Integrand size = 34, antiderivative size = 34 \[ \int \frac {(a+b \text {arcsinh}(c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\text {Int}\left (\frac {(a+b \text {arcsinh}(c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}},x\right ) \]
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Not integrable
Time = 0.13 (sec) , antiderivative size = 34, 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 = {} \[ \int \frac {(a+b \text {arcsinh}(c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int \frac {(a+b \text {arcsinh}(c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b \text {arcsinh}(c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx \\ \end{align*}
Not integrable
Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b \text {arcsinh}(c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int \frac {(a+b \text {arcsinh}(c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx \]
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Not integrable
Time = 2.50 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94
\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{n} \ln \left (h \left (g x +f \right )^{m}\right )}{\sqrt {c^{2} x^{2}+1}}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arcsinh}(c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {c^{2} x^{2} + 1}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.92 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arcsinh}(c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {c^{2} x^{2} + 1}} \,d x } \]
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Not integrable
Time = 73.33 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arcsinh}(c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {c^{2} x^{2} + 1}} \,d x } \]
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Not integrable
Time = 2.65 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arcsinh}(c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n}{\sqrt {c^2\,x^2+1}} \,d x \]
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