Integrand size = 28, antiderivative size = 28 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx=-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{b c x^2 (a+b \text {arccosh}(c x))}-\frac {2 \sqrt {1-c x} \text {Int}\left (\frac {-1+c^2 x^2}{x^3 (a+b \text {arccosh}(c x))},x\right )}{b c \sqrt {-1+c x}}-\frac {2 c \sqrt {1-c x} \text {Int}\left (\frac {-1+c^2 x^2}{x (a+b \text {arccosh}(c x))},x\right )}{b \sqrt {-1+c x}} \]
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Not integrable
Time = 0.30 (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 = {} \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{b c x^2 (a+b \text {arccosh}(c x))}-\frac {\left (2 \sqrt {1-c x}\right ) \int \frac {(-1+c x) (1+c x)}{x^3 (a+b \text {arccosh}(c x))} \, dx}{b c \sqrt {-1+c x}}-\frac {\left (2 c \sqrt {1-c x}\right ) \int \frac {(-1+c x) (1+c x)}{x (a+b \text {arccosh}(c x))} \, dx}{b \sqrt {-1+c x}} \\ & = -\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{b c x^2 (a+b \text {arccosh}(c x))}-\frac {\left (2 \sqrt {1-c x}\right ) \int \frac {-1+c^2 x^2}{x^3 (a+b \text {arccosh}(c x))} \, dx}{b c \sqrt {-1+c x}}-\frac {\left (2 c \sqrt {1-c x}\right ) \int \frac {-1+c^2 x^2}{x (a+b \text {arccosh}(c x))} \, dx}{b \sqrt {-1+c x}} \\ \end{align*}
Not integrable
Time = 20.94 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx \]
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Not integrable
Time = 1.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
\[\int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}{x^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]
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Not integrable
Time = 65.61 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
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Not integrable
Time = 0.94 (sec) , antiderivative size = 484, normalized size of antiderivative = 17.29 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]
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Not integrable
Time = 2.92 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {{\left (1-c^2\,x^2\right )}^{3/2}}{x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]
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