Integrand size = 28, antiderivative size = 28 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\frac {c \sqrt {-1+c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b \sqrt {1-c x}}-\frac {3 c \sqrt {-1+c x} \log (a+b \text {arccosh}(c x))}{2 b \sqrt {1-c x}}-\frac {c \sqrt {-1+c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b \sqrt {1-c x}}+\text {Int}\left (\frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))},x\right ) \]
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Not integrable
Time = 0.43 (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 = {} \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 c^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))}+\frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))}+\frac {c^4 x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))}\right ) \, dx \\ & = -\left (\left (2 c^2\right ) \int \frac {1}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx\right )+c^4 \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \\ & = -\frac {2 c \sqrt {-1+c x} \log (a+b \text {arccosh}(c x))}{b \sqrt {1-c x}}+\frac {\left (c \sqrt {-1+c x}\right ) \text {Subst}\left (\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b \sqrt {1-c x}}+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \\ & = -\frac {2 c \sqrt {-1+c x} \log (a+b \text {arccosh}(c x))}{b \sqrt {1-c x}}+\frac {\left (c \sqrt {-1+c x}\right ) \text {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b \sqrt {1-c x}}+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \\ & = -\frac {3 c \sqrt {-1+c x} \log (a+b \text {arccosh}(c x))}{2 b \sqrt {1-c x}}+\frac {\left (c \sqrt {-1+c x}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b \sqrt {1-c x}}+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \\ & = -\frac {3 c \sqrt {-1+c x} \log (a+b \text {arccosh}(c x))}{2 b \sqrt {1-c x}}+\frac {\left (c \sqrt {-1+c x} \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b \sqrt {1-c x}}-\frac {\left (c \sqrt {-1+c x} \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b \sqrt {1-c x}}+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \\ & = \frac {c \sqrt {-1+c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b \sqrt {1-c x}}-\frac {3 c \sqrt {-1+c x} \log (a+b \text {arccosh}(c x))}{2 b \sqrt {1-c x}}-\frac {c \sqrt {-1+c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b \sqrt {1-c x}}+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \\ \end{align*}
Not integrable
Time = 2.41 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx \]
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Not integrable
Time = 0.90 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
\[\int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}{x^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]
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Not integrable
Time = 11.35 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]
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Not integrable
Time = 3.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int \frac {{\left (1-c^2\,x^2\right )}^{3/2}}{x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )} \,d x \]
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