Integrand size = 24, antiderivative size = 177 \[ \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {40 \sqrt {1-a x} \sqrt {1+a x}}{27 a^4}-\frac {2 x^2 \sqrt {1-a x} \sqrt {1+a x}}{27 a^2}-\frac {4 x \sqrt {-1+a x} \text {arccosh}(a x)}{3 a^3 \sqrt {1-a x}}-\frac {2 x^3 \sqrt {-1+a x} \text {arccosh}(a x)}{9 a \sqrt {1-a x}}-\frac {2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{3 a^2} \]
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Time = 0.15 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5938, 5914, 5879, 75, 5883, 102, 12} \[ \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {40 \sqrt {1-a x} \sqrt {a x+1}}{27 a^4}-\frac {4 x \sqrt {a x-1} \text {arccosh}(a x)}{3 a^3 \sqrt {1-a x}}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{3 a^2}-\frac {2 x^2 \sqrt {1-a x} \sqrt {a x+1}}{27 a^2}-\frac {2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{3 a^4}-\frac {2 x^3 \sqrt {a x-1} \text {arccosh}(a x)}{9 a \sqrt {1-a x}} \]
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Rule 12
Rule 75
Rule 102
Rule 5879
Rule 5883
Rule 5914
Rule 5938
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{3 a^2}+\frac {2 \int \frac {x \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{3 a^2}-\frac {\left (2 \sqrt {-1+a x}\right ) \int x^2 \text {arccosh}(a x) \, dx}{3 a \sqrt {1-a x}} \\ & = -\frac {2 x^3 \sqrt {-1+a x} \text {arccosh}(a x)}{9 a \sqrt {1-a x}}-\frac {2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{3 a^2}+\frac {\left (2 \sqrt {-1+a x}\right ) \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{9 \sqrt {1-a x}}-\frac {\left (4 \sqrt {-1+a x}\right ) \int \text {arccosh}(a x) \, dx}{3 a^3 \sqrt {1-a x}} \\ & = -\frac {2 x^2 \sqrt {1-a x} \sqrt {1+a x}}{27 a^2}-\frac {4 x \sqrt {-1+a x} \text {arccosh}(a x)}{3 a^3 \sqrt {1-a x}}-\frac {2 x^3 \sqrt {-1+a x} \text {arccosh}(a x)}{9 a \sqrt {1-a x}}-\frac {2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{3 a^2}+\frac {\left (2 \sqrt {-1+a x}\right ) \int \frac {2 x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{27 a^2 \sqrt {1-a x}}+\frac {\left (4 \sqrt {-1+a x}\right ) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{3 a^2 \sqrt {1-a x}} \\ & = -\frac {4 \sqrt {1-a x} \sqrt {1+a x}}{3 a^4}-\frac {2 x^2 \sqrt {1-a x} \sqrt {1+a x}}{27 a^2}-\frac {4 x \sqrt {-1+a x} \text {arccosh}(a x)}{3 a^3 \sqrt {1-a x}}-\frac {2 x^3 \sqrt {-1+a x} \text {arccosh}(a x)}{9 a \sqrt {1-a x}}-\frac {2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{3 a^2}+\frac {\left (4 \sqrt {-1+a x}\right ) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{27 a^2 \sqrt {1-a x}} \\ & = -\frac {40 \sqrt {1-a x} \sqrt {1+a x}}{27 a^4}-\frac {2 x^2 \sqrt {1-a x} \sqrt {1+a x}}{27 a^2}-\frac {4 x \sqrt {-1+a x} \text {arccosh}(a x)}{3 a^3 \sqrt {1-a x}}-\frac {2 x^3 \sqrt {-1+a x} \text {arccosh}(a x)}{9 a \sqrt {1-a x}}-\frac {2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{3 a^2} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.69 \[ \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\left (-\frac {40}{27 a^4}-\frac {2 x^2}{27 a^2}\right ) \sqrt {1-a^2 x^2}+\frac {2 x \sqrt {1-a^2 x^2} \left (6+a^2 x^2\right ) \text {arccosh}(a x)}{9 a^3 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\sqrt {1-a^2 x^2} \left (2+a^2 x^2\right ) \text {arccosh}(a x)^2}{3 a^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(342\) vs. \(2(145)=290\).
Time = 1.12 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.94
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (4 a^{4} x^{4}-5 a^{2} x^{2}+4 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}-3 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +1\right ) \left (9 \operatorname {arccosh}\left (a x \right )^{2}-6 \,\operatorname {arccosh}\left (a x \right )+2\right )}{216 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \left (\sqrt {a x -1}\, \sqrt {a x +1}\, a x +a^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (a x \right )^{2}-2 \,\operatorname {arccosh}\left (a x \right )+2\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x -1}\, \sqrt {a x +1}\, a x -1\right ) \left (\operatorname {arccosh}\left (a x \right )^{2}+2 \,\operatorname {arccosh}\left (a x \right )+2\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (4 a^{4} x^{4}-5 a^{2} x^{2}-4 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}+3 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +1\right ) \left (9 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )+2\right )}{216 a^{4} \left (a^{2} x^{2}-1\right )}\) | \(343\) |
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none
Time = 0.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.85 \[ \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {9 \, {\left (a^{4} x^{4} + a^{2} x^{2} - 2\right )} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 6 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + 2 \, {\left (a^{4} x^{4} + 19 \, a^{2} x^{2} - 20\right )} \sqrt {-a^{2} x^{2} + 1}}{27 \, {\left (a^{6} x^{2} - a^{4}\right )}} \]
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\[ \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^{3} \operatorname {acosh}^{2}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.59 \[ \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {1}{3} \, {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arcosh}\left (a x\right )^{2} + \frac {2 \, {\left (-i \, \sqrt {a^{2} x^{2} - 1} x^{2} - \frac {20 i \, \sqrt {a^{2} x^{2} - 1}}{a^{2}}\right )}}{27 \, a^{2}} + \frac {2 \, {\left (i \, a^{2} x^{3} + 6 i \, x\right )} \operatorname {arcosh}\left (a x\right )}{9 \, a^{3}} \]
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Exception generated. \[ \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^3\,{\mathrm {acosh}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \]
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