Integrand size = 28, antiderivative size = 28 \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=-\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c \left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))}+\frac {2 \sqrt {-1+c x} \text {Int}\left (\frac {x}{\left (-1+c^2 x^2\right )^2 (a+b \text {arccosh}(c x))},x\right )}{b c \sqrt {1-c x}} \]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c \left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))}+\frac {\left (2 \sqrt {-1+c x}\right ) \int \frac {x}{(-1+c x)^2 (1+c x)^2 (a+b \text {arccosh}(c x))} \, dx}{b c \sqrt {1-c x}} \\ & = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c \left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))}+\frac {\left (2 \sqrt {-1+c x}\right ) \int \frac {x}{\left (-1+c^2 x^2\right )^2 (a+b \text {arccosh}(c x))} \, dx}{b c \sqrt {1-c x}} \\ \end{align*}
Not integrable
Time = 5.62 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx \]
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Not integrable
Time = 0.54 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
\[\int \frac {x^{2}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.79 \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {x^{2}}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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Not integrable
Time = 71.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^{2}}{\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
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Not integrable
Time = 0.95 (sec) , antiderivative size = 503, normalized size of antiderivative = 17.96 \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {x^{2}}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {x^{2}}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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Not integrable
Time = 3.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^2}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (1-c^2\,x^2\right )}^{3/2}} \,d x \]
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