Integrand size = 35, antiderivative size = 35 \[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\frac {\sqrt {-1+c x} \sqrt {1+c x} \text {Int}\left (\frac {\log \left (h (f+g x)^m\right )}{\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))},x\right )}{\sqrt {1-c^2 x^2}} \]
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Not integrable
Time = 0.48 (sec) , antiderivative size = 35, 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 = {} \[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))} \, dx}{\sqrt {1-c^2 x^2}} \\ \end{align*}
Not integrable
Time = 0.37 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \]
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Not integrable
Time = 4.99 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94
\[\int \frac {\ln \left (h \left (g x +f \right )^{m}\right )}{\left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) \sqrt {-c^{2} x^{2}+1}}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.63 \[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}} \,d x } \]
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Not integrable
Time = 13.49 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {\log {\left (h \left (f + g x\right )^{m} \right )}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \]
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Not integrable
Time = 0.50 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}} \,d x } \]
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Not integrable
Time = 0.77 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}} \,d x } \]
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Not integrable
Time = 4.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )}{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {1-c^2\,x^2}} \,d x \]
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