Integrand size = 24, antiderivative size = 188 \[ \int \frac {x^2 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {3 x^2 \sqrt {-1+a x}}{8 a \sqrt {1-a x}}-\frac {3 x \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{4 a^2}+\frac {3 \sqrt {-1+a x} \text {arccosh}(a x)^2}{8 a^3 \sqrt {1-a x}}-\frac {3 x^2 \sqrt {-1+a x} \text {arccosh}(a x)^2}{4 a \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 a^2}+\frac {\sqrt {-1+a x} \text {arccosh}(a x)^4}{8 a^3 \sqrt {1-a x}} \]
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Time = 0.28 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5938, 5892, 5883, 5939, 5893, 30} \[ \int \frac {x^2 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {a x-1} \text {arccosh}(a x)^4}{8 a^3 \sqrt {1-a x}}+\frac {3 \sqrt {a x-1} \text {arccosh}(a x)^2}{8 a^3 \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 a^2}-\frac {3 x \sqrt {1-a x} \sqrt {a x+1} \text {arccosh}(a x)}{4 a^2}-\frac {3 x^2 \sqrt {a x-1} \text {arccosh}(a x)^2}{4 a \sqrt {1-a x}}-\frac {3 x^2 \sqrt {a x-1}}{8 a \sqrt {1-a x}} \]
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Rule 30
Rule 5883
Rule 5892
Rule 5893
Rule 5938
Rule 5939
Rubi steps \begin{align*} \text {integral}& = -\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 a^2}+\frac {\int \frac {\text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{2 a^2}-\frac {\left (3 \sqrt {-1+a x}\right ) \int x \text {arccosh}(a x)^2 \, dx}{2 a \sqrt {1-a x}} \\ & = -\frac {3 x^2 \sqrt {-1+a x} \text {arccosh}(a x)^2}{4 a \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 a^2}+\frac {\sqrt {-1+a x} \text {arccosh}(a x)^4}{8 a^3 \sqrt {1-a x}}+\frac {\left (3 \sqrt {-1+a x}\right ) \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 \sqrt {1-a x}} \\ & = -\frac {3 x \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{4 a^2}-\frac {3 x^2 \sqrt {-1+a x} \text {arccosh}(a x)^2}{4 a \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 a^2}+\frac {\sqrt {-1+a x} \text {arccosh}(a x)^4}{8 a^3 \sqrt {1-a x}}+\frac {\left (3 \sqrt {-1+a x}\right ) \int \frac {\text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{4 a^2 \sqrt {1-a x}}-\frac {\left (3 \sqrt {-1+a x}\right ) \int x \, dx}{4 a \sqrt {1-a x}} \\ & = -\frac {3 x^2 \sqrt {-1+a x}}{8 a \sqrt {1-a x}}-\frac {3 x \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{4 a^2}+\frac {3 \sqrt {-1+a x} \text {arccosh}(a x)^2}{8 a^3 \sqrt {1-a x}}-\frac {3 x^2 \sqrt {-1+a x} \text {arccosh}(a x)^2}{4 a \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 a^2}+\frac {\sqrt {-1+a x} \text {arccosh}(a x)^4}{8 a^3 \sqrt {1-a x}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.52 \[ \int \frac {x^2 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {-((-1+a x) (1+a x))} \left (-3 \left (1+2 \text {arccosh}(a x)^2\right ) \cosh (2 \text {arccosh}(a x))+2 \text {arccosh}(a x) \left (\text {arccosh}(a x)^3+\left (3+2 \text {arccosh}(a x)^2\right ) \sinh (2 \text {arccosh}(a x))\right )\right )}{16 a^3 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \]
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Time = 0.70 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.36
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{4}}{8 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \operatorname {arccosh}\left (a x \right )^{3}-6 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )-3\right )}{32 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-2 a x -2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \operatorname {arccosh}\left (a x \right )^{3}+6 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )+3\right )}{32 a^{3} \left (a^{2} x^{2}-1\right )}\) | \(255\) |
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\[ \int \frac {x^2 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{2} \operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]
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\[ \int \frac {x^2 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^{2} \operatorname {acosh}^{3}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
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Exception generated. \[ \int \frac {x^2 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {x^2 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{2} \operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^2\,{\mathrm {acosh}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \]
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