Integrand size = 10, antiderivative size = 154 \[ \int x^m \text {arccosh}(a x)^2 \, dx=\frac {x^{1+m} \text {arccosh}(a x)^2}{1+m}-\frac {2 a x^{2+m} \sqrt {1-a x} \text {arccosh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{\left (2+3 m+m^2\right ) \sqrt {-1+a x}}-\frac {2 a^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};a^2 x^2\right )}{6+11 m+6 m^2+m^3} \]
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Time = 0.15 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5883, 5949} \[ \int x^m \text {arccosh}(a x)^2 \, dx=-\frac {2 a^2 x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};a^2 x^2\right )}{m^3+6 m^2+11 m+6}-\frac {2 a \sqrt {1-a x} x^{m+2} \text {arccosh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},a^2 x^2\right )}{\left (m^2+3 m+2\right ) \sqrt {a x-1}}+\frac {x^{m+1} \text {arccosh}(a x)^2}{m+1} \]
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Rule 5883
Rule 5949
Rubi steps \begin{align*} \text {integral}& = \frac {x^{1+m} \text {arccosh}(a x)^2}{1+m}-\frac {(2 a) \int \frac {x^{1+m} \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{1+m} \\ & = \frac {x^{1+m} \text {arccosh}(a x)^2}{1+m}-\frac {2 a x^{2+m} \sqrt {1-a x} \text {arccosh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{\left (2+3 m+m^2\right ) \sqrt {-1+a x}}-\frac {2 a^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};a^2 x^2\right )}{6+11 m+6 m^2+m^3} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.93 \[ \int x^m \text {arccosh}(a x)^2 \, dx=\frac {x^{1+m} \left (\text {arccosh}(a x)^2-\frac {2 a x \left (\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{\sqrt {-1+a x} \sqrt {1+a x}}+\frac {a x \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};a^2 x^2\right )}{3+m}\right )}{2+m}\right )}{1+m} \]
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\[\int x^{m} \operatorname {arccosh}\left (a x \right )^{2}d x\]
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\[ \int x^m \text {arccosh}(a x)^2 \, dx=\int { x^{m} \operatorname {arcosh}\left (a x\right )^{2} \,d x } \]
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\[ \int x^m \text {arccosh}(a x)^2 \, dx=\int x^{m} \operatorname {acosh}^{2}{\left (a x \right )}\, dx \]
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\[ \int x^m \text {arccosh}(a x)^2 \, dx=\int { x^{m} \operatorname {arcosh}\left (a x\right )^{2} \,d x } \]
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\[ \int x^m \text {arccosh}(a x)^2 \, dx=\int { x^{m} \operatorname {arcosh}\left (a x\right )^{2} \,d x } \]
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Timed out. \[ \int x^m \text {arccosh}(a x)^2 \, dx=\int x^m\,{\mathrm {acosh}\left (a\,x\right )}^2 \,d x \]
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