Integrand size = 22, antiderivative size = 79 \[ \int \frac {x \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {2 \sqrt {1-a x} \sqrt {1+a x}}{a^2}-\frac {2 x \sqrt {-1+a x} \text {arccosh}(a x)}{a \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{a^2} \]
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Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {5914, 5879, 75} \[ \int \frac {x \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{a^2}-\frac {2 \sqrt {1-a x} \sqrt {a x+1}}{a^2}-\frac {2 x \sqrt {a x-1} \text {arccosh}(a x)}{a \sqrt {1-a x}} \]
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Rule 75
Rule 5879
Rule 5914
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{a^2}-\frac {\left (2 \sqrt {-1+a x}\right ) \int \text {arccosh}(a x) \, dx}{a \sqrt {1-a x}} \\ & = -\frac {2 x \sqrt {-1+a x} \text {arccosh}(a x)}{a \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{a^2}+\frac {\left (2 \sqrt {-1+a x}\right ) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a x}} \\ & = -\frac {2 \sqrt {1-a x} \sqrt {1+a x}}{a^2}-\frac {2 x \sqrt {-1+a x} \text {arccosh}(a x)}{a \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{a^2} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.68 \[ \int \frac {x \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (-2+\frac {2 a x \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}}-\text {arccosh}(a x)^2\right )}{a^2} \]
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Time = 0.52 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.76
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (\sqrt {a x -1}\, \sqrt {a x +1}\, a x +a^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (a x \right )^{2}-2 \,\operatorname {arccosh}\left (a x \right )+2\right )}{2 a^{2} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x -1}\, \sqrt {a x +1}\, a x -1\right ) \left (\operatorname {arccosh}\left (a x \right )^{2}+2 \,\operatorname {arccosh}\left (a x \right )+2\right )}{2 a^{2} \left (a^{2} x^{2}-1\right )}\) | \(139\) |
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none
Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.44 \[ \int \frac {x \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {2 \, \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 2 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-a^{2} x^{2} + 1}}{a^{4} x^{2} - a^{2}} \]
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\[ \int \frac {x \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x \operatorname {acosh}^{2}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.63 \[ \int \frac {x \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {2 i \, x \operatorname {arcosh}\left (a x\right )}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )^{2}}{a^{2}} - \frac {2 i \, \sqrt {a^{2} x^{2} - 1}}{a^{2}} \]
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Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.96 \[ \int \frac {x \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2}}{a^{2}} - \frac {2 i \, {\left (x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {\sqrt {a^{2} x^{2} - 1}}{a}\right )}}{a} \]
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Timed out. \[ \int \frac {x \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x\,{\mathrm {acosh}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \]
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