x^{2}}{a^{2}} + \frac {9 \, \sqrt {a^{2} x^{2} - 1} c}{a^{2}} + \frac {2 \, \sqrt {a^{2} x^{2} - 1} d}{a^{4}}\right )} a + \frac {1}{3} \, {\left (d x^{3} + 3 \, c x\right )} \operatorname {arcosh}\left (a x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.83 \[ \int \left (c+d x^2\right ) \text {arccosh}(a x) \, dx=\frac {1}{3} \, {\left (d x^{3} + 3 \, c x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {{\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} d}{9 \, a^{3}} - \frac {\sqrt {a^{2} x^{2} - 1} {\left (3 \, a^{2} c + d\right )}}{3 \, a^{3}} \]
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Timed out. \[ \int \left (c+d x^2\right ) \text {arccosh}(a x) \, dx=\int \mathrm {acosh}\left (a\,x\right )\,\left (d\,x^2+c\right ) \,d x \]
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