Integrand size = 23, antiderivative size = 23 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^2} \, dx=\frac {\text {Int}\left (\frac {1}{(c+d x) (a+b \text {arcsinh}(c+d x))^2},x\right )}{e} \]
[Out]
Not integrable
Time = 0.04 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^2} \, dx=\int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^2} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{e x (a+b \text {arcsinh}(x))^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x (a+b \text {arcsinh}(x))^2} \, dx,x,c+d x\right )}{d e} \\ \end{align*}
Not integrable
Time = 1.37 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^2} \, dx=\int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^2} \, dx \]
[In]
[Out]
Not integrable
Time = 0.17 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00
\[\int \frac {1}{\left (d e x +c e \right ) \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}d x\]
[In]
[Out]
Not integrable
Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.65 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^2} \, dx=\int { \frac {1}{{\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
[In]
[Out]
Not integrable
Time = 1.89 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.17 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^2} \, dx=\frac {\int \frac {1}{a^{2} c + a^{2} d x + 2 a b c \operatorname {asinh}{\left (c + d x \right )} + 2 a b d x \operatorname {asinh}{\left (c + d x \right )} + b^{2} c \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{2} d x \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx}{e} \]
[In]
[Out]
Not integrable
Time = 1.63 (sec) , antiderivative size = 1086, normalized size of antiderivative = 47.22 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^2} \, dx=\int { \frac {1}{{\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
[In]
[Out]
Not integrable
Time = 0.35 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^2} \, dx=\int { \frac {1}{{\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
[In]
[Out]
Not integrable
Time = 2.68 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))^2} \, dx=\int \frac {1}{\left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2} \,d x \]
[In]
[Out]