Integrand size = 27, antiderivative size = 27 \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\text {Int}\left (\frac {1}{x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2},x\right ) \]
[Out]
Not integrable
Time = 0.09 (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 = {} \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx \\ \end{align*}
Timed out. \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\text {\$Aborted} \]
[In]
[Out]
Not integrable
Time = 0.33 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
\[\int \frac {1}{x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}d x\]
[In]
[Out]
Not integrable
Time = 0.26 (sec) , antiderivative size = 139, normalized size of antiderivative = 5.15 \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x} \,d x } \]
[In]
[Out]
Not integrable
Time = 8.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{x \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
Not integrable
Time = 0.62 (sec) , antiderivative size = 584, normalized size of antiderivative = 21.63 \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Not integrable
Time = 2.68 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (c^2\,x^2+1\right )}^{5/2}} \,d x \]
[In]
[Out]