Integrand size = 27, antiderivative size = 27 \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x (a+b \text {arcsinh}(c x))^2} \, dx=-\frac {\left (1+c^2 x^2\right )^2}{b c x (a+b \text {arcsinh}(c x))}+\frac {9 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b^2}+\frac {3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b^2}-\frac {9 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b^2}-\frac {3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b^2}-\frac {\text {Int}\left (\frac {1+c^2 x^2}{x^2 (a+b \text {arcsinh}(c x))},x\right )}{b c} \]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 27, 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 = {} \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\left (1+c^2 x^2\right )^{3/2}}{x (a+b \text {arcsinh}(c x))^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {\left (1+c^2 x^2\right )^2}{b c x (a+b \text {arcsinh}(c x))}-\frac {\int \frac {1+c^2 x^2}{x^2 (a+b \text {arcsinh}(c x))} \, dx}{b c}+\frac {(3 c) \int \frac {1+c^2 x^2}{a+b \text {arcsinh}(c x)} \, dx}{b} \\ & = -\frac {\left (1+c^2 x^2\right )^2}{b c x (a+b \text {arcsinh}(c x))}+\frac {3 \text {Subst}\left (\int \frac {\cosh ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2}-\frac {\int \frac {1+c^2 x^2}{x^2 (a+b \text {arcsinh}(c x))} \, dx}{b c} \\ & = -\frac {\left (1+c^2 x^2\right )^2}{b c x (a+b \text {arcsinh}(c x))}+\frac {3 \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 x}+\frac {3 \cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 x}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2}-\frac {\int \frac {1+c^2 x^2}{x^2 (a+b \text {arcsinh}(c x))} \, dx}{b c} \\ & = -\frac {\left (1+c^2 x^2\right )^2}{b c x (a+b \text {arcsinh}(c x))}+\frac {3 \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b^2}+\frac {9 \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b^2}-\frac {\int \frac {1+c^2 x^2}{x^2 (a+b \text {arcsinh}(c x))} \, dx}{b c} \\ & = -\frac {\left (1+c^2 x^2\right )^2}{b c x (a+b \text {arcsinh}(c x))}-\frac {\int \frac {1+c^2 x^2}{x^2 (a+b \text {arcsinh}(c x))} \, dx}{b c}+\frac {\left (9 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b^2}+\frac {\left (3 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b^2}-\frac {\left (9 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b^2}-\frac {\left (3 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b^2} \\ & = -\frac {\left (1+c^2 x^2\right )^2}{b c x (a+b \text {arcsinh}(c x))}+\frac {9 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b^2}+\frac {3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b^2}-\frac {9 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b^2}-\frac {3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b^2}-\frac {\int \frac {1+c^2 x^2}{x^2 (a+b \text {arcsinh}(c x))} \, dx}{b c} \\ \end{align*}
Not integrable
Time = 7.86 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\left (1+c^2 x^2\right )^{3/2}}{x (a+b \text {arcsinh}(c x))^2} \, dx \]
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Not integrable
Time = 0.12 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
\[\int \frac {\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{x \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x} \,d x } \]
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Not integrable
Time = 3.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}{x \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]
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Not integrable
Time = 0.58 (sec) , antiderivative size = 433, normalized size of antiderivative = 16.04 \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x} \,d x } \]
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Exception generated. \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x (a+b \text {arcsinh}(c x))^2} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 2.72 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {{\left (c^2\,x^2+1\right )}^{3/2}}{x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]
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